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<title>Two-state quantum system</title>

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Two-state quantum system

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In <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, a two-state system (also known as a two-level system) is a <a href="Quantum_system" class="mw-redirect" title="Quantum system">quantum system</a> that can exist in any <a href="Quantum_superposition" title="Quantum superposition">quantum superposition</a> of two independent (physically distinguishable) <a href="Quantum_state" title="Quantum state">quantum states</a>. The <a href="Hilbert_space" title="Hilbert space">Hilbert space</a> describing such a system is two-<a href="Dimension" title="Dimension">dimensional</a>. Therefore, a complete <a href="Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> spanning the space will consist of two independent states.<a href="#cite_note-1">[1]</a> Any two-state system can also be seen as a <a href="Qubit" title="Qubit">qubit</a>.

Two-state systems are the simplest quantum systems that are of interest, since the dynamics of a one-state system is trivial (as there are no other states in which the system can exist). The mathematical framework required for the analysis of two-state systems is that of <a href="Linear_differential_equation" title="Linear differential equation">linear differential equations</a> and <a href="Linear_algebra" title="Linear algebra">linear algebra</a> of two-dimensional spaces. As a result, the dynamics of a two-state system can be solved analytically without any approximation. The generic behavior of the system is that the wavefunction's amplitude oscillates between the two states.

A well known example of a two-state system is the <a href="Spin_(physics)" title="Spin (physics)">spin</a> of a <a href="Spin-1/2" title="Spin-1/2">spin-1/2</a> particle such as an electron, whose spin can have values +ħ/2 or −ħ/2, where ħ is the <a href="Reduced_Planck_constant" class="mw-redirect" title="Reduced Planck constant">reduced Planck constant</a>.

The two-state system cannot be used as a description of absorption or decay, because such processes require coupling to a continuum. Such processes would involve exponential decay of the amplitudes, but the solutions of the two-state system are oscillatory.

<div class="mw-heading mw-heading2"><h2 id="Analytical_solutions_for_stationary_state_energies_and_time-dependence">Analytical solutions for stationary state energies and time-dependence</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Representation">Representation</h3></div>

Supposing the two available basis states of the system are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |1\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle |1\rangle }</annotation>

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</math><img src="./2f53021ca18e77477ee5bd3c1523e5830189ec5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle |1\rangle }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |2\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle |2\rangle }</annotation>

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</math><img src="./32a7263d5f0cfb565e59ca2e03a5e93ca603414f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle |2\rangle }" loading="lazy">, in general the state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle }">

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<annotation encoding="application/x-tex">{\displaystyle |\psi \rangle }</annotation>

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</math><img src="./cc27f1893b769a08cd6b296e115a29e61cab675e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle }" loading="lazy"> can be written as a <a href="Superposition_(quantum_mechanics)" class="mw-redirect" title="Superposition (quantum mechanics)">superposition</a> of these two states with <a href="Probability_amplitude" title="Probability amplitude">probability amplitudes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{1},c_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle c_{1},c_{2}}</annotation>

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</math><img src="./15ae7f7452e416f7b70940ca260b2582f10382cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.156ex; height:2.009ex;" alt="{\displaystyle c_{1},c_{2}}" loading="lazy">,<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle =c_{1}|1\rangle +c_{2}|2\rangle .}">

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<annotation encoding="application/x-tex">{\displaystyle |\psi \rangle =c_{1}|1\rangle +c_{2}|2\rangle .}</annotation>

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Since the basis states are <a href="Orthonormality" title="Orthonormality">orthonormal</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle i|j\rangle =\delta _{ij}}">

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<annotation encoding="application/x-tex">{\displaystyle \langle i|j\rangle =\delta _{ij}}</annotation>

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</math><img src="./736111fc263c81855a75434f1da580d8060fe952.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.825ex; height:3.009ex;" alt="{\displaystyle \langle i|j\rangle =\delta _{ij}}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,j\in {1,2}}">

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<annotation encoding="application/x-tex">{\displaystyle i,j\in {1,2}}</annotation>

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</math><img src="./7be9e92bdcd2a18b4ab0dd55b70489169e8d7ab8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.994ex; height:2.509ex;" alt="{\displaystyle i,j\in {1,2}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{ij}}">

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<annotation encoding="application/x-tex">{\displaystyle \delta _{ij}}</annotation>

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</math><img src="./fa75d04c11480d976e1396951e02cbb3c4f71568.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.51ex; height:3.009ex;" alt="{\displaystyle \delta _{ij}}" loading="lazy"> is the <a href="Kronecker_delta" title="Kronecker delta">Kronecker delta</a>, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{i}=\langle i|\psi \rangle }">

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<annotation encoding="application/x-tex">{\displaystyle c_{i}=\langle i|\psi \rangle }</annotation>

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</math><img src="./5433e2a3f86297834d6e4ecbc256456c9c22c30f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.677ex; height:2.843ex;" alt="{\displaystyle c_{i}=\langle i|\psi \rangle }" loading="lazy">. These two <a href="Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a> may be considered coordinates in a two-dimensional <a href="Hilbert_Space" class="mw-redirect" title="Hilbert Space">complex Hilbert space</a>.<a href="#cite_note-griffiths353-2">[2]</a> Thus the <a href="Quantum_state" title="Quantum state">state vector</a> corresponding to the state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle }">

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<annotation encoding="application/x-tex">{\displaystyle |\psi \rangle }</annotation>

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</math><img src="./cc27f1893b769a08cd6b296e115a29e61cab675e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle }" loading="lazy"> is<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle \equiv {\begin{pmatrix}\langle 1|\psi \rangle \\\langle 2|\psi \rangle \end{pmatrix}}={\begin{pmatrix}c_{1}\\c_{2}\end{pmatrix}}=c_{1}{\begin{pmatrix}1\\0\end{pmatrix}}+c_{2}{\begin{pmatrix}0\\1\end{pmatrix}}=\mathbf {c} ,}">

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<annotation encoding="application/x-tex">{\displaystyle |\psi \rangle \equiv {\begin{pmatrix}\langle 1|\psi \rangle \\\langle 2|\psi \rangle \end{pmatrix}}={\begin{pmatrix}c_{1}\\c_{2}\end{pmatrix}}=c_{1}{\begin{pmatrix}1\\0\end{pmatrix}}+c_{2}{\begin{pmatrix}0\\1\end{pmatrix}}=\mathbf {c} ,}</annotation>

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</math> and the basis states correspond to the basis vectors, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |1\rangle \equiv {\begin{pmatrix}\langle 1|1\rangle \\\langle 2|1\rangle \end{pmatrix}}={\begin{pmatrix}1\\0\end{pmatrix}}}">

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<annotation encoding="application/x-tex">{\displaystyle |1\rangle \equiv {\begin{pmatrix}\langle 1|1\rangle \\\langle 2|1\rangle \end{pmatrix}}={\begin{pmatrix}1\\0\end{pmatrix}}}</annotation>

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</math> and <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |2\rangle \equiv {\begin{pmatrix}\langle 1|2\rangle \\\langle 2|2\rangle \end{pmatrix}}={\begin{pmatrix}0\\1\end{pmatrix}}.}">

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<annotation encoding="application/x-tex">{\displaystyle |2\rangle \equiv {\begin{pmatrix}\langle 1|2\rangle \\\langle 2|2\rangle \end{pmatrix}}={\begin{pmatrix}0\\1\end{pmatrix}}.}</annotation>

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If the state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle }">

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<annotation encoding="application/x-tex">{\displaystyle |\psi \rangle }</annotation>

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</math><img src="./cc27f1893b769a08cd6b296e115a29e61cab675e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle }" loading="lazy"> is <a href="Probability_amplitude" title="Probability amplitude">normalized</a>, the <a href="Norm_(mathematics)" title="Norm (mathematics)">norm</a> of the state vector is unity, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {|c_{1}|}^{2}+{|c_{2}|}^{2}=1}">

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<annotation encoding="application/x-tex">{\displaystyle {|c_{1}|}^{2}+{|c_{2}|}^{2}=1}</annotation>

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</math><img src="./c6445cf84da7729080571ba8b7a26ddae167f41e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.919ex; height:3.343ex;" alt="{\displaystyle {|c_{1}|}^{2}+{|c_{2}|}^{2}=1}" loading="lazy">.

All <a href="Observable" title="Observable">observable physical quantities</a>, such as energy, are associated with <a href="Self-adjoint_operator" title="Self-adjoint operator">hermitian operators</a>. In the case of energy and the corresponding <a href="Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a>, H, this means<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{ij}=\langle i|H|j\rangle =\langle j|H|i\rangle ^{*}=H_{ji}^{*},}">

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<annotation encoding="application/x-tex">{\displaystyle H_{ij}=\langle i|H|j\rangle =\langle j|H|i\rangle ^{*}=H_{ji}^{*},}</annotation>

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i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{11}}">

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<annotation encoding="application/x-tex">{\displaystyle H_{11}}</annotation>

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</math><img src="./89e8b960c5dfdc59018b9c1cca714a9867ddf70e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.807ex; height:2.509ex;" alt="{\displaystyle H_{11}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{22}}">

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<annotation encoding="application/x-tex">{\displaystyle H_{22}}</annotation>

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</math><img src="./012e51956ee289b5052c2631946a589522d85367.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.807ex; height:2.509ex;" alt="{\displaystyle H_{22}}" loading="lazy"> are real, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{12}=H_{21}^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle H_{12}=H_{21}^{*}}</annotation>

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</math><img src="./5ca76416693bd73850707debf7aaf3c0e78f472c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.713ex; height:2.843ex;" alt="{\displaystyle H_{12}=H_{21}^{*}}" loading="lazy">. Thus, these four matrix elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{ij}}">

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<annotation encoding="application/x-tex">{\displaystyle H_{ij}}</annotation>

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</math><img src="./cb13feb15799a5df604f075b03c85cf59572d2bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.408ex; height:2.843ex;" alt="{\displaystyle H_{ij}}" loading="lazy"> produce a 2×2 <a href="Hermitian_matrices" class="mw-redirect" title="Hermitian matrices">hermitian matrix</a>,

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {H} ={\begin{pmatrix}\langle 1|H|1\rangle &\langle 1|H|2\rangle \\\langle 2|H|1\rangle &\langle 2|H|2\rangle \end{pmatrix}}={\begin{pmatrix}H_{11}&H_{12}\\H_{12}^{*}&H_{22}\end{pmatrix}}.}</annotation>

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The <a href="Time-independent_Schr%C3%B6dinger_equation" class="mw-redirect" title="Time-independent Schrödinger equation">time-independent Schrödinger equation</a> states that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H|\psi \rangle =E|\psi \rangle }">

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<annotation encoding="application/x-tex">{\displaystyle H|\psi \rangle =E|\psi \rangle }</annotation>

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</math><img src="./cc77c7b70ba83674fba950469fb78624ba4fb234.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.067ex; height:2.843ex;" alt="{\displaystyle H|\psi \rangle =E|\psi \rangle }" loading="lazy">; substituting for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle }">

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</math><img src="./cc27f1893b769a08cd6b296e115a29e61cab675e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle }" loading="lazy"> in terms of the basis states from above, and multiplying both sides by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle 1|}">

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<annotation encoding="application/x-tex">{\displaystyle \langle 1|}</annotation>

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</math><img src="./d3462e8d2246986ff092caaa7074962585ff0ca1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle \langle 1|}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle 2|}">

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</math><img src="./46eb64cde5c8eb37446802dce5962607ed15d04a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle \langle 2|}" loading="lazy"> produces a <a href="System_of_linear_equations" title="System of linear equations">system of two linear equations</a> that can be written in matrix form,<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}H_{11}&H_{12}\\H_{12}^{*}&H_{22}\end{pmatrix}}{\begin{pmatrix}c_{1}\\c_{2}\end{pmatrix}}=E{\begin{pmatrix}c_{1}\\c_{2}\end{pmatrix}},}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}H_{11}&H_{12}\\H_{12}^{*}&H_{22}\end{pmatrix}}{\begin{pmatrix}c_{1}\\c_{2}\end{pmatrix}}=E{\begin{pmatrix}c_{1}\\c_{2}\end{pmatrix}},}</annotation>

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or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Hc} =E\mathbf {c} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {Hc} =E\mathbf {c} }</annotation>

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</math><img src="./8e54d8747e37e9ea45f39a08bf0aeb9a7a1dc7e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.342ex; height:2.176ex;" alt="{\displaystyle \mathbf {Hc} =E\mathbf {c} }" loading="lazy"> which is a 2×2 matrix <a href="Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvalues and eigenvectors</a> problem. As mentioned above, this equation comes from plugging a general state into the time-independent Schrödinger equation. Remember that the time-independent Schrödinger equation is a restrictive condition used to specify the eigenstates. Therefore, when plugging a general state into it, we are seeing what form the general state must take to be an eigenstate. Doing so, and distributing, we get <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{1}H|1\rangle +c_{2}H|2\rangle =c_{1}E|1\rangle +c_{2}E|2\rangle }">

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</math><img src="./453175f483471f03e90376edf8b31faeccd83e28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{2}}" loading="lazy">, the energies of the individual states, which are by definition different). Upon setting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{1}}">

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</math><img src="./0b30ba1b247fb8d334580cec68561e749d24aff2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle c_{2}}" loading="lazy"> to be 0, only one state remains, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

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</math><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> is the energy of the surviving state. This result is a redundant reminder that the time-independent Schrödinger equation is only satisfied by eigenstates of H, which are (by definition of the state vector) the states where all except one coefficient are zero. Now, if we follow the same derivation, but before acting with the Hamiltonian on the individual states, we multiply both sides by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle 1|}">

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</math><img src="./46eb64cde5c8eb37446802dce5962607ed15d04a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle \langle 2|}" loading="lazy">, we get a system of two linear equations that can be combined into the above matrix equation. Like before, this can only be satisfied if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{1}}">

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</math><img src="./0b30ba1b247fb8d334580cec68561e749d24aff2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle c_{2}}" loading="lazy"> is zero, and when this happens, the constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

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</math><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> will be the energy of the remaining state. The above matrix equation should thus be interpreted as a restrictive condition on a general state vector to yield an eigenvector of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

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</math><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy">, exactly analogous to the time-independent Schrödinger equation.

Of course, in general, commuting the matrix with a state vector will not result in the same vector multiplied by a constant E. For general validity, one has to write the equation in the form<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}H_{11}&H_{12}\\H_{12}^{*}&H_{22}\end{pmatrix}}{\begin{pmatrix}c_{1}\\c_{2}\end{pmatrix}}={\begin{pmatrix}\varepsilon _{1}c_{1}\\\varepsilon _{2}c_{2}\end{pmatrix}},}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}H_{11}&H_{12}\\H_{12}^{*}&H_{22}\end{pmatrix}}{\begin{pmatrix}c_{1}\\c_{2}\end{pmatrix}}={\begin{pmatrix}\varepsilon _{1}c_{1}\\\varepsilon _{2}c_{2}\end{pmatrix}},}</annotation>

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</math> with the individual eigenstate energies still inside the product vector. In either case, the <a href="Hamiltonian_matrix" title="Hamiltonian matrix">Hamiltonian matrix</a> can be derived using the method specified above, or via the more traditional method of constructing a matrix using boundary conditions; specifically, by using the requirement that when it acts on either basis state, it must return that state multiplied by the energy of that state. (There are no boundary conditions on how it acts on a general state.) This results in a diagonal matrix with the diagonal elements being the energies of the eigenstates and the off-diagonal elements being zero. The form of the matrix above that uses bra-ket-enclosed Hamiltonians is a more generalized version of this matrix.

One might ask why it is necessary to write the Hamiltonian matrix in such a general form with bra-ket-enclosed Hamiltonians, since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{ij},i\neq j}">

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<annotation encoding="application/x-tex">{\displaystyle H_{ij},i\neq j}</annotation>

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</math><img src="./1641d6e7ecca587dd32c62727ea73d53c370d796.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.301ex; height:2.843ex;" alt="{\displaystyle H_{ij},i\neq j}" loading="lazy"> should always equal zero and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{ii}}">

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</math><img src="./81943c716067a2aa974a24fba9aaf480b5e76a9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.298ex; height:2.509ex;" alt="{\displaystyle H_{ii}}" loading="lazy"> should always equal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{i}}">

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</math><img src="./00e1b6ad3cbad4af49bf21a3ad2dc379ff045079.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.883ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{i}}" loading="lazy">. The reason is that, in some more complex problems, the state vectors may not be eigenstates of the Hamiltonian used in the matrix. One place where this occurs is in <a href="Perturbation_theory_(quantum_mechanics)" title="Perturbation theory (quantum mechanics)">degenerate perturbation theory</a>, where the off-diagonal elements are nonzero until the problem is solved by <a href="Diagonalization_of_a_matrix" class="mw-redirect" title="Diagonalization of a matrix">diagonalization</a>.

Because of the hermiticity of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {H} }">

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</math><img src="./f017b876ed763037d8818ec5dfbbdc6703e0f683.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.091ex; height:2.176ex;" alt="{\displaystyle \mathbf {H} }" loading="lazy"> the eigenvalues are real; or, rather, conversely, it is the requirement that the energies are real that implies the hermiticity of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {H} }">

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<div class="mw-heading mw-heading3"><h3 id="Eigenvalues_of_the_Hamiltonian">Eigenvalues of the Hamiltonian</h3></div>

The most general form of a 2×2 Hermitian matrix such as the Hamiltonian of a two-state system is given by<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {H} ={\begin{pmatrix}\varepsilon _{1}&\beta -i\gamma \\\beta +i\gamma &\varepsilon _{2}\end{pmatrix}},}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {H} ={\begin{pmatrix}\varepsilon _{1}&\beta -i\gamma \\\beta +i\gamma &\varepsilon _{2}\end{pmatrix}},}</annotation>

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where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{1},\varepsilon _{2},\beta }">

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{1},\varepsilon _{2},\beta }</annotation>

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</math><img src="./b86c7957b014e7ec0050e68c07ad2b3552725af1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.675ex; height:2.509ex;" alt="{\displaystyle \varepsilon _{1},\varepsilon _{2},\beta }" loading="lazy"> and γ are real numbers with units of energy. The allowed energy levels of the system, namely the <a href="Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvalues</a> of the Hamiltonian matrix, can be found in the usual way.

Equivalently, this matrix can be decomposed as,<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {H} =\alpha \cdot \sigma _{0}+\beta \cdot \sigma _{1}+\gamma \cdot \sigma _{2}+\delta \cdot \sigma _{3}={\begin{pmatrix}\alpha +\delta &\beta -i\gamma \\\beta +i\gamma &\alpha -\delta \end{pmatrix}}.}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {H} =\alpha \cdot \sigma _{0}+\beta \cdot \sigma _{1}+\gamma \cdot \sigma _{2}+\delta \cdot \sigma _{3}={\begin{pmatrix}\alpha +\delta &\beta -i\gamma \\\beta +i\gamma &\alpha -\delta \end{pmatrix}}.}</annotation>

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</math> Here, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \alpha ={\frac {1}{2}}\left(\varepsilon _{1}+\varepsilon _{2}\right)}">

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<annotation encoding="application/x-tex">{\textstyle \alpha ={\frac {1}{2}}\left(\varepsilon _{1}+\varepsilon _{2}\right)}</annotation>

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</math><img src="./e2720d06d6e9d724a75dd49d0e6cd12f51dbc11a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.556ex; height:3.509ex;" alt="{\textstyle \alpha ={\frac {1}{2}}\left(\varepsilon _{1}+\varepsilon _{2}\right)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \delta ={\frac {1}{2}}\left(\varepsilon _{1}-\varepsilon _{2}\right)}">

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<annotation encoding="application/x-tex">{\textstyle \delta ={\frac {1}{2}}\left(\varepsilon _{1}-\varepsilon _{2}\right)}</annotation>

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</math><img src="./fb4ff5058bb415f58c08abe7a722668f482f456c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.117ex; height:3.509ex;" alt="{\textstyle \delta ={\frac {1}{2}}\left(\varepsilon _{1}-\varepsilon _{2}\right)}" loading="lazy"> are real numbers. The matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{0}}">

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</math><img src="./b2f330c7d4a4ee92e5d43dfe3e23f0de3406ec78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.382ex; height:2.009ex;" alt="{\displaystyle \sigma _{0}}" loading="lazy"> is the 2×2 identity matrix and the matrices <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{k}}">

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</math><img src="./330415aef0a06818b843c55814fda645b793811b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.416ex; height:2.009ex;" alt="{\displaystyle \sigma _{k}}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1,2,3}">

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</math><img src="./f91eb32681441793a4f3edee319505ca8afaa4f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.865ex; height:2.509ex;" alt="{\displaystyle k=1,2,3}" loading="lazy"> are the <a href="Pauli_matrices" title="Pauli matrices">Pauli matrices</a>. This decomposition simplifies the analysis of the system, especially in the time-independent case, where the values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ,\beta ,\gamma }">

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</math><img src="./301cc1b37ba8f0fb0c9bedee5efa5e0b5bc9e791.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.15ex; height:2.676ex;" alt="{\displaystyle \alpha ,\beta ,\gamma }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }">

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</math><img src="./c5321cfa797202b3e1f8620663ff43c4660ea03a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" loading="lazy"> are constants.

The Hamiltonian can be further condensed as<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {H} =\alpha \cdot \sigma _{0}+\mathbf {r} \cdot {\boldsymbol {\sigma }}.}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {H} =\alpha \cdot \sigma _{0}+\mathbf {r} \cdot {\boldsymbol {\sigma }}.}</annotation>

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The vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }">

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</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"> is given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\sigma _{1},\sigma _{2},\sigma _{3})}">

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</math><img src="./e69e2c2e8f41a8f7fb8244eb2218f78485c8aaab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.022ex; height:2.843ex;" alt="{\displaystyle (\sigma _{1},\sigma _{2},\sigma _{3})}" loading="lazy">. This representation simplifies the analysis of the time evolution of the system and is easier to use with other specialized representations such as the <a href="Bloch_sphere" title="Bloch sphere">Bloch sphere</a>.

If the two-state system's time-independent Hamiltonian H is defined as above, then its <a href="Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> are given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\pm }=\alpha \pm |\mathbf {r} |}">

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<annotation encoding="application/x-tex">{\displaystyle E_{\pm }=\alpha \pm |\mathbf {r} |}</annotation>

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</math><img src="./51309fe486396f3843af024903a6b3235be34afb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.048ex; height:2.843ex;" alt="{\displaystyle E_{\pm }=\alpha \pm |\mathbf {r} |}" loading="lazy">. Evidently, α is the average energy of the two levels, and the <a href="Norm_(mathematics)" title="Norm (mathematics)">norm</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }">

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</math><img src="./eca0f46511c4c986c48b254073732c0bd98ae0c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }" loading="lazy"> is the splitting between them. The corresponding eigenvectors are denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |+\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle |+\rangle }</annotation>

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</math><img src="./6f39109e5dd0d63a152bd539e71a233b78e23eb5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.36ex; height:2.843ex;" alt="{\displaystyle |+\rangle }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |-\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle |-\rangle }</annotation>

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</math><img src="./7da62e696eae51c2d89944b647e24b41cafb2f1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.36ex; height:2.843ex;" alt="{\displaystyle |-\rangle }" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Time_dependence">Time dependence</h3></div>

We now assume that the <a href="Probability_amplitude" title="Probability amplitude">probability amplitudes</a> are time-dependent, though the basis states are not. The <a href="Time-dependent_Schr%C3%B6dinger_equation" class="mw-redirect" title="Time-dependent Schrödinger equation">Time-dependent Schrödinger equation</a> states <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle i\hbar \partial _{t}|\psi \rangle =H|\psi \rangle }">

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<annotation encoding="application/x-tex">{\textstyle i\hbar \partial _{t}|\psi \rangle =H|\psi \rangle }</annotation>

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</math><img src="./478c99ed28f9fe70d6a4e967584af80b32b97548.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.461ex; height:2.843ex;" alt="{\textstyle i\hbar \partial _{t}|\psi \rangle =H|\psi \rangle }" loading="lazy">, and proceeding as before (substituting for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle }">

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</math><img src="./cc27f1893b769a08cd6b296e115a29e61cab675e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle }" loading="lazy"> and premultiplying by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle 1|,\langle 2|}">

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</math><img src="./001283ca6e49f1b7fa5f65f96f43608e559c90df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.462ex; height:2.843ex;" alt="{\displaystyle \langle 1|,\langle 2|}" loading="lazy"> again produces a pair of coupled linear equations, but this time they are first order partial differential equations: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle i\hbar \partial _{t}\mathbf {c} =\mathbf {Hc} }">

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<annotation encoding="application/x-tex">{\textstyle i\hbar \partial _{t}\mathbf {c} =\mathbf {Hc} }</annotation>

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</math><img src="./5900e49d406ee5953538b7e5b739fc8e85a81fe8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.735ex; height:2.509ex;" alt="{\textstyle i\hbar \partial _{t}\mathbf {c} =\mathbf {Hc} }" loading="lazy">. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {H} }">

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</math><img src="./f017b876ed763037d8818ec5dfbbdc6703e0f683.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.091ex; height:2.176ex;" alt="{\displaystyle \mathbf {H} }" loading="lazy"> is time independent there are several approaches to find the time dependence of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{1},c_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle c_{1},c_{2}}</annotation>

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</math><img src="./15ae7f7452e416f7b70940ca260b2582f10382cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.156ex; height:2.009ex;" alt="{\displaystyle c_{1},c_{2}}" loading="lazy">, such as <a href="Normal_mode" title="Normal mode">normal modes</a>. The result is that<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {c} (t)=e^{-i\mathbf {H} t/\hbar }\mathbf {c} _{0}=\mathbf {U} (t)\mathbf {c} _{0}.}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {c} (t)=e^{-i\mathbf {H} t/\hbar }\mathbf {c} _{0}=\mathbf {U} (t)\mathbf {c} _{0}.}</annotation>

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</math> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {c} _{0}=\mathbf {c} (0)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {c} _{0}=\mathbf {c} (0)}</annotation>

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</math><img src="./a2760401c04d6188a89cf6dda337b8d8821f3671.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.5ex; height:2.843ex;" alt="{\displaystyle \mathbf {c} _{0}=\mathbf {c} (0)}" loading="lazy"> is the statevector at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0}">

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<annotation encoding="application/x-tex">{\displaystyle t=0}</annotation>

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</math><img src="./43469ec032d858feae5aa87029e22eaaf0109e9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=0}" loading="lazy">. Here the <a href="Matrix_exponential" title="Matrix exponential">exponential of a matrix</a> may be found from the series expansion. The matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {U} (t)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {U} (t)}</annotation>

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</math><img src="./57eacf7315a9fb17d51b5e08780d61ac0ed9dafa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.706ex; height:2.843ex;" alt="{\displaystyle \mathbf {U} (t)}" loading="lazy"> is called the time evolution matrix (which comprises the matrix elements of the corresponding time evolution operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(t)}">

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<annotation encoding="application/x-tex">{\displaystyle U(t)}</annotation>

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</math><img src="./666c639df532e88616357c4991cabce9a57b5611.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.431ex; height:2.843ex;" alt="{\displaystyle U(t)}" loading="lazy">). It is easily proved that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {U} (t)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {U} (t)}</annotation>

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</math><img src="./57eacf7315a9fb17d51b5e08780d61ac0ed9dafa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.706ex; height:2.843ex;" alt="{\displaystyle \mathbf {U} (t)}" loading="lazy"> is <a href="Unitary_matrix" title="Unitary matrix">unitary</a>, meaning that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {U} ^{\dagger }\mathbf {U} =1}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {U} ^{\dagger }\mathbf {U} =1}</annotation>

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</math><img src="./1631a08e6dd874bfe04600e30571944d6d684d2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.336ex; height:2.676ex;" alt="{\displaystyle \mathbf {U} ^{\dagger }\mathbf {U} =1}" loading="lazy">.

It can be shown that<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {U} (t)=e^{-i\mathbf {H} t/\hbar }=e^{-i\alpha t/\hbar }\left(\cos \left({\frac {|\mathbf {r} |}{\hbar }}t\right)\sigma _{0}-i\sin \left({\frac {|\mathbf {r} |}{\hbar }}t\right){\hat {r}}\cdot {\boldsymbol {\sigma }}\right),}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {U} (t)=e^{-i\mathbf {H} t/\hbar }=e^{-i\alpha t/\hbar }\left(\cos \left({\frac {|\mathbf {r} |}{\hbar }}t\right)\sigma _{0}-i\sin \left({\frac {|\mathbf {r} |}{\hbar }}t\right){\hat {r}}\cdot {\boldsymbol {\sigma }}\right),}</annotation>

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<annotation encoding="application/x-tex">{\textstyle {\hat {r}}={\frac {\mathbf {r} }{|\mathbf {r} |}}.}</annotation>

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</math><img src="./6cc1c3e0c2aa693968f99a5b21a0fdaa23fe996a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.567ex; height:3.843ex;" alt="{\textstyle {\hat {r}}={\frac {\mathbf {r} }{|\mathbf {r} |}}.}" loading="lazy">

When one changes the basis to the eigenvectors of the Hamiltonian, in other words, if the basis states <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |1\rangle ,|2\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle |1\rangle ,|2\rangle }</annotation>

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</math><img src="./557b3417efe4dde0cfb4ad64c1fe5d76edbb54c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.462ex; height:2.843ex;" alt="{\displaystyle |1\rangle ,|2\rangle }" loading="lazy"> are chosen to be the eigenvectors, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon _{1}=H_{11}=\langle 1|H|1\rangle =E_{1}\langle 1|1\rangle =E_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle \epsilon _{1}=H_{11}=\langle 1|H|1\rangle =E_{1}\langle 1|1\rangle =E_{1}}</annotation>

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</math><img src="./ab8298d8e98e2e8f83bdefaa9f9438f2dffec516.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.011ex; height:2.843ex;" alt="{\displaystyle \epsilon _{1}=H_{11}=\langle 1|H|1\rangle =E_{1}\langle 1|1\rangle =E_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta +i\gamma =H_{21}=\langle 2|H|1\rangle =E_{1}\langle 2|1\rangle =0}">

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<annotation encoding="application/x-tex">{\displaystyle \beta +i\gamma =H_{21}=\langle 2|H|1\rangle =E_{1}\langle 2|1\rangle =0}</annotation>

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</math><img src="./5d50332c1dd74f647a400d036efafe132f679961.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.643ex; height:2.843ex;" alt="{\displaystyle \beta +i\gamma =H_{21}=\langle 2|H|1\rangle =E_{1}\langle 2|1\rangle =0}" loading="lazy"> and so the Hamiltonian is diagonal, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathbf {r} |=\delta }">

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<annotation encoding="application/x-tex">{\displaystyle |\mathbf {r} |=\delta }</annotation>

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</math><img src="./44d277d7b19bff31c5ec4422a165997ee32c29f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.543ex; height:2.843ex;" alt="{\displaystyle |\mathbf {r} |=\delta }" loading="lazy"> and is of the form,<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {H} ={\begin{pmatrix}E_{1}&0\\0&E_{2}\end{pmatrix}}.}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {H} ={\begin{pmatrix}E_{1}&0\\0&E_{2}\end{pmatrix}}.}</annotation>

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Now, the unitary time evolution operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

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</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> is easily seen to be given by:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {U} (t)=e^{-i\mathbf {H} t/\hbar }={\begin{pmatrix}e^{-iE_{1}t/\hbar }&0\\0&e^{-iE_{2}vt/\hbar }\end{pmatrix}}=e^{-i\alpha t/\hbar }{\begin{pmatrix}e^{-i\delta t/\hbar }&0\\0&e^{i\delta t/\hbar }\end{pmatrix}}=e^{-i\alpha t/\hbar }\left(\cos \left({\frac {\delta }{\hbar }}t\right)\sigma _{0}-i\sin \left({\frac {\delta }{\hbar }}t\right){\boldsymbol {\sigma }}_{3}\right).}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {U} (t)=e^{-i\mathbf {H} t/\hbar }={\begin{pmatrix}e^{-iE_{1}t/\hbar }&0\\0&e^{-iE_{2}vt/\hbar }\end{pmatrix}}=e^{-i\alpha t/\hbar }{\begin{pmatrix}e^{-i\delta t/\hbar }&0\\0&e^{i\delta t/\hbar }\end{pmatrix}}=e^{-i\alpha t/\hbar }\left(\cos \left({\frac {\delta }{\hbar }}t\right)\sigma _{0}-i\sin \left({\frac {\delta }{\hbar }}t\right){\boldsymbol {\sigma }}_{3}\right).}</annotation>

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</math>The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-i\alpha t/\hbar }}">

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<annotation encoding="application/x-tex">{\displaystyle e^{-i\alpha t/\hbar }}</annotation>

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</math><img src="./96378daabb36060a2f4e97e365af67140d72bbb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.553ex; height:2.843ex;" alt="{\displaystyle e^{-i\alpha t/\hbar }}" loading="lazy"> factor merely contributes to the overall phase of the operator, and can usually be ignored to yield a new time evolution operator that is physically indistinguishable from the original operator. Moreover, any <a href="Perturbation_theory_(quantum_mechanics)" title="Perturbation theory (quantum mechanics)">perturbation</a> to the system (which will be of the same form as the Hamiltonian) can be added to the system in the eigenbasis of the unperturbed Hamiltonian and analysed in the same way as above. Therefore, for any perturbation the new eigenvectors of the perturbed system can be solved for exactly, as mentioned in the introduction.

<div class="mw-heading mw-heading3"><h3 id="Rabi_formula_for_a_static_perturbation">Rabi formula for a static perturbation</h3></div>

Suppose that the system starts in one of the basis states at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0}">

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<annotation encoding="application/x-tex">{\displaystyle t=0}</annotation>

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</math><img src="./43469ec032d858feae5aa87029e22eaaf0109e9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=0}" loading="lazy">, say <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |1\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle |1\rangle }</annotation>

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</math><img src="./2f53021ca18e77477ee5bd3c1523e5830189ec5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle |1\rangle }" loading="lazy"> so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {c} _{0}={\begin{pmatrix}1\\0\end{pmatrix}}}">

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<annotation encoding="application/x-tex">{\textstyle \mathbf {c} _{0}={\begin{pmatrix}1\\0\end{pmatrix}}}</annotation>

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</math><img src="./9aab45e0a4633c9edbe9c2efc86ece6a07ff3cac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.676ex; height:6.176ex;" alt="{\textstyle \mathbf {c} _{0}={\begin{pmatrix}1\\0\end{pmatrix}}}" loading="lazy">, and we are interested in the probability of occupation of each of the basis states as a function of time when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {H} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {H} }</annotation>

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</math><img src="./f017b876ed763037d8818ec5dfbbdc6703e0f683.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.091ex; height:2.176ex;" alt="{\displaystyle \mathbf {H} }" loading="lazy"> is the time-independent Hamiltonian.<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {c} (t)=\mathbf {U} (t)\mathbf {c} _{0}={\begin{pmatrix}U_{11}(t)&U_{12}(t)\\U_{21}(t)&U_{22}(t)\end{pmatrix}}{\begin{pmatrix}1\\0\end{pmatrix}}={\begin{pmatrix}U_{11}(t)\\U_{21}(t)\end{pmatrix}}.}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {c} (t)=\mathbf {U} (t)\mathbf {c} _{0}={\begin{pmatrix}U_{11}(t)&U_{12}(t)\\U_{21}(t)&U_{22}(t)\end{pmatrix}}{\begin{pmatrix}1\\0\end{pmatrix}}={\begin{pmatrix}U_{11}(t)\\U_{21}(t)\end{pmatrix}}.}</annotation>

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</math>

The probability of occupation of state i is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{i}(t)=|c_{i}(t)|^{2}=|U_{i1}(t)|^{2}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle P_{i}(t)=|c_{i}(t)|^{2}=|U_{i1}(t)|^{2}}</annotation>

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</math><img src="./b0c00673df62c2d9b07ed550f56014f12cd5a25e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.147ex; height:3.343ex;" alt="{\displaystyle P_{i}(t)=|c_{i}(t)|^{2}=|U_{i1}(t)|^{2}}" loading="lazy">. In the case of the starting state, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}(t)=|c_{1}(t)|^{2}=|U_{11}(t)|^{2}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle P_{1}(t)=|c_{1}(t)|^{2}=|U_{11}(t)|^{2}}</annotation>

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</math><img src="./f1edede00f213503ab276f600e660c82a19bcf6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.911ex; height:3.343ex;" alt="{\displaystyle P_{1}(t)=|c_{1}(t)|^{2}=|U_{11}(t)|^{2}}" loading="lazy">, and from above, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{11}(t)=e^{\frac {-i\alpha t}{\hbar }}\left(\cos \left({\frac {|\mathbf {r} |}{\hbar }}t\right)-i\sin \left({\frac {|\mathbf {r} |}{\hbar }}t\right){\frac {\delta }{|\mathbf {r} |}}\right).}">

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<annotation encoding="application/x-tex">{\displaystyle U_{11}(t)=e^{\frac {-i\alpha t}{\hbar }}\left(\cos \left({\frac {|\mathbf {r} |}{\hbar }}t\right)-i\sin \left({\frac {|\mathbf {r} |}{\hbar }}t\right){\frac {\delta }{|\mathbf {r} |}}\right).}</annotation>

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</math> Hence,<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}(t)=\cos ^{2}(\Omega t)+\sin ^{2}(\Omega t){\frac {\Delta ^{2}}{\Omega ^{2}}}.}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle P_{1}(t)=\cos ^{2}(\Omega t)+\sin ^{2}(\Omega t){\frac {\Delta ^{2}}{\Omega ^{2}}}.}</annotation>

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</math>

Obviously, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}(0)=1}">

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<annotation encoding="application/x-tex">{\displaystyle P_{1}(0)=1}</annotation>

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</math><img src="./ea29a42cae72b54bb8f198c9d90149047862f4d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.779ex; height:2.843ex;" alt="{\displaystyle P_{1}(0)=1}" loading="lazy"> due to the initial condition. The frequency <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega ={\frac {|\mathbf {r} |}{\hbar }}={\frac {1}{\hbar }}{\sqrt {\beta ^{2}+\gamma ^{2}+\delta ^{2}}}={\sqrt {|\Omega _{R}|^{2}+\Delta ^{2}}}}">

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \Omega ={\frac {|\mathbf {r} |}{\hbar }}={\frac {1}{\hbar }}{\sqrt {\beta ^{2}+\gamma ^{2}+\delta ^{2}}}={\sqrt {|\Omega _{R}|^{2}+\Delta ^{2}}}}</annotation>

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</math> is called the generalised Rabi frequency, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{R}=(\beta +i\gamma )/\hbar }">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \Omega _{R}=(\beta +i\gamma )/\hbar }</annotation>

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</math><img src="./1c43e1415c621d0cc7ae0d414cbbe4a5adeb2396.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.772ex; height:2.843ex;" alt="{\displaystyle \Omega _{R}=(\beta +i\gamma )/\hbar }" loading="lazy"> is called the Rabi frequency, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta =\delta /\hbar }">

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<annotation encoding="application/x-tex">{\displaystyle \Delta =\delta /\hbar }</annotation>

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</math><img src="./e1e914aa2e934b92980a75f8e4d3893c033748ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.552ex; height:2.843ex;" alt="{\displaystyle \Delta =\delta /\hbar }" loading="lazy"> is called the detuning.

At zero detuning, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}(t)=\cos ^{2}(|\Omega _{R}|t)}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle P_{1}(t)=\cos ^{2}(|\Omega _{R}|t)}</annotation>

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</math><img src="./1eac04b2872031d982e2ea35d8b093f901e829a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.559ex; height:3.176ex;" alt="{\displaystyle P_{1}(t)=\cos ^{2}(|\Omega _{R}|t)}" loading="lazy">, i.e., there is Rabi flopping from guaranteed occupation of state 1, to guaranteed occupation of state 2, and back to state 1, etc., with frequency <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Omega _{R}|}">

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<annotation encoding="application/x-tex">{\displaystyle |\Omega _{R}|}</annotation>

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</math><img src="./14cff21faf4a64d7cc6646dea4a828ce44bad0a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.451ex; height:2.843ex;" alt="{\displaystyle |\Omega _{R}|}" loading="lazy">. As the detuning is increased away from zero, the frequency of the flopping increases (to Ω) and the amplitude of exciting the electron decreases to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega ^{2}/\Delta ^{2}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \Omega ^{2}/\Delta ^{2}}</annotation>

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</math><img src="./7f5fe441cc109ebfc9db6e5359c56fc66d2c90db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.885ex; height:3.176ex;" alt="{\displaystyle \Omega ^{2}/\Delta ^{2}}" loading="lazy">.

For time dependent Hamiltonians induced by light waves, see the articles on <a href="Rabi_cycle" title="Rabi cycle">Rabi cycle</a> and <a href="Rotating_wave_approximation" class="mw-redirect" title="Rotating wave approximation">rotating wave approximation</a>.

<div class="mw-heading mw-heading2"><h2 id="Some_important_two-state_systems">Some important two-state systems</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Precession_in_a_field">Precession in a field</h3></div>

Consider the case of a <a href="Spin-1/2" title="Spin-1/2">spin-1/2</a> particle in a magnetic field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} =B\mathbf {\hat {n}} }">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {B} =B\mathbf {\hat {n}} }</annotation>

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</math><img src="./84d83fd78861bfaaa5a2d4dadb66e534f720ab57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.249ex; height:2.343ex;" alt="{\displaystyle \mathbf {B} =B\mathbf {\hat {n}} }" loading="lazy">. The interaction Hamiltonian for this system is<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=-{\boldsymbol {\mu }}\cdot \mathbf {B} =-\mu {\boldsymbol {\sigma }}\cdot \mathbf {B} ,}">

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<annotation encoding="application/x-tex">{\displaystyle H=-{\boldsymbol {\mu }}\cdot \mathbf {B} =-\mu {\boldsymbol {\sigma }}\cdot \mathbf {B} ,}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

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</math><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> is the magnitude of the particle's <a href="Magnetic_moment" title="Magnetic moment">magnetic moment</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\sigma }}}">

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\sigma }}}</annotation>

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</math><img src="./e45fe1b9d8dcbc3103fc7805d69798bfe5ca5b16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.594ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {\sigma }}}" loading="lazy"> is the vector of <a href="Pauli_matrices" title="Pauli matrices">Pauli matrices</a>. Solving the time dependent Schrödinger equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\psi =i\hbar \partial _{t}\psi }">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle H\psi =i\hbar \partial _{t}\psi }</annotation>

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</math><img src="./95df883f2e55294f1ce95f2415d7a1f0455cbc43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.358ex; height:2.509ex;" alt="{\displaystyle H\psi =i\hbar \partial _{t}\psi }" loading="lazy"> yields<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (t)=e^{i\omega t{\boldsymbol {\sigma }}\cdot \mathbf {\hat {n}} }\psi (0),}">

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<annotation encoding="application/x-tex">{\displaystyle \psi (t)=e^{i\omega t{\boldsymbol {\sigma }}\cdot \mathbf {\hat {n}} }\psi (0),}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \omega =\mu B/\hbar }</annotation>

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</math><img src="./61fa4e34bab1f2e76c421d3629dedca8ab3cabb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.179ex; height:2.843ex;" alt="{\displaystyle \omega =\mu B/\hbar }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\omega t{\boldsymbol {\sigma }}\cdot \mathbf {\hat {n}} }=\cos {\left(\omega t\right)}I+i\;\mathbf {\hat {n}} \cdot {\boldsymbol {\sigma }}\sin {\left(\omega t\right)}}">

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<annotation encoding="application/x-tex">{\displaystyle e^{i\omega t{\boldsymbol {\sigma }}\cdot \mathbf {\hat {n}} }=\cos {\left(\omega t\right)}I+i\;\mathbf {\hat {n}} \cdot {\boldsymbol {\sigma }}\sin {\left(\omega t\right)}}</annotation>

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</math><img src="./1c347b93d1a612ba0dce162e87a5db4f6ac18ddd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.769ex; height:3.509ex;" alt="{\displaystyle e^{i\omega t{\boldsymbol {\sigma }}\cdot \mathbf {\hat {n}} }=\cos {\left(\omega t\right)}I+i\;\mathbf {\hat {n}} \cdot {\boldsymbol {\sigma }}\sin {\left(\omega t\right)}}" loading="lazy">. Physically, this corresponds to the <a href="Bloch_vector" class="mw-redirect" title="Bloch vector">Bloch vector</a> precessing around <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {n}} }">

<mover>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {n}} }</annotation>

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</math><img src="./4eb84e133d15551d660800ec29b44783ff36e19d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.343ex;" alt="{\displaystyle \mathbf {\hat {n}} }" loading="lazy"> with angular frequency <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\omega }">

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<annotation encoding="application/x-tex">{\displaystyle 2\omega }</annotation>

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</math><img src="./fd713c0f268229c4d960abe6b57f9205e4725cf1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.608ex; height:2.176ex;" alt="{\displaystyle 2\omega }" loading="lazy">. Without loss of generality, assume the field is uniform and points in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {z}} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {z}} }</annotation>

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</math><img src="./5bdb354dce52452a8b65ebca5427d3012427412f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.343ex;" alt="{\displaystyle \mathbf {\hat {z}} }" loading="lazy">, so that the time evolution operator is given as<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\omega t{\boldsymbol {\sigma }}\cdot \mathbf {\hat {n}} }={\begin{pmatrix}e^{i\omega t}&0\\0&e^{-i\omega t}\end{pmatrix}}.}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle e^{i\omega t{\boldsymbol {\sigma }}\cdot \mathbf {\hat {n}} }={\begin{pmatrix}e^{i\omega t}&0\\0&e^{-i\omega t}\end{pmatrix}}.}</annotation>

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</math>

It can be seen that such a time evolution operator acting on a general spin state of a spin-1/2 particle will lead to the precession about the axis defined by the applied magnetic field (this is the quantum mechanical equivalent of <a href="Larmor_precession" title="Larmor precession">Larmor precession</a>)<a href="#cite_note-feynman_1-3">[3]</a>

The above method can be applied to the analysis of any generic two-state system that is interacting with some field (equivalent to the magnetic field in the previous case) if the interaction is given by an appropriate coupling term that is analogous to the magnetic moment. The precession of the state vector (which need not be a physical spinning as in the previous case) can be viewed as the precession of the state vector on the <a href="Bloch_sphere" title="Bloch sphere">Bloch sphere</a>.

The representation on the Bloch sphere for a state vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (0)}">

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<annotation encoding="application/x-tex">{\displaystyle \psi (0)}</annotation>

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</math><img src="./fcfdf3fefa6feee8b2192815ac15a2650d8a8db2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.485ex; height:2.843ex;" alt="{\displaystyle \psi (0)}" loading="lazy"> will simply be the vector of expectation values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} =\left(\langle \sigma _{x}\rangle ,\langle \sigma _{y}\rangle ,\langle \sigma _{z}\rangle \right)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {R} =\left(\langle \sigma _{x}\rangle ,\langle \sigma _{y}\rangle ,\langle \sigma _{z}\rangle \right)}</annotation>

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</math><img src="./a95c9b809b0679ecf8dcdd9b9fd74f992a7c72cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.612ex; height:3.009ex;" alt="{\displaystyle \mathbf {R} =\left(\langle \sigma _{x}\rangle ,\langle \sigma _{y}\rangle ,\langle \sigma _{z}\rangle \right)}" loading="lazy">. As an example, consider a state vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (0)}">

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<annotation encoding="application/x-tex">{\displaystyle \psi (0)}</annotation>

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</math><img src="./fcfdf3fefa6feee8b2192815ac15a2650d8a8db2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.485ex; height:2.843ex;" alt="{\displaystyle \psi (0)}" loading="lazy"> that is a normalized superposition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\uparrow \right\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle \left|\uparrow \right\rangle }</annotation>

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</math><img src="./b63af311377b1ea64952f5b9389a8a8f9d994a32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle \left|\uparrow \right\rangle }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\downarrow \right\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle \left|\downarrow \right\rangle }</annotation>

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</math><img src="./c2925aef9464d9e4f3393dc2a2e0e20dca0a2be4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle \left|\downarrow \right\rangle }" loading="lazy">, that is, a vector that can be represented in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{z}}">

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle \sigma _{z}}</annotation>

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</math><img src="./61ea9b1af0a90b25f6dcdc140820ff2428fd92c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.329ex; height:2.009ex;" alt="{\displaystyle \sigma _{z}}" loading="lazy"> basis as<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (0)={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\1\end{pmatrix}}}">

<mfrac>

<msqrt>

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<mrow>

<mtr>

<mtd>

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<mtr>

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<annotation encoding="application/x-tex">{\displaystyle \psi (0)={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\1\end{pmatrix}}}</annotation>

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</math>

The components of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (t)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \psi (t)}</annotation>

</semantics>

</math><img src="./6cf4a36b5f945be90a527b3dbe3d55d3f0439cdb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.162ex; height:2.843ex;" alt="{\displaystyle \psi (t)}" loading="lazy"> on the Bloch sphere will simply be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} =\left(\cos {2\omega t},-\sin {2\omega t},0\right)}">

</mrow>

<mrow>

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {R} =\left(\cos {2\omega t},-\sin {2\omega t},0\right)}</annotation>

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</math><img src="./91b580d2321a31883e1db3f32891c0d484f3c623.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.973ex; height:2.843ex;" alt="{\displaystyle \mathbf {R} =\left(\cos {2\omega t},-\sin {2\omega t},0\right)}" loading="lazy">. This is a unit vector that begins pointing along <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {x}} }">

<mover>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {x}} }</annotation>

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</math><img src="./8f3b8566d86d1eaf41596d59a93dc903cb3bb755.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:2.343ex;" alt="{\displaystyle \mathbf {\hat {x}} }" loading="lazy"> and precesses around <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {z}} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {z}} }</annotation>

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</math><img src="./5bdb354dce52452a8b65ebca5427d3012427412f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.343ex;" alt="{\displaystyle \mathbf {\hat {z}} }" loading="lazy"> in a left-handed manner. In general, by a rotation around <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {z}} }">

<mover>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {z}} }</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \psi (0)}</annotation>

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</math><img src="./fcfdf3fefa6feee8b2192815ac15a2650d8a8db2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.485ex; height:2.843ex;" alt="{\displaystyle \psi (0)}" loading="lazy"> can be represented as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\left|\uparrow \right\rangle +b\left|\downarrow \right\rangle }">

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle a\left|\uparrow \right\rangle +b\left|\downarrow \right\rangle }</annotation>

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</math><img src="./a08388734599c7a2c88d549a969c1df0e0bbbe2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.27ex; height:2.843ex;" alt="{\displaystyle a\left|\uparrow \right\rangle +b\left|\downarrow \right\rangle }" loading="lazy"> with real coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

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<annotation encoding="application/x-tex">{\displaystyle b}</annotation>

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</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy">. Such a state vector corresponds to a <a href="Bloch_vector" class="mw-redirect" title="Bloch vector">Bloch vector</a> in the xz-plane making an angle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(\theta /2)=b/a}">

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<annotation encoding="application/x-tex">{\displaystyle \tan(\theta /2)=b/a}</annotation>

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</math><img src="./d273a9e38501912ff85929893aa2089de88427b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.073ex; height:2.843ex;" alt="{\displaystyle \tan(\theta /2)=b/a}" loading="lazy"> with the z-axis. This vector will proceed to precess around <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {z}} }">

<mover>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {z}} }</annotation>

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</math><img src="./5bdb354dce52452a8b65ebca5427d3012427412f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.343ex;" alt="{\displaystyle \mathbf {\hat {z}} }" loading="lazy">. In theory, by allowing the system to interact with the field of a particular direction and strength for precise durations, it is possible to obtain any orientation of the <a href="Bloch_vector" class="mw-redirect" title="Bloch vector">Bloch vector</a>, which is equivalent to obtaining any complex superposition. This is the basis for numerous technologies including <a href="Quantum_computing" title="Quantum computing">quantum computing</a> and <a href="MRI" class="mw-redirect" title="MRI">MRI</a>.

<div class="mw-heading mw-heading3"><h3 id="Evolution_in_a_time-dependent_field:_Nuclear_magnetic_resonance">Evolution in a time-dependent field: Nuclear magnetic resonance</h3></div>

<a href="Nuclear_magnetic_resonance" title="Nuclear magnetic resonance">Nuclear magnetic resonance</a> (NMR) is an important example in the dynamics of two-state systems because it involves the exact solution to a time dependent Hamiltonian. The NMR phenomenon is achieved by placing a nucleus in a strong, static field B0 (the "holding field") and then applying a weak, transverse field B1 that oscillates at some radiofrequency ωr.<a href="#cite_note-griffiths377-4">[4]</a> Explicitly, consider a <a href="Spin-1/2" title="Spin-1/2">spin-1/2</a> particle in a holding field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{0}\mathbf {\hat {z}} }">

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<annotation encoding="application/x-tex">{\displaystyle B_{0}\mathbf {\hat {z}} }</annotation>

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</math><img src="./8bd5ac71d6acb82646ad1bb575104700dbe934fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.155ex; height:2.676ex;" alt="{\displaystyle B_{0}\mathbf {\hat {z}} }" loading="lazy"> and a transverse rf field B1 rotating in the xy-plane in a right-handed fashion around B0:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} ={\begin{pmatrix}B_{1}\cos \omega _{\mathrm {r} }t\\B_{1}\sin \omega _{\mathrm {r} }t\\B_{0}\end{pmatrix}}.}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {B} ={\begin{pmatrix}B_{1}\cos \omega _{\mathrm {r} }t\\B_{1}\sin \omega _{\mathrm {r} }t\\B_{0}\end{pmatrix}}.}</annotation>

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As in the free precession case, the Hamiltonian is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=-\mu {\boldsymbol {\sigma }}\cdot \mathbf {B} }">

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<annotation encoding="application/x-tex">{\displaystyle H=-\mu {\boldsymbol {\sigma }}\cdot \mathbf {B} }</annotation>

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</math><img src="./20bf49245260d35a9b593fee41cedfd19cc67635.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.547ex; height:2.676ex;" alt="{\displaystyle H=-\mu {\boldsymbol {\sigma }}\cdot \mathbf {B} }" loading="lazy">, and the evolution of a state vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (t)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \psi (t)}</annotation>

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</math><img src="./6cf4a36b5f945be90a527b3dbe3d55d3f0439cdb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.162ex; height:2.843ex;" alt="{\displaystyle \psi (t)}" loading="lazy"> is found by solving the time-dependent Schrödinger equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\psi =i\hbar \,\partial \psi /\partial t}">

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle H\psi =i\hbar \,\partial \psi /\partial t}</annotation>

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</math><img src="./8a88a866b42d92ec189c30c7b8387503f1e2dbd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.323ex; height:2.843ex;" alt="{\displaystyle H\psi =i\hbar \,\partial \psi /\partial t}" loading="lazy">. After some manipulation (given in the collapsed section below), it can be shown that the Schrödinger equation becomes<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \psi }{\partial t}}=i\left(\omega _{1}\sigma _{x}+\left(w_{0}+{\frac {\omega _{r}}{2}}\right)\sigma _{z}\right)\psi ,}">

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \psi }{\partial t}}=i\left(\omega _{1}\sigma _{x}+\left(w_{0}+{\frac {\omega _{r}}{2}}\right)\sigma _{z}\right)\psi ,}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \omega _{0}=\mu B_{0}/\hbar }</annotation>

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</math><img src="./8bc4f1d12c06073ccf0a5ab5461eab2a0d80e8c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.287ex; height:2.843ex;" alt="{\displaystyle \omega _{0}=\mu B_{0}/\hbar }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{1}=\mu B_{1}/\hbar }">

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<annotation encoding="application/x-tex">{\displaystyle \omega _{1}=\mu B_{1}/\hbar }</annotation>

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</math><img src="./5c98f85376d83096f601104154691b174a3323f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.287ex; height:2.843ex;" alt="{\displaystyle \omega _{1}=\mu B_{1}/\hbar }" loading="lazy">.

As per the previous section, the solution to this equation has the <a href="Bloch_vector" class="mw-redirect" title="Bloch vector">Bloch vector</a> precessing around <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\omega _{1},0,\omega _{0}+\omega _{r}/2)}">

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<annotation encoding="application/x-tex">{\displaystyle (\omega _{1},0,\omega _{0}+\omega _{r}/2)}</annotation>

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</math><img src="./8e03d1178b4bcc60d96a889f9dc54384bb1ccda0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.625ex; height:2.843ex;" alt="{\displaystyle (\omega _{1},0,\omega _{0}+\omega _{r}/2)}" loading="lazy"> with a frequency that is twice the magnitude of the vector. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{0}}">

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<annotation encoding="application/x-tex">{\displaystyle \omega _{0}}</annotation>

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</math><img src="./9a713d16c489051d4f515e12b1f86061c6be799b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{0}}" loading="lazy"> is sufficiently strong, some proportion of the spins will be pointing directly down prior to the introduction of the rotating field. If the angular frequency of the rotating magnetic field is chosen such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{r}=-2\omega _{0}}">

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<annotation encoding="application/x-tex">{\displaystyle \omega _{r}=-2\omega _{0}}</annotation>

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</math><img src="./eb7e5e296bdd3988eefa32628a8cf29cbf22e73b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.989ex; height:2.509ex;" alt="{\displaystyle \omega _{r}=-2\omega _{0}}" loading="lazy">, in the rotating frame the state vector will precess around <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {x}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {x}}}</annotation>

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</math><img src="./18d95a7845e4e16ffb7e18ab37a208d0ab18e0e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.176ex;" alt="{\displaystyle {\hat {x}}}" loading="lazy"> with frequency <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\omega _{1}}">

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<annotation encoding="application/x-tex">{\displaystyle 2\omega _{1}}</annotation>

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</math><img src="./be907252c2c25c6926ab1e470f1570f2de81caf2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.663ex; height:2.509ex;" alt="{\displaystyle 2\omega _{1}}" loading="lazy">, and will thus flip from down to up releasing energy in the form of detectable photons. This is the fundamental basis for <a href="NMR" class="mw-redirect" title="NMR">NMR</a>, and in practice is accomplished by scanning <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{r}}">

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<annotation encoding="application/x-tex">{\displaystyle \omega _{r}}</annotation>

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</math><img src="./fcf70e3a6975ab8aaaa19a091ec2cdb04d299a3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.42ex; height:2.009ex;" alt="{\displaystyle \omega _{r}}" loading="lazy"> until the resonant frequency is found at which point the sample will emit light. Similar calculations are done in atomic physics, and in the case that the field is not rotating, but oscillating with a complex amplitude, use is made of the <a href="Rotating_wave_approximation" class="mw-redirect" title="Rotating wave approximation">rotating wave approximation</a> in deriving such results.

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</style><div class="math_proof" style="">Derivation of above expression for the NMR Schrödinger equation

Here the Schrödinger equation reads

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<annotation encoding="application/x-tex">{\displaystyle -\mu {\boldsymbol {\sigma }}\cdot \mathbf {B} \psi =i\hbar {\frac {\partial \psi }{\partial t}}.}</annotation>

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Expanding the dot product and dividing by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\hbar }">

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<annotation encoding="application/x-tex">{\displaystyle i\hbar }</annotation>

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</math><img src="./3d724d600903f701570a9f97c2a8378a6f3a80e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.109ex; height:2.176ex;" alt="{\displaystyle i\hbar }" loading="lazy"> yields

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \psi }{\partial t}}=i\left(\omega _{1}\sigma _{x}\cos {\omega _{r}t}+\omega _{1}\sigma _{y}\sin {\omega _{r}t}+\omega _{0}\sigma _{z}\right)\psi .}</annotation>

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To remove the time dependence from the problem, the wave function is transformed according to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi \rightarrow e^{-i\sigma _{z}\omega _{r}t/2}\psi }">

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<annotation encoding="application/x-tex">{\displaystyle \psi \rightarrow e^{-i\sigma _{z}\omega _{r}t/2}\psi }</annotation>

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</math><img src="./d0a5e931a380e911a2c1f49b96a22580f43e2901.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.556ex; height:3.176ex;" alt="{\displaystyle \psi \rightarrow e^{-i\sigma _{z}\omega _{r}t/2}\psi }" loading="lazy">. The time dependent Schrödinger equation becomes

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</msub>

</mrow>

</mrow>

<msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -i\sigma _{z}{\frac {\omega _{r}}{2}}e^{-i\sigma _{z}\omega _{r}t/2}\psi +e^{-i\sigma _{z}\omega _{r}t/2}{\frac {\partial \psi }{\partial t}}=i\left(\omega _{1}\sigma _{x}\cos {\omega _{r}t}+\omega _{1}\sigma _{y}\sin {\omega _{r}t}+\omega _{0}\sigma _{z}\right)e^{-i\sigma _{z}\omega _{r}t/2}\psi ,}</annotation>

</semantics>

</math>

which after some rearrangement yields

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \psi }{\partial t}}=ie^{i\sigma _{z}\omega _{r}t/2}\left(\omega _{1}\sigma _{x}\cos {\omega _{r}t}+\omega _{1}\sigma _{y}\sin {\omega _{r}t}+\left(\omega _{0}+{\frac {\omega _{r}}{2}}\right)\sigma _{z}\right)e^{-i\sigma _{z}\omega _{r}t/2}\psi }</annotation>

</semantics>

</math>

Evaluating each term on the right hand side of the equation

<msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

<mrow>

<mtr>

<mtd>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{i\sigma _{z}\omega _{r}t/2}\sigma _{x}e^{-i\sigma _{z}\omega _{r}t/2}={\begin{pmatrix}e^{i\omega _{r}t/2}&0\\0&e^{-i\omega _{r}t/2}\end{pmatrix}}{\begin{pmatrix}0&1\\1&0\end{pmatrix}}{\begin{pmatrix}e^{-i\omega _{r}t/2}&0\\0&e^{i\omega _{r}t/2}\end{pmatrix}}={\begin{pmatrix}0&e^{i\omega _{r}t}\\e^{-i\omega _{r}t}&0\end{pmatrix}}}</annotation>

</semantics>

</math>

<msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

<mrow>

<mtr>

<mtd>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{i\sigma _{z}\omega _{r}t/2}\sigma _{y}e^{-i\sigma _{z}\omega _{r}t/2}={\begin{pmatrix}e^{i\omega _{r}t/2}&0\\0&e^{-i\omega _{r}t/2}\end{pmatrix}}{\begin{pmatrix}0&-i\\i&0\end{pmatrix}}{\begin{pmatrix}e^{-i\omega _{r}t/2}&0\\0&e^{i\omega _{r}t/2}\end{pmatrix}}={\begin{pmatrix}0&-ie^{i\omega _{r}t}\\ie^{-i\omega _{r}t}&0\end{pmatrix}}}</annotation>

</semantics>

</math>

<msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

<mrow>

<mtr>

<mtd>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{i\sigma _{z}\omega _{r}t/2}\sigma _{z}e^{-i\sigma _{z}\omega _{r}t/2}={\begin{pmatrix}e^{i\omega _{r}t/2}&0\\0&e^{-i\omega _{r}t/2}\end{pmatrix}}{\begin{pmatrix}1&0\\0&-1\end{pmatrix}}{\begin{pmatrix}e^{-i\omega _{r}t/2}&0\\0&e^{i\omega _{r}t/2}\end{pmatrix}}=\sigma _{z}}</annotation>

</semantics>

</math>

The equation now reads

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \psi }{\partial t}}=i\left(\omega _{1}{\begin{pmatrix}0&e^{i\omega _{r}t}\left(\cos {\omega _{r}t}-i\sin {\omega _{r}t}\right)\\e^{-i\omega _{r}t}\left(\cos {\omega _{r}t}+i\sin {\omega _{r}t}\right)&0\end{pmatrix}}+\left(w_{0}+{\frac {\omega _{r}}{2}}\right)\sigma _{z}\right)\psi ,}</annotation>

</semantics>

</math>

which by <a href="Euler's_identity" title="Euler's identity">Euler's identity</a> becomes

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

</semantics>

</math>

</div>

<div class="mw-heading mw-heading3"><h3 id="Relation_to_Bloch_equations">Relation to Bloch equations</h3></div>

The <a href="Optical_Bloch_equations" class="mw-redirect" title="Optical Bloch equations">optical Bloch equations</a> for a collection of <a href="Spin-1/2" title="Spin-1/2">spin-1/2</a> particles can be derived from the time dependent Schrödinger equation for a two level system. Starting with the previously stated Hamiltonian <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\hbar \partial _{t}\psi =-\mu {\boldsymbol {\sigma }}\cdot \mathbf {B} \psi }">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i\hbar \partial _{t}\psi =-\mu {\boldsymbol {\sigma }}\cdot \mathbf {B} \psi }</annotation>

</semantics>

</math><img src="./8cd417bf68054e6cda5dd5f9f399cbaa3ee80d4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.679ex; height:2.676ex;" alt="{\displaystyle i\hbar \partial _{t}\psi =-\mu {\boldsymbol {\sigma }}\cdot \mathbf {B} \psi }" loading="lazy">, it can be written in summation notation after some rearrangement as<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \psi }{\partial t}}=i{\frac {\mu }{\hbar }}\sigma _{i}B_{i}\psi }">

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \psi }{\partial t}}=i{\frac {\mu }{\hbar }}\sigma _{i}B_{i}\psi }</annotation>

</semantics>

</math>

Multiplying by a <a href="Pauli_matrix" class="mw-redirect" title="Pauli matrix">Pauli matrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{i}}</annotation>

</semantics>

</math><img src="./6ab3208a7d0c634ef720e03ff5a9949e8310edc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.127ex; height:2.009ex;" alt="{\displaystyle \sigma _{i}}" loading="lazy"> and the conjugate transpose of the wavefunction, and subsequently expanding the product of two Pauli matrices yields<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ^{\dagger }\sigma _{j}{\frac {\partial \psi }{\partial t}}=i{\frac {\mu }{\hbar }}\psi ^{\dagger }\sigma _{j}\sigma _{i}B_{i}\psi =i{\frac {\mu }{\hbar }}\psi ^{\dagger }\left(I\delta _{ij}-i\sigma _{k}\varepsilon _{ijk}\right)B_{i}\psi ={\frac {\mu }{\hbar }}\psi ^{\dagger }\left(iI\delta _{ij}+\sigma _{k}\varepsilon _{ijk}\right)B_{i}\psi }">

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi ^{\dagger }\sigma _{j}{\frac {\partial \psi }{\partial t}}=i{\frac {\mu }{\hbar }}\psi ^{\dagger }\sigma _{j}\sigma _{i}B_{i}\psi =i{\frac {\mu }{\hbar }}\psi ^{\dagger }\left(I\delta _{ij}-i\sigma _{k}\varepsilon _{ijk}\right)B_{i}\psi ={\frac {\mu }{\hbar }}\psi ^{\dagger }\left(iI\delta _{ij}+\sigma _{k}\varepsilon _{ijk}\right)B_{i}\psi }</annotation>

</semantics>

</math>

Adding this equation to its own conjugate transpose yields a left hand side of the form<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ^{\dagger }\sigma _{j}{\frac {\partial \psi }{\partial t}}+{\frac {\partial \psi ^{\dagger }}{\partial t}}\sigma _{j}\psi ={\frac {\partial \left(\psi ^{\dagger }\sigma _{j}\psi \right)}{\partial t}}}">

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi ^{\dagger }\sigma _{j}{\frac {\partial \psi }{\partial t}}+{\frac {\partial \psi ^{\dagger }}{\partial t}}\sigma _{j}\psi ={\frac {\partial \left(\psi ^{\dagger }\sigma _{j}\psi \right)}{\partial t}}}</annotation>

</semantics>

</math>

And a right hand side of the form<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mu }{\hbar }}\psi ^{\dagger }\left(iI\delta _{ij}+\sigma _{k}\varepsilon _{ijk}\right)B_{i}\psi +{\frac {\mu }{\hbar }}\psi ^{\dagger }\left(-iI\delta _{ij}+\sigma _{k}\varepsilon _{ijk}\right)B_{i}\psi ={\frac {2\mu }{\hbar }}\left(\psi ^{\dagger }\sigma _{k}\psi \right)B_{i}\varepsilon _{ijk}}">

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\mu }{\hbar }}\psi ^{\dagger }\left(iI\delta _{ij}+\sigma _{k}\varepsilon _{ijk}\right)B_{i}\psi +{\frac {\mu }{\hbar }}\psi ^{\dagger }\left(-iI\delta _{ij}+\sigma _{k}\varepsilon _{ijk}\right)B_{i}\psi ={\frac {2\mu }{\hbar }}\left(\psi ^{\dagger }\sigma _{k}\psi \right)B_{i}\varepsilon _{ijk}}</annotation>

</semantics>

</math>

As previously mentioned, the expectation value of each <a href="Pauli_matrix" class="mw-redirect" title="Pauli matrix">Pauli matrix</a> is a component of the <a href="Bloch_vector" class="mw-redirect" title="Bloch vector">Bloch vector</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \sigma _{i}\rangle =\psi ^{\dagger }\sigma _{i}\psi =R_{i}}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle \sigma _{i}\rangle =\psi ^{\dagger }\sigma _{i}\psi =R_{i}}</annotation>

</semantics>

</math><img src="./3a8577751e899f94b9b0b56794b3f14b79012295.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.813ex; height:3.176ex;" alt="{\displaystyle \langle \sigma _{i}\rangle =\psi ^{\dagger }\sigma _{i}\psi =R_{i}}" loading="lazy">. Equating the left and right hand sides, and noting that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2\mu }{\hbar }}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {2\mu }{\hbar }}}</annotation>

</semantics>

</math><img src="./79d9132dc878efb382c41530ce0e9e138d78f10c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.4ex; height:5.509ex;" alt="{\displaystyle {\frac {2\mu }{\hbar }}}" loading="lazy"> is the <a href="Gyromagnetic_ratio" title="Gyromagnetic ratio">gyromagnetic ratio</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

</semantics>

</math><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy">, yields another form for the equations of motion of the <a href="Bloch_vector" class="mw-redirect" title="Bloch vector">Bloch vector</a><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial R_{j}}{\partial t}}=\gamma R_{k}B_{i}\varepsilon _{kij}}">

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial R_{j}}{\partial t}}=\gamma R_{k}B_{i}\varepsilon _{kij}}</annotation>

</semantics>

</math> where the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{ijk}=\varepsilon _{kij}}">

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{ijk}=\varepsilon _{kij}}</annotation>

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</math><img src="./41c7932b8a32e759e60c003f6cf1a44deec5c560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.933ex; height:2.343ex;" alt="{\displaystyle \varepsilon _{ijk}=\varepsilon _{kij}}" loading="lazy"> has been used. In vector form these three equations can be expressed in terms of a <a href="Cross_product" title="Cross product">cross product</a><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {R} }{\partial t}}=\gamma \mathbf {R} \times \mathbf {B} }">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {R} }{\partial t}}=\gamma \mathbf {R} \times \mathbf {B} }</annotation>

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</math> Classically, this equation describes the dynamics of a spin in a magnetic field. An ideal magnet consists of a collection of identical spins behaving independently, and thus the total <a href="Magnetization" title="Magnetization">magnetization</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {M} }</annotation>

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</math><img src="./e499ae5946af9c09777ada933051b3669d3372c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.537ex; height:2.176ex;" alt="{\displaystyle \mathbf {M} }" loading="lazy"> is proportional to the <a href="Bloch_vector" class="mw-redirect" title="Bloch vector">Bloch vector</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {R} }</annotation>

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</math><img src="./5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }" loading="lazy">. All that is left to obtain the final form of the <a href="Optical_Bloch_equations" class="mw-redirect" title="Optical Bloch equations">optical Bloch equations</a> is the inclusion of the phenomenological <a href="Relaxation_(NMR)" title="Relaxation (NMR)">relaxation</a> terms.

As a final aside, the above equation can be derived by considering the time evolution of the <a href="Angular_momentum_operator" title="Angular momentum operator">angular momentum operator</a> in the <a href="Heisenberg_picture" title="Heisenberg picture">Heisenberg picture</a>.

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle i\hbar {\frac {d\sigma _{j}}{dt}}=\left[\sigma _{j},H\right]=\left[\sigma _{j},-\mu \sigma _{i}B_{i}\right]=-\mu \left(\sigma _{j}\sigma _{i}B_{i}-\sigma _{i}\sigma _{j}B_{i}\right)=\mu [\sigma _{i},\sigma _{j}]B_{i}=2\mu i\varepsilon _{ijk}\sigma _{k}B_{i}}</annotation>

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</math>

When coupled with the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} _{i}=\langle \sigma _{i}\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {R} _{i}=\langle \sigma _{i}\rangle }</annotation>

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</math><img src="./cdc57c461e3c1f62303c073db50b8352538dd98b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.838ex; height:2.843ex;" alt="{\displaystyle \mathbf {R} _{i}=\langle \sigma _{i}\rangle }" loading="lazy">, this equation is the same equation as before.

<div class="mw-heading mw-heading2"><h2 id="Validity">Validity</h2></div>

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Two-state systems are the simplest non-trivial quantum systems that occur in nature, but the above-mentioned methods of analysis are not just valid for simple two-state systems. Any general multi-state quantum system can be treated as a two-state system as long as the observable of interest has two eigenvalues. For example, a spin-1/2 particle may in reality have additional translational or even rotational degrees of freedom, but those degrees of freedom are irrelevant to the preceding analysis. Mathematically, the neglected degrees of freedom correspond to the degeneracy of the spin eigenvalues.

Another case where the effective two-state formalism is valid is when the system under consideration has two levels that are effectively decoupled from the system. This is the case in the analysis of the spontaneous or stimulated emission of light by atoms and that of <a href="Charge_qubit" title="Charge qubit">charge qubits</a>. In this case it should be kept in mind that the perturbations (interactions with an external field) are in the right range and do not cause transitions to states other than the ones of interest.

<div class="mw-heading mw-heading2"><h2 id="Significance_and_other_examples">Significance and other examples</h2></div>

Pedagogically, the two-state formalism is among the simplest of mathematical techniques used for the analysis of quantum systems. It can be used to illustrate fundamental quantum mechanical phenomena such as the <a href="Double-slit_experiment#Interpretations_of_the_experiment" title="Double-slit experiment">interference</a> exhibited by particles of the polarization states of the photon,<a href="#cite_note-feynman_2-5">[5]</a> but also more complex phenomena such as <a href="Neutrino_oscillation" title="Neutrino oscillation">neutrino oscillation</a> or the <a href="Kaon" title="Kaon">neutral K-meson</a> oscillation.

Two-state formalism can be used to describe simple mixing of states, which leads to phenomena such as <a href="Resonance_(chemistry)" title="Resonance (chemistry)">resonance</a> stabilization and other <a href="Avoided_crossing" title="Avoided crossing">level crossing</a> related symmetries. Such phenomena have a wide variety of application in chemistry. Phenomena with tremendous industrial applications such as the <a href="Maser" title="Maser">maser</a> and <a href="Laser" title="Laser">laser</a> can be explained using the two-state formalism.

The two-state formalism also forms the basis of <a href="Quantum_computing" title="Quantum computing">quantum computing</a>. <a href="Qubit" title="Qubit">Qubits</a>, which are the building blocks of a quantum computer, are nothing but two-state systems. Any quantum computational operation is a unitary operation that rotates the state vector on the Bloch sphere.

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li>A treatment of the two-state formalism, presented in the third volume of <a href="The_Feynman_Lectures_on_Physics" title="The Feynman Lectures on Physics">The Feynman Lectures on Physics</a>.</li>

<li>Lecture notes:

<ul><li>from the Quantum mechanics II course offered at <a href="MIT" class="mw-redirect" title="MIT">MIT</a>, <a rel="nofollow" class="external free" href="http://web.mit.edu/8.05/handouts/Twostates_03.pdf">http://web.mit.edu/8.05/handouts/Twostates_03.pdf</a></li>

<li>from the same course dealing with neutral particle oscillation, <a rel="nofollow" class="external free" href="http://web.mit.edu/8.05/handouts/nukaon_07.pdf">http://web.mit.edu/8.05/handouts/nukaon_07.pdf</a></li>

<li>from the Quantum mechanics I course offered at <a href="TIFR" class="mw-redirect" title="TIFR">TIFR</a>, <a rel="nofollow" class="external free" href="http://theory.tifr.res.in/~sgupta/courses/qm2013/hand4.pdf">http://theory.tifr.res.in/~sgupta/courses/qm2013/hand4.pdf</a> covers the essential mathematics</li>

<li><a rel="nofollow" class="external free" href="http://theory.tifr.res.in/~sgupta/courses/qm2013/hand5.pdf">http://theory.tifr.res.in/~sgupta/courses/qm2013/hand5.pdf</a>; from the same course deals with the some physical two-state systems and other important aspects of the formalism</li>

<li>the mathematical in the initial section is done in a manner similar to these notes <a rel="nofollow" class="external free" href="http://www.math.columbia.edu/~woit/QM/qubit.pdf">http://www.math.columbia.edu/~woit/QM/qubit.pdf</a>, which are from the Quantum Mechanics for Mathematicians course offered at University of Columbia.</li>

<li>a book version of the same ; <a rel="nofollow" class="external free" href="http://www.math.columbia.edu/~woit/QM/qmbook.pdf">http://www.math.columbia.edu/~woit/QM/qmbook.pdf</a></li>

<li>Two-state systems and the two-sphere, R. J. Plymen, Il Nuovo Cimento B 13 (1973), pp. 55–58</li></ul></li></ul>

<ul><li><a href="Rabi_cycle" title="Rabi cycle">Rabi cycle</a></li>

<li><a href="Doublet_(physics)" class="mw-redirect" title="Doublet (physics)">Doublet</a></li>

<li><a href="Nuclear_magnetic_resonance" title="Nuclear magnetic resonance">Nuclear magnetic resonance</a></li>

<li><a href="Quantum_optics" title="Quantum optics">Quantum optics</a></li>

<li><a href="Qubit" title="Qubit">Qubit</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFViolaLloyd1998" class="citation journal cs1">Viola, Lorenza; Lloyd, Seth (October 1998). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/PhysRevA.58.2733">"Dynamical suppression of decoherence in two-state quantum systems"</a>. Physical Review A. 58 (4). American Physical Society: 2733–2744. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/9803057">quant-ph/9803057</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevA.58.2733">10.1103/PhysRevA.58.2733</a>.</cite>

</li>

<li id="cite_note-griffiths353-2"><a href="#cite_ref-griffiths353_2-0">^</a> <cite id="CITEREFGriffiths2005" class="citation book cs1"><a href="David_J._Griffiths" title="David J. Griffiths">Griffiths, David</a> (2005). Introduction to Quantum Mechanics (2nd ed.). p. 353.</cite>

</li>

<li id="cite_note-feynman_1-3"><a href="#cite_ref-feynman_1_3-0">^</a> <cite id="CITEREFFeynman1965" class="citation book cs1"><a href="Richard_Feynman" title="Richard Feynman">Feynman, R.P.</a> (1965). "7-5 and 10-7". The Feynman Lectures on Physics: Volume 3. Addison Wesley.</cite>

</li>

<li id="cite_note-griffiths377-4"><a href="#cite_ref-griffiths377_4-0">^</a> Griffiths, p. 377.

</li>

<li id="cite_note-feynman_2-5"><a href="#cite_ref-feynman_2_5-0">^</a> <cite id="CITEREFFeynman1965" class="citation book cs1">Feynman, R.P. (1965). "11-4". The Feynman Lectures on Physics: Volume 3. Addison Wesley.</cite>

</li>

</ol></div></div>

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