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Probability amplitude

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about probability amplitude in quantum mechanics. For other uses, see <a href="Amplitude_(disambiguation)" class="mw-disambig" title="Amplitude (disambiguation)">Amplitude (disambiguation)</a>.</div>

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In <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, a probability amplitude is a <a href="Complex_number" title="Complex number">complex number</a> used for describing the behaviour of systems. The square of the <a href="Absolute_value" title="Absolute value">modulus</a> of this quantity at a point in space represents a <a href="Probability_density_function" title="Probability density function">probability density</a> at that point.

Probability amplitudes provide a relationship between the <a href="Quantum_state" title="Quantum state">quantum state</a> vector of a system and the results of observations of that system, a link that was first proposed by <a href="Max_Born" title="Max Born">Max Born</a>, in 1926. Interpretation of values of a wave function as the probability amplitude is a pillar of the <a href="Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen interpretation</a> of quantum mechanics. In fact, the properties of the space of wave functions were being used to make physical predictions (such as <a href="Atomic_emission_spectroscopy" title="Atomic emission spectroscopy">emissions from atoms</a> being at certain discrete energies) before any physical interpretation of a particular function was offered. Born was awarded half of the 1954 <a href="Nobel_Prize_in_Physics" title="Nobel Prize in Physics">Nobel Prize in Physics</a> for this understanding, and the probability thus calculated is sometimes called the "Born probability". These probabilistic concepts, namely the probability density and <a href="Quantum_measurement" class="mw-redirect" title="Quantum measurement">quantum measurements</a>, were vigorously contested at the time by the original physicists working on the theory, such as <a href="Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Schrödinger</a> and <a href="Albert_Einstein" title="Albert Einstein">Einstein</a>. It is the source of the mysterious consequences and philosophical difficulties in the <a href="Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">interpretations of quantum mechanics</a>—topics that continue to be debated even today.

<div class="mw-heading mw-heading2"><h2 id="Physical_overview">Physical overview</h2></div>

Neglecting some technical complexities, the problem of <a href="Quantum_measurement" class="mw-redirect" title="Quantum measurement">quantum measurement</a> is the behaviour of a quantum state, for which the value of the <a href="Observable" title="Observable">observable</a> Q to be measured is <a href="Uncertainty_principle" title="Uncertainty principle">uncertain</a>. Such a state is thought to be a <a href="Quantum_superposition" title="Quantum superposition">coherent superposition</a> of the observable's <a href="Eigenstate" class="mw-redirect" title="Eigenstate">eigenstates</a>, states on which the value of the observable is uniquely defined, for different possible values of the observable.

When a measurement of Q is made, the system (under the <a href="Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen interpretation</a>) <a href="State_vector_reduction" class="mw-redirect" title="State vector reduction">jumps to one of the eigenstates</a>, returning the eigenvalue belonging to that eigenstate. The system may always be described by a <a href="Linear_combination" title="Linear combination">linear combination</a> or <a href="Quantum_superposition" title="Quantum superposition">superposition</a> of these eigenstates with unequal <a href="Weight_function" title="Weight function">"weights"</a>. Intuitively it is clear that eigenstates with heavier "weights" are more "likely" to be produced. Indeed, which of the above eigenstates the system jumps to is given by a probabilistic law: the probability of the system jumping to the state is proportional to the absolute value of the corresponding numerical weight squared. These numerical weights are called probability amplitudes, and this relationship used to calculate probabilities from given pure quantum states (such as wave functions) is called the <a href="Born_rule" title="Born rule">Born rule</a>.

Clearly, the sum of the probabilities, which equals the sum of the <a href="Square_(algebra)#In_complex_numbers" title="Square (algebra)">absolute squares</a> of the probability amplitudes, must equal 1. This is the <a href="#Normalization">normalization</a> requirement.

If the system is known to be in some eigenstate of Q (e.g. after an observation of the corresponding eigenvalue of Q) the probability of observing that eigenvalue becomes equal to 1 (certain) for all subsequent measurements of Q (so long as no other important forces act between the measurements). In other words, the probability amplitudes are zero for all the other eigenstates, and remain zero for the future measurements. If the set of eigenstates to which the system can jump upon measurement of Q is the same as the set of eigenstates for measurement of R, then subsequent measurements of either Q or R always produce the same values with probability of 1, no matter the order in which they are applied. The probability amplitudes are unaffected by either measurement, and the observables are said to <a href="Commutator" title="Commutator">commute</a>.

By contrast, if the eigenstates of Q and R are different, then measurement of R produces a jump to a state that is not an eigenstate of Q. Therefore, if the system is known to be in some eigenstate of Q (all probability amplitudes zero except for one eigenstate), then when R is observed the probability amplitudes are changed. A second, subsequent observation of Q no longer certainly produces the eigenvalue corresponding to the starting state. In other words, the probability amplitudes for the second measurement of Q depend on whether it comes before or after a measurement of R, and the two observables <a href="Commutator" title="Commutator">do not commute</a>.

<div class="mw-heading mw-heading2"><h2 id="Mathematical_formulation">Mathematical formulation</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Bound_state#Definition" title="Bound state">Bound state § Definition</a></div>

In a formal setup, the state of an isolated physical system in <a href="Mathematical_formulation_of_quantum_mechanics#Description_of_the_state_of_a_system" title="Mathematical formulation of quantum mechanics">quantum mechanics</a> is represented, at a fixed time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">, by a <a href="Quantum_state" title="Quantum state">state vector</a> |Ψ⟩ belonging to a <a href="Hilbert_space#Separable_spaces" title="Hilbert space">separable</a> complex <a href="Hilbert_space#Quantum_mechanics" title="Hilbert space">Hilbert space</a>. Using <a href="Bra%E2%80%93ket_notation#Usage_in_quantum_mechanics" title="Bra–ket notation">bra–ket notation</a> the relation between state vector and "position <a href="Quantum_state#Basis_states_of_one-particle_systems" title="Quantum state">basis</a>" <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{|x\rangle \}}">

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

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</math><img src="./4bc9fa9ee80fce884a6cff63eaeba9d5b737985d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.206ex; height:2.843ex;" alt="{\displaystyle \{|x\rangle \}}" loading="lazy"> of the Hilbert space can be written as<a href="#cite_note-1">[1]</a>

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

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</math><img src="./79e8879bad59256ad2a4849de2bf31075f84773c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.345ex; height:2.843ex;" alt="{\displaystyle \psi (x)=\langle x|\Psi \rangle }" loading="lazy">.</dd></dl>

Its relation with an <a href="Observable" title="Observable">observable</a> can be elucidated by generalizing the quantum state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

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

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</math><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"> to a <a href="Measurable_function" title="Measurable function">measurable function</a> and its <a href="Partial_function" title="Partial function">domain of definition</a> to a given <a href="Measure_space#Important_classes_of_measure_spaces" title="Measure space">σ-finite measure space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,{\mathcal {A}},\mu )}">

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<annotation encoding="application/x-tex">{\displaystyle (X,{\mathcal {A}},\mu )}</annotation>

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</math><img src="./d634d210e57700027029694595ffea10410bf0d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.162ex; height:2.843ex;" alt="{\displaystyle (X,{\mathcal {A}},\mu )}" loading="lazy">. This allows for a refinement of <a href="Lebesgue's_decomposition_theorem" title="Lebesgue's decomposition theorem">Lebesgue's decomposition theorem</a>, decomposing μ into three mutually singular parts

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<annotation encoding="application/x-tex">{\displaystyle \mu =\mu _{\mathrm {ac} }+\mu _{\mathrm {sc} }+\mu _{\mathrm {pp} }}</annotation>

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</math><img src="./77887a69f80d57e11727948b21095b664e6efd5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.841ex; height:2.676ex;" alt="{\displaystyle \mu =\mu _{\mathrm {ac} }+\mu _{\mathrm {sc} }+\mu _{\mathrm {pp} }}" loading="lazy"></dd></dl>

where μac is absolutely continuous with respect to the Lebesgue measure, μsc is singular with respect to the Lebesgue measure and atomless, and μpp is a pure point measure.<a href="#cite_note-FOOTNOTESimon200543-2">[2]</a><a href="#cite_note-FOOTNOTETeschl2014114-119-3">[3]</a>

<div class="mw-heading mw-heading3"><h3 id="Continuous_amplitudes">Continuous amplitudes</h3></div>

A usual presentation of the probability amplitude is that of a <a href="Wave_function" title="Wave function">wave function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

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

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</math><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"> belonging to the L2 space of (<a href="Equivalence_class" title="Equivalence class">equivalence classes</a> of) <a href="Square-integrable_function" title="Square-integrable function">square integrable functions</a>, i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

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

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</math><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"> belongs to L2(X) if and only if

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<annotation encoding="application/x-tex">{\displaystyle \|\psi \|^{2}=\int _{X}|\psi (x)|^{2}\,dx<\infty }</annotation>

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</math><img src="./e16169ced69c21d97e5053f26ecc41b7bf248a56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.658ex; height:5.676ex;" alt="{\displaystyle \|\psi \|^{2}=\int _{X}|\psi (x)|^{2}\,dx<\infty }" loading="lazy">.</dd></dl>

If the <a href="Normed_vector_space" title="Normed vector space">norm</a> is equal to 1 and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi (x)|^{2}\in \mathbb {R} _{\geq 0}}">

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<annotation encoding="application/x-tex">{\displaystyle |\psi (x)|^{2}\in \mathbb {R} _{\geq 0}}</annotation>

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</math><img src="./b6d9a303a7fef588df82a338758e3fea55730c7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.852ex; height:3.343ex;" alt="{\displaystyle |\psi (x)|^{2}\in \mathbb {R} _{\geq 0}}" loading="lazy"> such that

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<annotation encoding="application/x-tex">{\displaystyle \int _{X}|\psi (x)|^{2}\,dx\equiv \int _{X}\,d\mu _{ac}(x)=1}</annotation>

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</math><img src="./477806cdfbfade0b7692a1f4c055a841364fd667.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.873ex; height:5.676ex;" alt="{\displaystyle \int _{X}|\psi (x)|^{2}\,dx\equiv \int _{X}\,d\mu _{ac}(x)=1}" loading="lazy">,</dd></dl>

then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi (x)|^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle |\psi (x)|^{2}}</annotation>

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</math><img src="./99516be05fd9132a16e5f2dcd0cb03bd632eab12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7ex; height:3.343ex;" alt="{\displaystyle |\psi (x)|^{2}}" loading="lazy"> is the <a href="Probability_density_function" title="Probability density function">probability density function</a> for a measurement of the particle's position at a given time, defined as the <a href="Radon%E2%80%93Nikodym_derivative" class="mw-redirect" title="Radon–Nikodym derivative">Radon–Nikodym derivative</a> with respect to the <a href="Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a> (e.g. on the set R of all <a href="Real_number" title="Real number">real numbers</a>). As probability is a dimensionless quantity, |ψ(x)|2 must have the inverse dimension of the variable of integration x. For example, the above amplitude has <a href="Dimensional_analysis" title="Dimensional analysis">dimension</a> [L−1/2], where L represents <a href="Length" title="Length">length</a>.

Whereas a Hilbert space is separable if and only if it admits a <a href="Countable" class="mw-redirect" title="Countable">countable</a> orthonormal basis, the <a href="Range_of_a_function" title="Range of a function">range</a> of a <a href="Random_variable#Continuous_random_variable" title="Random variable">continuous random variable</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.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 x}" loading="lazy"> is an <a href="Uncountable_set" title="Uncountable set">uncountable set</a> (i.e. the probability that the system is "at position <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.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 x}" loading="lazy">" will always <a href="Almost_never" class="mw-redirect" title="Almost never">be zero</a>). As such, <a href="Eigenstate" class="mw-redirect" title="Eigenstate">eigenstates</a> of an observable need not necessarily be measurable functions belonging to L2(X) (see <a href="#Normalization">normalization condition</a> below). A <a href="Expectation_value_(quantum_mechanics)#Example_in_configuration_space" title="Expectation value (quantum mechanics)">typical example</a> is the <a href="Position_operator" title="Position operator">position operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathrm {x} }}}">

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

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</math><img src="./4e557e942cfff0e071c543f9a43d650cd7b90e0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.227ex; height:2.176ex;" alt="{\displaystyle {\hat {\mathrm {x} }}}" loading="lazy"> defined as

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<annotation encoding="application/x-tex">{\displaystyle \langle x|{\hat {\mathrm {x} }}|\Psi \rangle ={\hat {\mathrm {x} }}\langle x|\Psi \rangle =x_{0}\psi (x),\quad x\in \mathbb {R} ,}</annotation>

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</math><img src="./a977255ebcc82f7e647d27f800160d19df102322.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.374ex; height:2.843ex;" alt="{\displaystyle \langle x|{\hat {\mathrm {x} }}|\Psi \rangle ={\hat {\mathrm {x} }}\langle x|\Psi \rangle =x_{0}\psi (x),\quad x\in \mathbb {R} ,}" loading="lazy"></dd></dl>

whose eigenfunctions are <a href="Dirac_delta_function#Quantum_mechanics" title="Dirac delta function">Dirac delta functions</a>

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

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</math><img src="./aa87a227061c0dcf971d392f868a8192d2792cd3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.163ex; height:2.843ex;" alt="{\displaystyle \psi (x)=\delta (x-x_{0})}" loading="lazy"></dd></dl>

which clearly do not belong to L2(X). By replacing the state space by a suitable <a href="Rigged_Hilbert_space" title="Rigged Hilbert space">rigged Hilbert space</a>, however, the rigorous notion of eigenstates from <a href="Self-adjoint_operator#Spectral_theorem" title="Self-adjoint operator">spectral theorem</a> as well as <a href="Decomposition_of_spectrum_(functional_analysis)#Quantum_physics" title="Decomposition of spectrum (functional analysis)">spectral decomposition</a> is preserved.<a href="#cite_note-FOOTNOTEde_la_Madrid_Modino200197-4">[4]</a>

<div class="mw-heading mw-heading3"><h3 id="Discrete_amplitudes">Discrete amplitudes</h3></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{pp}}">

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

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</math><img src="./e51b806230ae169e7b50c4a77288f365aa6fa242.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.288ex; height:2.343ex;" alt="{\displaystyle \mu _{pp}}" loading="lazy"> be <a href="Atom_(measure_theory)" title="Atom (measure theory)">atomic</a> (i.e. the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subset X}">

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</math><img src="./826569be03f873b81cdc6f12637ef5520c369d21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.822ex; height:2.176ex;" alt="{\displaystyle A\subset X}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}}">

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

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</math><img src="./280ae03440942ab348c2ca9b8db6b56ffa9618f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}" loading="lazy"> is an atom); specifying the measure of any <a href="Continuous_or_discrete_variable#Discrete_variable" title="Continuous or discrete variable">discrete variable</a> x ∈ A equal to 1. The amplitudes are composed of state vector |Ψ⟩ <a href="Indexed_family" title="Indexed family">indexed</a> by A; its components are denoted by ψ(x) for uniformity with the previous case. If the <a href="Lp_space#General_ℓp-space" title="Lp space">ℓ2-norm</a> of |Ψ⟩ is equal to 1, then |ψ(x)|2 is a <a href="Probability_mass_function" title="Probability mass function">probability mass function</a>.

A convenient configuration space X is such that each point x produces some unique value of the observable Q. For discrete X it means that all elements of the standard basis are <a href="Eigenvector" class="mw-redirect" title="Eigenvector">eigenvectors</a> of Q. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x)}">

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

</semantics>

</math><img src="./a596a1fb4130a47f6b88c66150497338bd6cbccc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.652ex; height:2.843ex;" alt="{\displaystyle \psi (x)}" loading="lazy"> is the probability amplitude for the eigenstate |x⟩. If it corresponds to a non-<a href="Degenerate_energy_levels" title="Degenerate energy levels">degenerate</a> eigenvalue of Q, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi (x)|^{2}}">

</mrow>

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle |\psi (x)|^{2}}</annotation>

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</math><img src="./99516be05fd9132a16e5f2dcd0cb03bd632eab12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7ex; height:3.343ex;" alt="{\displaystyle |\psi (x)|^{2}}" loading="lazy"> gives the probability of the corresponding value of Q for the initial state |Ψ⟩.

|ψ(x)| = 1 if and only if |x⟩ is <a href="Ray_(quantum_theory)" class="mw-redirect" title="Ray (quantum theory)">the same quantum state</a> as |Ψ⟩. ψ(x) = 0 if and only if |x⟩ and |Ψ⟩ are <a href="Orthogonality_(mathematics)" title="Orthogonality (mathematics)">orthogonal</a>. Otherwise the modulus of ψ(x) is between 0 and 1.

A discrete probability amplitude may be considered as a <a href="Fundamental_frequency" title="Fundamental frequency">fundamental frequency</a> in the probability frequency domain (<a href="Spherical_harmonics" title="Spherical harmonics">spherical harmonics</a>) for the purposes of simplifying <a href="M-theory" title="M-theory">M-theory</a> transformation calculations. Discrete dynamical variables are used in such problems as a <a href="Particle_in_a_box" title="Particle in a box">particle in an idealized reflective box</a> and <a href="Quantum_harmonic_oscillator" title="Quantum harmonic oscillator">quantum harmonic oscillator</a>.

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

An example of the discrete case is a quantum system that can be in <a href="Two-state_quantum_system" title="Two-state quantum system">two possible states</a>, e.g. the <a href="Light_polarization" class="mw-redirect" title="Light polarization">polarization</a> of a <a href="Photon" title="Photon">photon</a>. When the polarization is measured, it could be the horizontal state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |H\rangle }">

</mrow>

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

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</math><img src="./f010e5c1fd1410fa37d26b3ae7c769db1df7ab57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.615ex; height:2.843ex;" alt="{\displaystyle |H\rangle }" loading="lazy"> or the vertical state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |V\rangle }">

</mrow>

</mstyle>

</mrow>

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

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</math><img src="./89463c7bdd8be86eae354b0fd9fc5ca09d423499.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.339ex; height:2.843ex;" alt="{\displaystyle |V\rangle }" loading="lazy">. Until its polarization is measured the photon can be in a <a href="Quantum_superposition" title="Quantum superposition">superposition</a> of both these states, so its state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle }">

</mrow>

</mstyle>

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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"> could be written as

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<annotation encoding="application/x-tex">{\displaystyle |\psi \rangle =\alpha |H\rangle +\beta |V\rangle }</annotation>

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</math><img src="./1b7579290f6018364de65fe3b84288c020db00f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.777ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle =\alpha |H\rangle +\beta |V\rangle }" loading="lazy">,</dd></dl>

with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

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

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</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

</mstyle>

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

</semantics>

</math><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"> the probability amplitudes for the states <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |H\rangle }">

</mrow>

</mstyle>

</mrow>

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

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</math><img src="./f010e5c1fd1410fa37d26b3ae7c769db1df7ab57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.615ex; height:2.843ex;" alt="{\displaystyle |H\rangle }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |V\rangle }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./89463c7bdd8be86eae354b0fd9fc5ca09d423499.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.339ex; height:2.843ex;" alt="{\displaystyle |V\rangle }" loading="lazy"> respectively. When the photon's polarization is measured, the resulting state is either horizontal or vertical. But in a random experiment, the probability of being horizontally polarized is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\alpha |^{2}}">

</mrow>

<msup>

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</msup>

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

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</math><img src="./fa34618537661f2d4d710cc26e8afe891f50f7b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.836ex; height:3.343ex;" alt="{\displaystyle |\alpha |^{2}}" loading="lazy">, and the probability of being vertically polarized is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\beta |^{2}}">

</mrow>

<msup>

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</msup>

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

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</math><img src="./4caecb560883af3b9c0c3a1d0e13aae75f121d0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.68ex; height:3.343ex;" alt="{\displaystyle |\beta |^{2}}" loading="lazy">.

Hence, a photon in a state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle |\psi \rangle ={\sqrt {\frac {1}{3}}}|H\rangle -i{\sqrt {\frac {2}{3}}}|V\rangle }">

</mrow>

<msqrt>

<mfrac>

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<mfrac>

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<annotation encoding="application/x-tex">{\textstyle |\psi \rangle ={\sqrt {\frac {1}{3}}}|H\rangle -i{\sqrt {\frac {2}{3}}}|V\rangle }</annotation>

</semantics>

</math><img src="./4b065180e966e557f9981a1d17c4209b798a28c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.724ex; height:4.843ex;" alt="{\textstyle |\psi \rangle ={\sqrt {\frac {1}{3}}}|H\rangle -i{\sqrt {\frac {2}{3}}}|V\rangle }" loading="lazy"> would have a probability of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{3}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\textstyle {\frac {1}{3}}}</annotation>

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</math><img src="./26ea1944c3f21fcd8a37aa679a3bce05d5cd6e1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\textstyle {\frac {1}{3}}}" loading="lazy"> to come out horizontally polarized, and a probability of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {2}{3}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\textstyle {\frac {2}{3}}}</annotation>

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</math><img src="./ab0e30d9a795c6a0635f89ec69e4ef7ed13a0d14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\textstyle {\frac {2}{3}}}" loading="lazy"> to come out vertically polarized when an <a href="Statistical_ensemble_(mathematical_physics)" class="mw-redirect" title="Statistical ensemble (mathematical physics)">ensemble</a> of measurements are made. The order of such results, is, however, completely random.

Another example is quantum spin. If a spin-measuring apparatus is pointing along the z-axis and is therefore able to measure the z-component of the spin (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sigma _{z}}">

<msub>

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

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</math><img src="./a6d68a7e5a5e5f002ce6b823576bdda6ef7a895b.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="{\textstyle \sigma _{z}}" loading="lazy">), the following must be true for the measurement of spin "up" and "down":

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<annotation encoding="application/x-tex">{\displaystyle \sigma _{z}|u\rangle =(+1)|u\rangle }</annotation>

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</math><img src="./6208b238c82eac2742d9c1f78618cecb0dd865cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.97ex; height:2.843ex;" alt="{\displaystyle \sigma _{z}|u\rangle =(+1)|u\rangle }" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \sigma _{z}|d\rangle =(-1)|d\rangle }</annotation>

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</math><img src="./36f2a7744b43834607dee5d9f27d78b2cf757513.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.742ex; height:2.843ex;" alt="{\displaystyle \sigma _{z}|d\rangle =(-1)|d\rangle }" loading="lazy"></dd></dl>

If one assumes that system is prepared, so that +1 is registered in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sigma _{x}}">

<msub>

</mrow>

</msub>

</mstyle>

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

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</math><img src="./5ebd64ceccea373875cb5082f36cbd1131f1b9ba.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="{\textstyle \sigma _{x}}" loading="lazy"> and then the apparatus is rotated to measure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sigma _{z}}">

<msub>

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</msub>

</mstyle>

</mrow>

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

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</math><img src="./a6d68a7e5a5e5f002ce6b823576bdda6ef7a895b.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="{\textstyle \sigma _{z}}" loading="lazy">, the following holds:

<mtr>

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<mrow>

<mfrac>

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<mfrac>

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<mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\langle r|u\rangle &=\left({\frac {1}{\sqrt {2}}}\langle u|+{\frac {1}{\sqrt {2}}}\langle d|\right)\cdot |u\rangle \\&=\left({\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\0\end{pmatrix}}+{\frac {1}{\sqrt {2}}}{\begin{pmatrix}0\\1\end{pmatrix}}\right)\cdot {\begin{pmatrix}1\\0\end{pmatrix}}\\&={\frac {1}{\sqrt {2}}}\end{aligned}}}</annotation>

</semantics>

</math><img src="./934c87591de81719be44195b9fd801dc5269393b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.813ex; margin-bottom: -0.192ex; width:40.499ex; height:19.176ex;" alt="{\displaystyle {\begin{aligned}\langle r|u\rangle &=\left({\frac {1}{\sqrt {2}}}\langle u|+{\frac {1}{\sqrt {2}}}\langle d|\right)\cdot |u\rangle \\&=\left({\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\0\end{pmatrix}}+{\frac {1}{\sqrt {2}}}{\begin{pmatrix}0\\1\end{pmatrix}}\right)\cdot {\begin{pmatrix}1\\0\end{pmatrix}}\\&={\frac {1}{\sqrt {2}}}\end{aligned}}}" loading="lazy"></dd></dl>

The probability amplitude of measuring spin up is given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \langle r|u\rangle }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \langle r|u\rangle }</annotation>

</semantics>

</math><img src="./acc43e4cdb86b47776d651c4772c75aca81720ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.834ex; height:2.843ex;" alt="{\textstyle \langle r|u\rangle }" loading="lazy">, since the system had the initial state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle |r\rangle }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8c0ab4367dbc63ca7fd567a14fafaea79858b174.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.6ex; height:2.843ex;" alt="{\textstyle |r\rangle }" loading="lazy">. The probability of measuring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle |u\rangle }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./35b63759d17f70dd8a50b5c58b1872b15b03d9d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.881ex; height:2.843ex;" alt="{\textstyle |u\rangle }" loading="lazy"> is given by

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<msup>

<mrow>

<mfrac>

<msqrt>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle P(|u\rangle )=\langle r|u\rangle \langle u|r\rangle =\left({\frac {1}{\sqrt {2}}}\right)^{2}={\frac {1}{2}}}</annotation>

</semantics>

</math><img src="./908930b8d3da1b2ab7ca23076d298b5d7974429d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:35.809ex; height:6.843ex;" alt="{\displaystyle P(|u\rangle )=\langle r|u\rangle \langle u|r\rangle =\left({\frac {1}{\sqrt {2}}}\right)^{2}={\frac {1}{2}}}" loading="lazy"></dd></dl>

Which agrees with experiment.

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

In the example above, the measurement must give either | H ⟩ or | V ⟩, so the total probability of measuring | H ⟩ or | V ⟩ must be 1. This leads to a constraint that α2 + β2 = 1; more generally the sum of the squared moduli of the probability amplitudes of all the possible states is equal to one. If to understand "all the possible states" as an <a href="Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a>, that makes sense in the discrete case, then this condition is the same as the norm-1 condition explained <a href="#Mathematical">above</a>.

One can always divide any non-zero element of a Hilbert space by its norm and obtain a normalized state vector. Not every wave function belongs to the Hilbert space L2(X), though. Wave functions that fulfill this constraint are called <a href="Normalizable_wave_function" class="mw-redirect" title="Normalizable wave function">normalizable</a>.

The <a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a>, describing states of quantum particles, has solutions that describe a system and determine precisely how the state <a href="Time_evolution_operator" class="mw-redirect" title="Time evolution operator">changes with time</a>. Suppose a <a href="Wave_function" title="Wave function">wave function</a> ψ(x, t) gives a description of the particle (position x at a given time t). A wave function is <a href="Square_integrable" class="mw-redirect" title="Square integrable">square integrable</a> if

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<annotation encoding="application/x-tex">{\displaystyle \int |\psi (\mathbf {x} ,t)|^{2}\,\mathrm {d\mathbf {x} } =a^{2}<\infty .}</annotation>

</semantics>

</math><img src="./3223e005f76195426622406c89eacd425716aeb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.078ex; height:5.676ex;" alt="{\displaystyle \int |\psi (\mathbf {x} ,t)|^{2}\,\mathrm {d\mathbf {x} } =a^{2}<\infty .}" loading="lazy"></dd></dl>

After <a href="Wave_function#Normalization_condition" title="Wave function">normalization</a> the wave function still represents the same state and is therefore equal by definition to<a href="#cite_note-FOOTNOTEBäuerlede_Kerf1990330-5">[5]</a><a href="#cite_note-6">[6]</a>

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<annotation encoding="application/x-tex">{\displaystyle \psi (\mathbf {x} ,t):={\frac {\psi (\mathbf {x} ,t)}{a}}.}</annotation>

</semantics>

</math><img src="./5d201d85365f4fed311318e49cb41cb4c739c748.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.442ex; height:5.676ex;" alt="{\displaystyle \psi (\mathbf {x} ,t):={\frac {\psi (\mathbf {x} ,t)}{a}}.}" loading="lazy"></dd></dl>

Under the standard <a href="Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen interpretation</a>, the normalized wavefunction gives probability amplitudes for the position of the particle. Hence, ρ(x) = |ψ(x, t)|2 is a <a href="Probability_density_function" title="Probability density function">probability density function</a> and the probability that the particle is in the volume V at fixed time t is given by

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<annotation encoding="application/x-tex">{\displaystyle P_{\mathbf {x} \in V}(t)=\int _{V}|\psi (\mathbf {x} ,t)|^{2}\,\mathrm {d\mathbf {x} } =\int _{V}\rho (\mathbf {x} )\,\mathrm {d\mathbf {x} } .}</annotation>

</semantics>

</math><img src="./4567b973559bfae2eff07f1f5740f7953cd38302.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:40.485ex; height:5.676ex;" alt="{\displaystyle P_{\mathbf {x} \in V}(t)=\int _{V}|\psi (\mathbf {x} ,t)|^{2}\,\mathrm {d\mathbf {x} } =\int _{V}\rho (\mathbf {x} )\,\mathrm {d\mathbf {x} } .}" loading="lazy"></dd></dl>

The probability density function does not vary with time as the evolution of the wave function is dictated by the Schrödinger equation and is therefore entirely deterministic.<a href="#cite_note-FOOTNOTEZwiebach2022170-7">[7]</a> This is key to understanding the importance of this interpretation: for a given particle constant <a href="Mass" title="Mass">mass</a>, initial ψ(x, t0) and <a href="Potential_energy" title="Potential energy">potential</a>, the Schrödinger equation fully determines subsequent wavefunctions. The above then gives probabilities of locations of the particle at all subsequent times.

<div class="mw-heading mw-heading2"><h2 id="In_the_context_of_the_double-slit_experiment">In the context of the double-slit experiment</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Double-slit_experiment" title="Double-slit experiment">Double-slit experiment</a></div>

Probability amplitudes have special significance because they act in quantum mechanics as the equivalent of conventional probabilities, with many analogous laws, as described above. For example, in the classic <a href="Double-slit_experiment" title="Double-slit experiment">double-slit experiment</a>, electrons are fired randomly at two slits, and the probability distribution of detecting electrons at all parts on a large screen placed behind the slits, is questioned. An intuitive answer is that P(through either slit) = P(through first slit) + P(through second slit), where P(event) is the probability of that event. This is obvious if one assumes that an electron passes through either slit. When no measurement apparatus that determines through which slit the electrons travel is installed, the observed probability distribution on the screen reflects the <a href="Interference_(wave_propagation)" class="mw-redirect" title="Interference (wave propagation)">interference pattern</a> that is common with light waves. If one assumes the above law to be true, then this pattern cannot be explained. The particles cannot be said to go through either slit and the simple explanation does not work. The correct explanation is, however, by the association of probability amplitudes to each event. The complex amplitudes which represent the electron passing each slit (ψfirst and ψsecond) follow the law of precisely the form expected: ψtotal = ψfirst + ψsecond. This is the principle of <a href="Quantum_superposition" title="Quantum superposition">quantum superposition</a>. The probability, which is the <a href="Modulus_squared" class="mw-redirect" title="Modulus squared">modulus squared</a> of the probability amplitude, then, follows the interference pattern under the requirement that amplitudes are complex:

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<mtext>first</mtext>

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<mtext>second</mtext>

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<mtext>first</mtext>

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<mtext>second</mtext>

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<annotation encoding="application/x-tex">{\displaystyle P=\left|\psi _{\text{first}}+\psi _{\text{second}}\right|^{2}=\left|\psi _{\text{first}}\right|^{2}+\left|\psi _{\text{second}}\right|^{2}+2\left|\psi _{\text{first}}\right|\left|\psi _{\text{second}}\right|\cos(\varphi _{1}-\varphi _{2}).}</annotation>

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

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

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</math><img src="./d7daf493c8f6ef669c04c7b9715532fc35d12d60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.574ex; height:2.176ex;" alt="{\displaystyle \varphi _{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{2}}">

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

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</math><img src="./c08631714273b6c8edaa9573ef3d8c548314a930.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.574ex; height:2.176ex;" alt="{\displaystyle \varphi _{2}}" loading="lazy">

are the <a href="Argument_(complex_analysis)" title="Argument (complex analysis)">arguments</a> of ψfirst and ψsecond respectively. A purely real formulation has too few dimensions to describe the system's state when superposition is taken into account. That is, without the arguments of the amplitudes, we cannot describe the phase-dependent interference. The crucial term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 2\left|\psi _{\text{first}}\right|\left|\psi _{\text{second}}\right|\cos(\varphi _{1}-\varphi _{2})}">

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<mtext>first</mtext>

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<annotation encoding="application/x-tex">{\textstyle 2\left|\psi _{\text{first}}\right|\left|\psi _{\text{second}}\right|\cos(\varphi _{1}-\varphi _{2})}</annotation>

</semantics>

</math><img src="./0acda04d3352bb5ecf0cc527b9f0adfb2652406d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.962ex; height:2.843ex;" alt="{\textstyle 2\left|\psi _{\text{first}}\right|\left|\psi _{\text{second}}\right|\cos(\varphi _{1}-\varphi _{2})}" loading="lazy"> is called the "interference term", and this would be missing if we had added the probabilities.

However, one may choose to devise an experiment in which the experimenter observes which slit each electron goes through. Then, due to <a href="Wavefunction_collapse" class="mw-redirect" title="Wavefunction collapse">wavefunction collapse</a>, the interference pattern is not observed on the screen.

One may go further in devising an experiment in which the experimenter gets rid of this "which-path information" by a <a href="Quantum_eraser_experiment" title="Quantum eraser experiment">"quantum eraser"</a>. Then, according to the <a href="Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen interpretation</a>, the case A applies again and the interference pattern is restored.<a href="#cite_note-8">[8]</a>

<div class="mw-heading mw-heading2"><h2 id="Conservation_of_probabilities_and_the_continuity_equation">Conservation of probabilities and the continuity equation</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Probability_current" title="Probability current">Probability current</a></div>

Intuitively, since a normalised wave function stays normalised while evolving according to the wave equation, there will be a relationship between the change in the probability density of the particle's position and the change in the amplitude at these positions.

Define the <a href="Probability_current" title="Probability current">probability current</a> (or flux) j as

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {j} ={\hbar \over m}{1 \over {2i}}\left(\psi ^{*}\nabla \psi -\psi \nabla \psi ^{*}\right)={\hbar \over m}\operatorname {Im} \left(\psi ^{*}\nabla \psi \right),}</annotation>

</semantics>

</math><img src="./ebfad5cd3dccd9a34e1030deacae409d7a665b6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.164ex; width:44.436ex; height:5.343ex;" alt="{\displaystyle \mathbf {j} ={\hbar \over m}{1 \over {2i}}\left(\psi ^{*}\nabla \psi -\psi \nabla \psi ^{*}\right)={\hbar \over m}\operatorname {Im} \left(\psi ^{*}\nabla \psi \right),}" loading="lazy"></dd></dl>

measured in units of (probability)/(area × time).

Then the current satisfies the equation

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<annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {j} +{\partial \over \partial t}|\psi |^{2}=0.}</annotation>

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</math><img src="./13a48758b57dbb51cb783037059b7abce23f48d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.034ex; height:5.509ex;" alt="{\displaystyle \nabla \cdot \mathbf {j} +{\partial \over \partial t}|\psi |^{2}=0.}" loading="lazy"></dd></dl>

The probability density is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =|\psi |^{2}}">

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

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</math><img src="./3102dd83fdcdaa7cb4ff7c44b81562624d1efc54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.161ex; height:3.343ex;" alt="{\displaystyle \rho =|\psi |^{2}}" loading="lazy">, this equation is exactly the <a href="Continuity_equation" title="Continuity equation">continuity equation</a>, appearing in many situations in physics where we need to describe the local conservation of quantities. The best example is in classical electrodynamics, where j corresponds to current density corresponding to electric charge, and the density is the charge-density. The corresponding continuity equation describes the local <a href="Charge_conservation" title="Charge conservation">conservation of charges</a>.

<div class="mw-heading mw-heading2"><h2 id="Composite_systems">Composite systems</h2></div>

For two quantum systems with spaces L2(X1) and L2(X2) and given states |Ψ1⟩ and |Ψ2⟩ respectively, their combined state |Ψ1⟩ <a href="Outer_product" title="Outer product">⊗</a> |Ψ2⟩ can be expressed as ψ1(x1) ψ2(x2) a function on X1 <a href="Direct_product" title="Direct product">×</a> X2, that gives the

<a href="Product_measure" title="Product measure">product of respective probability measures</a>. In other words, amplitudes of a non-<a href="Quantum_entanglement" title="Quantum entanglement">entangled</a> composite state are <a href="Multiplication" title="Multiplication">products</a> of original amplitudes, and <a href="#convenient">respective observables</a> on the systems 1 and 2 behave on these states as <a href="Independent_random_variables" class="mw-redirect" title="Independent random variables">independent random variables</a>. This strengthens the probabilistic interpretation explicated <a href="#The_laws_of_calculating_probabilities_of_events">above</a> .

<div class="mw-heading mw-heading2"><h2 id="Amplitudes_in_operators">Amplitudes in operators</h2></div>

The concept of amplitudes is also used in the context of <a href="Scattering_theory" class="mw-redirect" title="Scattering theory">scattering theory</a>, notably in the form of <a href="S-matrix" title="S-matrix">S-matrices</a>. Whereas moduli of vector components squared, for a given vector, give a fixed probability distribution, moduli of <a href="Matrix_element_(physics)" title="Matrix element (physics)">matrix elements</a> squared are interpreted as <a href="Transition_probabilities" class="mw-redirect" title="Transition probabilities">transition probabilities</a> just as in a random process. Like a finite-dimensional <a href="Unit_vector" title="Unit vector">unit vector</a> specifies a finite probability distribution, a finite-dimensional <a href="Unitary_matrix" title="Unitary matrix">unitary matrix</a> specifies transition probabilities between a finite number of states.

The "transitional" interpretation may be applied to L2s on non-discrete spaces as well.

<ul><li><a href="Expectation_value_(quantum_mechanics)" title="Expectation value (quantum mechanics)">Expectation value (quantum mechanics)</a></li>

<li><a href="Free_particle" title="Free particle">Free particle</a></li>

<li><a href="Finite_potential_barrier" class="mw-redirect" title="Finite potential barrier">Finite potential barrier</a></li>

<li><a href="Matter_wave" title="Matter wave">Matter wave</a></li>

<li><a href="Phase_space_formulation" class="mw-redirect" title="Phase space formulation">Phase space formulation</a></li>

<li><a href="Uncertainty_principle" title="Uncertainty principle">Uncertainty principle</a></li>

<li><a href="John_Clive_Ward" title="John Clive Ward">Ward's probability amplitude</a></li>

<li><a href="Wave_packet" title="Wave packet">Wave packet</a></li></ul>

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

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> The spanning set of a Hilbert space does not suffice for defining coordinates as wave functions form rays in a projective Hilbert space (rather than an ordinary Hilbert space). See: <a href="Projective_space#Frame" title="Projective space">Projective frame</a>

</li>

<li id="cite_note-FOOTNOTESimon200543-2"><a href="#cite_ref-FOOTNOTESimon200543_2-0">^</a> <a href="#CITEREFSimon2005">Simon 2005</a>, p. 43.

</li>

<li id="cite_note-FOOTNOTETeschl2014114-119-3"><a href="#cite_ref-FOOTNOTETeschl2014114-119_3-0">^</a> <a href="#CITEREFTeschl2014">Teschl 2014</a>, p. 114-119.

</li>

<li id="cite_note-FOOTNOTEde_la_Madrid_Modino200197-4"><a href="#cite_ref-FOOTNOTEde_la_Madrid_Modino200197_4-0">^</a> <a href="#CITEREFde_la_Madrid_Modino2001">de la Madrid Modino 2001</a>, p. 97.

</li>

<li id="cite_note-FOOTNOTEBäuerlede_Kerf1990330-5"><a href="#cite_ref-FOOTNOTEBäuerlede_Kerf1990330_5-0">^</a> <a href="#CITEREFBäuerlede_Kerf1990">Bäuerle & de Kerf 1990</a>, p. 330.

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> See also <a href="Wigner's_theorem" title="Wigner's theorem">Wigner's theorem</a>

</li>

<li id="cite_note-FOOTNOTEZwiebach2022170-7"><a href="#cite_ref-FOOTNOTEZwiebach2022170_7-0">^</a> <a href="#CITEREFZwiebach2022">Zwiebach 2022</a>, p. 170.

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> A recent 2013 experiment gives insight regarding the correct physical interpretation of such phenomena. The information can actually be obtained, but then the electron seemingly went through all the possible paths simultaneously. (Certain <a href="Ensemble_interpretation" title="Ensemble interpretation">ensemble-alike</a> realistic interpretations of the wavefunction may presume such coexistence in all the points of an orbital.) Cf. <style data-mw-deduplicate="TemplateStyles:r1238218222">

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</style><cite id="CITEREFSchmidtLowerJahnkeSchößler2013" class="citation journal cs1">Schmidt, L. Ph. H.; et al. (2013). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190307191633/http://pdfs.semanticscholar.org/e551/f885162ab3b25b16fb7fff48c87dbc1cbd02.pdf">"Momentum Transfer to a Free Floating Double Slit: Realization of a Thought Experiment from the Einstein-Bohr Debates"</a> (PDF). <a href="Physical_Review_Letters" title="Physical Review Letters">Physical Review Letters</a>. 111 (10): 103201. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013PhRvL.111j3201S">2013PhRvL.111j3201S</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%2FPhysRevLett.111.103201">10.1103/PhysRevLett.111.103201</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/25166663">25166663</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:2725093">2725093</a>. Archived from <a rel="nofollow" class="external text" href="http://pdfs.semanticscholar.org/e551/f885162ab3b25b16fb7fff48c87dbc1cbd02.pdf">the original</a> (PDF) on 2019-03-07.</cite>

</li>

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

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

<ul><li><cite id="CITEREFBäuerlede_Kerf1990" class="citation book cs1">Bäuerle, Gerard G. A.; de Kerf, Eddy A. (1990). Lie Algebras, Part 1: Finite and Infinite Dimensional Lie Algebras and Applications in Physics. Studies in Mathematical Physics. Amsterdam: North Holland. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-444-88776-8</bdi>.</cite></li>

<li><cite id="CITEREFde_la_Madrid_Modino2001" class="citation thesis cs1">de la Madrid Modino, R. (2001). <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?oi=bibs&cluster=2442809273695897641&btnI=1&hl=en">Quantum mechanics in rigged Hilbert space language</a> (PhD thesis). Universidad de Valladolid.</cite></li>

<li><cite id="CITEREFFeynmanLeightonSands1989" class="citation book cs1">Feynman, R. P.; Leighton, R. B.; Sands, M. (1989). <a rel="nofollow" class="external text" href="https://feynmanlectures.caltech.edu/III_03.html">"Probability Amplitudes"</a>. <a href="The_Feynman_Lectures_on_Physics" title="The Feynman Lectures on Physics">The Feynman Lectures on Physics</a>. Vol. 3. Redwood City: Addison-Wesley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-201-51005-7</bdi>.</cite></li>

<li><cite id="CITEREFGudder1988" class="citation book cs1">Gudder, Stanley P. (1988). Quantum Probability. San Diego: Academic Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-12-305340-4</bdi>.</cite></li>

<li><cite id="CITEREFSimon2005" class="citation book cs1"><a href="Barry_Simon" title="Barry Simon">Simon, Barry</a> (2005). Orthogonal polynomials on the unit circle. Part 1. Classical theory. American Mathematical Society Colloquium Publications. Vol. 54. Providence, R.I.: <a href="American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8218-3446-6</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2105088">2105088</a>.</cite></li>

<li><cite id="CITEREFTeschl2014" class="citation book cs1">Teschl, G. (2014). Mathematical Methods in Quantum Mechanics. Providence (R.I): American Mathematical Soc. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4704-1704-8</bdi>.</cite></li>

<li><cite id="CITEREFZwiebach2022" class="citation book cs1">Zwiebach, Barton (2022). Mastering Quantum Mechanics. Cambridge, Mass: MIT Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-262-04613-8</bdi>.</cite></li></ul>

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</style><div id="Quantum_mechanics328" style="font-size:114%;margin:0 4em"><a href="Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Background</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction</a></li>

<li><a href="History_of_quantum_mechanics" title="History of quantum mechanics">History</a>

<ul><li><a href="Timeline_of_quantum_mechanics" title="Timeline of quantum mechanics">Timeline</a></li></ul></li>

<li><a href="Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li>

<li><a href="Old_quantum_theory" title="Old quantum theory">Old quantum theory</a></li>

<li><a href="Glossary_of_elementary_quantum_mechanics" title="Glossary of elementary quantum mechanics">Glossary</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fundamentals</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a></li>

<li><a href="Complementarity_(physics)" title="Complementarity (physics)"> Complementarity</a></li>

<li><a href="Density_matrix" title="Density matrix">Density matrix</a></li>

<li><a href="Energy_level" title="Energy level">Energy level</a>

<ul><li><a href="Ground_state" title="Ground state">Ground state</a></li>

<li><a href="Excited_state" title="Excited state">Excited state</a></li>

<li><a href="Degenerate_energy_levels" title="Degenerate energy levels">Degenerate levels</a></li>

<li><a href="Zero-point_energy" title="Zero-point energy">Zero-point energy</a></li></ul></li>

<li><a href="Quantum_entanglement" title="Quantum entanglement">Entanglement</a></li>

<li><a href="Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a></li>

<li><a href="Wave_interference" title="Wave interference">Interference</a></li>

<li><a href="Quantum_decoherence" title="Quantum decoherence">Decoherence</a></li>

<li><a href="Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurement</a></li>

<li><a href="Quantum_nonlocality" title="Quantum nonlocality">Nonlocality</a></li>

<li><a href="Quantum_state" title="Quantum state">Quantum state</a></li>

<li><a href="Quantum_superposition" title="Quantum superposition">Superposition</a></li>

<li><a href="Quantum_tunnelling" title="Quantum tunnelling">Tunnelling</a></li>

<li><a href="Scattering#Theory" title="Scattering">Scattering theory</a></li>

<li><a href="Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry in quantum mechanics</a></li>

<li><a href="Uncertainty_principle" title="Uncertainty principle">Uncertainty</a></li>

<li><a href="Wave_function" title="Wave function">Wave function</a>

<ul><li><a href="Wave_function_collapse" title="Wave function collapse">Collapse</a></li>

<li><a href="Wave%E2%80%93particle_duality" title="Wave–particle duality">Wave–particle duality</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Formulations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Formulations</a></li>

<li><a href="Heisenberg_picture" title="Heisenberg picture">Heisenberg</a></li>

<li><a href="Interaction_picture" title="Interaction picture">Interaction</a></li>

<li><a href="Matrix_mechanics" title="Matrix mechanics">Matrix mechanics</a></li>

<li><a href="Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger</a></li>

<li><a href="Path_integral_formulation" title="Path integral formulation">Path integral formulation</a></li>

<li><a href="Phase-space_formulation" title="Phase-space formulation">Phase space</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Equations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon</a></li>

<li><a href="Dirac_equation" title="Dirac equation">Dirac</a></li>

<li><a href="Majorana_equation" title="Majorana equation">Majorana</a></li>

<li><a href="Rarita%E2%80%93Schwinger_equation" title="Rarita–Schwinger equation">Rarita–Schwinger</a></li>

<li><a href="Pauli_equation" title="Pauli equation">Pauli</a></li>

<li><a href="Rydberg_formula" title="Rydberg formula">Rydberg</a></li>

<li><a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Quantum_Bayesianism" title="Quantum Bayesianism">Bayesian</a></li>

<li><a href="Consciousness_causes_collapse" title="Consciousness causes collapse">Consciousness causes collapse</a></li>

<li><a href="Consistent_histories" title="Consistent histories">Consistent histories</a></li>

<li><a href="Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen</a></li>

<li><a href="De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">de Broglie–Bohm</a></li>

<li><a href="Ensemble_interpretation" title="Ensemble interpretation">Ensemble</a></li>

<li><a href="Hidden-variable_theory" title="Hidden-variable theory">Hidden-variable</a>

<ul><li><a href="Local_hidden-variable_theory" title="Local hidden-variable theory">Local</a>

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

<li><a href="Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds</a></li>

<li><a href="Objective-collapse_theory" title="Objective-collapse theory">Objective collapse</a></li>

<li><a href="Quantum_logic" title="Quantum logic">Quantum logic</a></li>

<li><a href="Relational_quantum_mechanics" title="Relational quantum mechanics">Relational</a></li>

<li><a href="Transactional_interpretation" title="Transactional interpretation">Transactional</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Experiments</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Davisson%E2%80%93Germer_experiment" title="Davisson–Germer experiment">Davisson–Germer</a></li>

<li><a href="Delayed-choice_quantum_eraser" title="Delayed-choice quantum eraser">Delayed-choice quantum eraser</a></li>

<li><a href="Double-slit_experiment" title="Double-slit experiment">Double-slit</a></li>

<li><a href="Franck%E2%80%93Hertz_experiment" title="Franck–Hertz experiment">Franck–Hertz</a></li>

<li><a href="Mach%E2%80%93Zehnder_interferometer" title="Mach–Zehnder interferometer">Mach–Zehnder interferometer</a></li>

<li><a href="Elitzur%E2%80%93Vaidman_bomb_tester" title="Elitzur–Vaidman bomb tester">Elitzur–Vaidman</a></li>

<li><a href="Popper's_experiment" title="Popper's experiment">Popper</a></li>

<li><a href="Quantum_eraser_experiment" title="Quantum eraser experiment">Quantum eraser</a></li>

<li><a href="Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach</a></li>

<li><a href="Wheeler's_delayed-choice_experiment" title="Wheeler's delayed-choice experiment">Wheeler's delayed choice</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Nanotechnology" title="Nanotechnology">Science</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Quantum_biology" title="Quantum biology">Quantum biology</a></li>

<li><a href="Quantum_chemistry" title="Quantum chemistry">Quantum chemistry</a></li>

<li><a href="Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li>

<li><a href="Quantum_cosmology" title="Quantum cosmology">Quantum cosmology</a></li>

<li><a href="Quantum_differential_calculus" title="Quantum differential calculus">Quantum differential calculus</a></li>

<li><a href="Quantum_dynamics" title="Quantum dynamics">Quantum dynamics</a></li>

<li><a href="Quantum_geometry" title="Quantum geometry">Quantum geometry</a></li>

<li><a href="Measurement_problem" title="Measurement problem">Quantum measurement problem</a></li>

<li><a href="Quantum_mind" title="Quantum mind">Quantum mind</a></li>

<li><a href="Quantum_stochastic_calculus" title="Quantum stochastic calculus">Quantum stochastic calculus</a></li>

<li><a href="Quantum_spacetime" title="Quantum spacetime">Quantum spacetime</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Quantum_engineering" title="Quantum engineering">Technology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Quantum_algorithm" title="Quantum algorithm">Quantum algorithms</a></li>

<li><a href="Quantum_amplifier" title="Quantum amplifier">Quantum amplifier</a></li>

<li><a href="Quantum_bus" title="Quantum bus">Quantum bus</a></li>

<li><a href="Quantum_cellular_automaton" title="Quantum cellular automaton">Quantum cellular automata</a>

<ul><li><a href="Quantum_finite_automaton" title="Quantum finite automaton">Quantum finite automata</a></li></ul></li>

<li><a href="Quantum_channel" title="Quantum channel">Quantum channel</a></li>

<li><a href="Quantum_circuit" title="Quantum circuit">Quantum circuit</a></li>

<li><a href="Quantum_complexity_theory" title="Quantum complexity theory">Quantum complexity theory</a></li>

<li><a href="Quantum_computing" title="Quantum computing">Quantum computing</a>

<ul><li><a href="Timeline_of_quantum_computing_and_communication" title="Timeline of quantum computing and communication">Timeline</a></li></ul></li>

<li><a href="Quantum_cryptography" title="Quantum cryptography">Quantum cryptography</a></li>

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

<li><a href="Quantum_error_correction" title="Quantum error correction">Quantum error correction</a></li>

<li><a href="Quantum_imaging" title="Quantum imaging">Quantum imaging</a></li>

<li><a href="Quantum_image_processing" title="Quantum image processing">Quantum image processing</a></li>

<li><a href="Quantum_information" title="Quantum information">Quantum information</a></li>

<li><a href="Quantum_key_distribution" title="Quantum key distribution">Quantum key distribution</a></li>

<li><a href="Quantum_logic" title="Quantum logic">Quantum logic</a></li>

<li><a href="Quantum_logic_gate" title="Quantum logic gate">Quantum logic gates</a></li>

<li><a href="Quantum_machine" title="Quantum machine">Quantum machine</a></li>

<li><a href="Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li>

<li><a href="Quantum_metamaterial" title="Quantum metamaterial">Quantum metamaterial</a></li>

<li><a href="Quantum_metrology" title="Quantum metrology">Quantum metrology</a></li>

<li><a href="Quantum_network" title="Quantum network">Quantum network</a></li>

<li><a href="Quantum_neural_network" title="Quantum neural network">Quantum neural network</a></li>

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

<li><a href="Quantum_programming" title="Quantum programming">Quantum programming</a></li>

<li><a href="Quantum_sensor" title="Quantum sensor">Quantum sensing</a></li>

<li><a href="Quantum_simulator" title="Quantum simulator">Quantum simulator</a></li>

<li><a href="Quantum_teleportation" title="Quantum teleportation">Quantum teleportation</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Extensions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Quantum_fluctuation" title="Quantum fluctuation">Quantum fluctuation</a></li>

<li><a href="Casimir_effect" title="Casimir effect">Casimir effect</a></li>

<li><a href="Quantum_statistical_mechanics" title="Quantum statistical mechanics">Quantum statistical mechanics</a></li>

<li><a href="Quantum_field_theory" title="Quantum field theory">Quantum field theory</a>

<ul><li><a href="History_of_quantum_field_theory" title="History of quantum field theory">History</a></li></ul></li>

<li><a href="Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li>

<li><a href="Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">Relativistic quantum mechanics</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Schr%C3%B6dinger's_cat" title="Schrödinger's cat">Schrödinger's cat</a>

<ul><li><a href="Schr%C3%B6dinger's_cat_in_popular_culture" title="Schrödinger's cat in popular culture">in popular culture</a></li></ul></li>

<li><a href="Wigner's_friend" title="Wigner's friend">Wigner's friend</a></li>

<li><a href="Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox" title="Einstein–Podolsky–Rosen paradox">EPR paradox</a></li>

<li><a href="Quantum_mysticism" title="Quantum mysticism">Quantum mysticism</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li> Category</li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

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