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<title>Rectangular potential barrier</title>

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Rectangular potential barrier

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In <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, the rectangular (or, at times, square) potential barrier is a standard one-dimensional problem that demonstrates the phenomena of <a href="Quantum_tunneling" class="mw-redirect" title="Quantum tunneling">wave-mechanical tunneling</a> (also called "quantum tunneling") and wave-mechanical reflection. The problem consists of solving the one-dimensional time-independent <a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a> for a particle encountering a rectangular <a href="Potential_energy" title="Potential energy">potential energy</a> barrier. It is usually assumed, as here, that a <a href="Free_particle" title="Free particle">free particle</a> impinges on the barrier from the left.

Although classically a particle behaving as a <a href="Point_mass" class="mw-redirect" title="Point mass">point mass</a> would be reflected if its energy is less than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{0}}">

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</math><img src="./7ae15ff9b845587dc4e1816f59c3fed0e71a132f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="{\displaystyle V_{0}}" loading="lazy">, a particle actually behaving as a matter wave has a non-zero probability of penetrating the barrier and continuing its travel as a wave on the other side. In classical wave-physics, this effect is known as <a href="Evanescent_wave_coupling" class="mw-redirect" title="Evanescent wave coupling">evanescent wave coupling</a>. The likelihood that the particle will pass through the barrier is given by the <a href="Transmission_coefficient" title="Transmission coefficient">transmission coefficient</a>, whereas the likelihood that it is reflected is given by the <a href="Reflection_coefficient" title="Reflection coefficient">reflection coefficient</a>. <a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger's wave-equation</a> allows these coefficients to be calculated.

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

The time-independent Schrödinger equation for the wave function <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>

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<annotation encoding="application/x-tex">{\displaystyle {\hat {H}}\psi (x)=\left[-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}+V(x)\right]\psi (x)=E\psi (x)}</annotation>

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

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

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</math><img src="./6bb06de5217295d7fbdbf68fb9c5309a513fc99e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.843ex;" alt="{\displaystyle {\hat {H}}}" loading="lazy"> is the <a href="Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar }">

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

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</math><img src="./de68de3a92517953436c93b5a76461d49160cc41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle \hbar }" loading="lazy"> is the (reduced)

<a href="Planck_constant" title="Planck constant">Planck constant</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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

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</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> is the <a href="Mass" title="Mass">mass</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

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<annotation encoding="application/x-tex">{\displaystyle V(x)=V_{0}[\Theta (x)-\Theta (x-a)]}</annotation>

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is the barrier potential with height <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{0}>0}">

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</math><img src="./8609b98cff53c5942c52ef2dbf8237c72e72adbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.67ex; height:2.509ex;" alt="{\displaystyle V_{0}>0}" loading="lazy"> and width <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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

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

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is the <a href="Heaviside_step_function" title="Heaviside step function">Heaviside step function</a>, i.e.,

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<annotation encoding="application/x-tex">{\displaystyle V(x)={\begin{cases}0&{\text{if }}x<0\\V_{0}&{\text{if }}0<x<a\\0&{\text{if }}a<x\end{cases}}}</annotation>

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The barrier is positioned between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}">

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

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</math><img src="./aaae23950e96a955ab5b07015a168fd931d4d82b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.658ex; height:1.676ex;" alt="{\displaystyle x=a}" loading="lazy">. The barrier can be shifted to any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\textstyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}\psi }</annotation>

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</math><img src="./a4737894fd1c2158ff9da44f736b78a120dfab38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:9.89ex; height:4.343ex;" alt="{\textstyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}\psi }" loading="lazy"> is the kinetic energy.

The barrier divides the space in three parts (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<0,0<x<a,x>a}">

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

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</math><img src="./e4d0087764149b33afa91392ba4f15dd80e84d1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.235ex; height:2.509ex;" alt="{\displaystyle x<0,0<x<a,x>a}" loading="lazy">). In any of these parts, the potential is constant, meaning that the particle is quasi-free, and the solution of the Schrödinger equation can be written as a <a href="Quantum_superposition" title="Quantum superposition">superposition</a> of left and right moving waves (see <a href="Free_particle" title="Free particle">free particle</a>). If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E>V_{0}}">

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<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}\psi _{L}(x)=A_{r}e^{ik_{0}x}+A_{l}e^{-ik_{0}x}&x<0\\\psi _{C}(x)=B_{r}e^{ik_{1}x}+B_{l}e^{-ik_{1}x}&0<x<a\\\psi _{R}(x)=C_{r}e^{ik_{0}x}+C_{l}e^{-ik_{0}x}&x>a\end{cases}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}\psi _{L}(x)=A_{r}e^{ik_{0}x}+A_{l}e^{-ik_{0}x}&x<0\\\psi _{C}(x)=B_{r}e^{ik_{1}x}+B_{l}e^{-ik_{1}x}&0<x<a\\\psi _{R}(x)=C_{r}e^{ik_{0}x}+C_{l}e^{-ik_{0}x}&x>a\end{cases}}}</annotation>

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where the <a href="Wave_number" class="mw-redirect" title="Wave number">wave numbers</a> are related to the energy via

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<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}k_{0}={\sqrt {2mE/\hbar ^{2}}}&x<0\quad {\text{or}}\quad x>a\\k_{1}={\sqrt {2m(E-V_{0})/\hbar ^{2}}}&0<x<a.\end{cases}}}</annotation>

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

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</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> denotes the direction of the velocity vector. Note that, if the energy of the particle is below the barrier height, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{1}}">

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</math><img src="./376315fd4983f01dada5ec2f7bebc48455b14a66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.265ex; height:2.509ex;" alt="{\displaystyle k_{1}}" loading="lazy"> becomes imaginary and the wave function is exponentially decaying within the barrier. Nevertheless, we keep the notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r/l}">

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The coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B,C}">

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</math><img src="./0ce2acf22b93dfbd22373336bd9c22dbd98a49d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.341ex; height:2.509ex;" alt="{\displaystyle A,B,C}" loading="lazy"> have to be found from the <a href="Boundary_condition" class="mw-redirect" title="Boundary condition">boundary conditions</a> of the wave function at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}">

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</math><img src="./aaae23950e96a955ab5b07015a168fd931d4d82b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.658ex; height:1.676ex;" alt="{\displaystyle x=a}" loading="lazy">. The wave function and its derivative have to be <a href="Continuous_function" title="Continuous function">continuous</a> everywhere, so

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\psi _{L}(0)&=\psi _{C}(0)\\\left.{\frac {d\psi _{L}}{dx}}\right|_{x=0}&=\left.{\frac {d\psi _{C}}{dx}}\right|_{x=0}\\\psi _{C}(a)&=\psi _{R}(a)\\\left.{\frac {d\psi _{C}}{dx}}\right|_{x=a}&=\left.{\frac {d\psi _{R}}{dx}}\right|_{x=a}.\end{aligned}}}</annotation>

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Inserting the wave functions, the boundary conditions give the following restrictions on the coefficients

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<annotation encoding="application/x-tex">{\displaystyle A_{r}+A_{l}=B_{r}+B_{l}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle ik_{0}(A_{r}-A_{l})=ik_{1}(B_{r}-B_{l})}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle B_{r}e^{iak_{1}}+B_{l}e^{-iak_{1}}=C_{r}e^{iak_{0}}+C_{l}e^{-iak_{0}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle ik_{1}\left(B_{r}e^{iak_{1}}-B_{l}e^{-iak_{1}}\right)=ik_{0}\left(C_{r}e^{iak_{0}}-C_{l}e^{-iak_{0}}\right).}</annotation>

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</math>

<div class="mw-heading mw-heading2"><h2 id="Transmission_and_reflection">Transmission and reflection</h2></div>

At this point, it is instructive to compare the situation to the classical case. In both cases, the particle behaves as a free particle outside of the barrier region. A classical particle with energy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

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

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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"> larger than the barrier height <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{0}}">

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

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

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</math><img src="./5a221b3eb6a427f497c86c68dc5c76fbc2e99b75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.284ex; height:2.509ex;" alt="{\displaystyle E<V_{0}}" loading="lazy"> incident on the barrier would always get reflected.

To study the quantum case, consider the following situation: a particle incident on the barrier from the left side (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{r}}">

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

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</math><img src="./bfc925d70377b6da729e56bcc6313a7825a19b52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.717ex; height:2.509ex;" alt="{\displaystyle A_{r}}" loading="lazy">). It may be reflected (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{l}}">

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

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</math><img src="./d5f77a3ce180d67724145786a553dc6cba2684b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.466ex; height:2.509ex;" alt="{\displaystyle A_{l}}" loading="lazy">) or transmitted (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{r}}">

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

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</math><img src="./feb55c7754335b4aa4e673b4f359a018fc062023.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.636ex; height:2.509ex;" alt="{\displaystyle C_{r}}" loading="lazy">).

To find the amplitudes for reflection and transmission for incidence from the left, we put in the above equations <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{r}=1}">

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

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</math><img src="./7bb6c3917523447fc16e0d94add01cdf4c42a3ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.978ex; height:2.509ex;" alt="{\displaystyle A_{r}=1}" loading="lazy"> (incoming particle), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{l}=r}">

<msub>

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

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</math><img src="./e2efb874921deacbabf4b7bc569183a77f1ccb72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.613ex; height:2.509ex;" alt="{\displaystyle A_{l}=r}" loading="lazy"> (reflection), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{l}=0}">

<msub>

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

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</math><img src="./93d297c68210e9438739f04bc023d74f6b57cec8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.645ex; height:2.509ex;" alt="{\displaystyle C_{l}=0}" loading="lazy"> (no incoming particle from the right), and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{r}=t}">

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

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</math><img src="./9762a5cc3b52513fc983fb37f7711b64a80f7896.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.574ex; height:2.509ex;" alt="{\displaystyle C_{r}=t}" loading="lazy"> (transmission). We then eliminate the coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{l},B_{r}}">

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<annotation encoding="application/x-tex">{\displaystyle B_{l},B_{r}}</annotation>

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</math><img src="./1a0e09ed72267bf386ba5ba4955f46117c36b667.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.258ex; height:2.509ex;" alt="{\displaystyle B_{l},B_{r}}" loading="lazy"> from the equation and solve for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

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

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</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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

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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">.

The result is:

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<annotation encoding="application/x-tex">{\displaystyle t={\frac {4k_{0}k_{1}e^{-ia(k_{0}-k_{1})}}{(k_{0}+k_{1})^{2}-e^{2iak_{1}}(k_{0}-k_{1})^{2}}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle r={\frac {(k_{0}^{2}-k_{1}^{2})\sin(ak_{1})}{2ik_{0}k_{1}\cos(ak_{1})+(k_{0}^{2}+k_{1}^{2})\sin(ak_{1})}}.}</annotation>

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Due to the mirror <a href="Symmetry" title="Symmetry">symmetry</a> of the model, the amplitudes for incidence from the right are the same as those from the left. Note that these expressions hold for any energy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E>0}">

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

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</math><img src="./8f04631216d2e15429be4a35d29eeca15ade4fe7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.037ex; height:2.176ex;" alt="{\displaystyle E>0}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\neq V_{0}}">

<msub>

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

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</math><img src="./5627fd67d25b2c04931860140187cb61231fa327.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.284ex; height:2.676ex;" alt="{\displaystyle E\neq V_{0}}" loading="lazy">. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=V_{0}}">

<msub>

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

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</math><img src="./8eecb56140c3e19d386e51d1efc854f7801c2575.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.284ex; height:2.509ex;" alt="{\displaystyle E=V_{0}}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{1}=0}">

<msub>

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

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</math><img src="./484a4883bb85f16fbeb4390c60e856b343676dbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.526ex; height:2.509ex;" alt="{\displaystyle k_{1}=0}" loading="lazy">, so there is a singularity in both of these expressions.

<div class="mw-heading mw-heading2"><h2 id="Analysis_of_the_obtained_expressions">Analysis of the obtained expressions</h2></div>

The surprising result is that for energies less than the barrier height, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E<V_{0}}">

<msub>

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

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<annotation encoding="application/x-tex">{\displaystyle T=|t|^{2}={\frac {1}{1+{\frac {V_{0}^{2}\sinh ^{2}(k_{1}a)}{4E(V_{0}-E)}}}}}</annotation>

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for the particle to be transmitted through the barrier, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle k_{1}={\sqrt {2m(V_{0}-E)/\hbar ^{2}}}}">

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<annotation encoding="application/x-tex">{\textstyle k_{1}={\sqrt {2m(V_{0}-E)/\hbar ^{2}}}}</annotation>

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</math><img src="./17b47bdb8367f0e130bf64e4561d6168d3e520d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.264ex; height:3.343ex;" alt="{\textstyle k_{1}={\sqrt {2m(V_{0}-E)/\hbar ^{2}}}}" loading="lazy">. This effect, which differs from the classical case, is called <a href="Quantum_tunneling" class="mw-redirect" title="Quantum tunneling">quantum tunneling</a>. The transmission is exponentially suppressed with the barrier width, which can be understood from the functional form of the wave function: Outside of the barrier it oscillates with wave vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{0}}">

<msub>

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

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</math><img src="./eafdfeca4ead3960fca92311910970eba7ec48a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.265ex; height:2.509ex;" alt="{\displaystyle k_{0}}" loading="lazy">, whereas within the barrier it is exponentially damped over a distance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/k_{1}}">

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

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</math><img src="./b70a1b068c84769b47d9835b747fce582f0ab5ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.59ex; height:2.843ex;" alt="{\displaystyle 1/k_{1}}" loading="lazy">. If the barrier is much wider than this decay length, the left and right part are virtually independent and tunneling as a consequence is suppressed.

In this case

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<annotation encoding="application/x-tex">{\displaystyle T=|t|^{2}={\frac {1}{1+{\frac {V_{0}^{2}\sin ^{2}(k_{1}a)}{4E(E-V_{0})}}}},}</annotation>

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where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle k_{1}={\sqrt {2m(E-V_{0})/\hbar ^{2}}}}">

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<annotation encoding="application/x-tex">{\textstyle k_{1}={\sqrt {2m(E-V_{0})/\hbar ^{2}}}}</annotation>

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</math><img src="./b8306242853e4e0076193cee94fe87ef88250a79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.264ex; height:3.343ex;" alt="{\textstyle k_{1}={\sqrt {2m(E-V_{0})/\hbar ^{2}}}}" loading="lazy">.

Equally surprising is that for energies larger than the barrier height, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E>V_{0}}">

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

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<annotation encoding="application/x-tex">{\displaystyle R=|r|^{2}=1-T.}</annotation>

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The transmission and reflection probabilities are in fact oscillating with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{1}a}">

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

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</math><img src="./bcaa5e7c8ffe9180170ed5a03fc256f2fb763478.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.495ex; height:2.509ex;" alt="{\displaystyle k_{1}a}" loading="lazy">. The classical result of perfect transmission without any reflection (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T=1}">

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

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</math><img src="./a6664a95bcd54fdd09a9178e106bd05b1c849856.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.897ex; height:2.176ex;" alt="{\displaystyle T=1}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=0}">

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

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</math><img src="./b7e38203eea12f5613721e2fc5e185b8eef9e800.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.025ex; height:2.176ex;" alt="{\displaystyle R=0}" loading="lazy">) is reproduced not only in the limit of high energy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\gg V_{0}}">

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

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</math><img src="./2fed04ab894efc164af1b8ff61f9e0855cc98047.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.799ex; height:2.509ex;" alt="{\displaystyle E\gg V_{0}}" loading="lazy"> but also when the energy and barrier width satisfy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{1}a=n\pi }">

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<annotation encoding="application/x-tex">{\displaystyle k_{1}a=n\pi }</annotation>

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</math><img src="./56e517bcd1829efdd7c923420514d632f8e76b84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.32ex; height:2.509ex;" alt="{\displaystyle k_{1}a=n\pi }" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1,2,\dots }">

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

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</math><img src="./ac3f39e31d25a3197ae7ee7d9daebac8feaf7644.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.609ex; height:2.509ex;" alt="{\displaystyle n=1,2,\dots }" loading="lazy"> (see peaks near <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E/V_{0}=1.2}">

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<annotation encoding="application/x-tex">{\displaystyle E/V_{0}=1.2}</annotation>

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</math><img src="./966e4e07fbe01c6aeae01242f14a282df139c40b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.418ex; height:2.843ex;" alt="{\displaystyle E/V_{0}=1.2}" loading="lazy"> and 1.8 in the above figure). Note that the probabilities and amplitudes as written are for any energy (above/below) the barrier height.

The transmission probability at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=V_{0}}">

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

</semantics>

</math><img src="./8eecb56140c3e19d386e51d1efc854f7801c2575.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.284ex; height:2.509ex;" alt="{\displaystyle E=V_{0}}" loading="lazy"> is <a href="#cite_note-1">[1]</a>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T={\frac {1}{1+ma^{2}V_{0}/2\hbar ^{2}}}.}</annotation>

</semantics>

</math>

This expression can be obtained by calculating the transmission coefficient from the constants stated <a class="mw-selflink-fragment" href="#E_=_V0">above</a> as for the other cases or by taking the limit of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

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

</semantics>

</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"> approaches <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math>

is defined, which is used in the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy">:

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msqrt>

</msqrt>

</mrow>

</mrow>

<msqrt>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {\sinh(v_{0}{\sqrt {1-x}})}{\sqrt {1-x}}}}</annotation>

</semantics>

</math>

In the last equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./60faad24775635f4722ccc438093dbbfe05f34ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.182ex; height:2.009ex;" alt="{\displaystyle v_{0}}" loading="lazy"> is defined as follows:

<msub>

</mrow>

</msub>

<msqrt>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

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</msqrt>

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle v_{0}={\sqrt {\frac {2mV_{0}a^{2}}{\hbar ^{2}}}}}</annotation>

</semantics>

</math>

These definitions can be inserted in the expression for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> which was obtained for the case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E<V_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

<mfrac>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mfrac>

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle T(x)={\frac {1}{1+{\frac {f(x)^{2}}{4x}}}}}</annotation>

</semantics>

</math>

Now, when calculating the <a href="Limit_of_a_function" title="Limit of a function">limit</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

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

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</mrow>

</munder>

</mrow>

</munder>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msqrt>

</msqrt>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</munder>

<mfrac>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msqrt>

</msqrt>

</mrow>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lim _{x\to 1}f(x)=\lim _{x\to 1}{\frac {\sinh(v_{0}{\sqrt {1-x}})}{(1-x)}}=\lim _{x\to 1}{\frac {{\frac {d}{dx}}\sinh(v_{0}{\sqrt {1-x}})}{{\frac {d}{dx}}{\sqrt {1-x}}}}=v_{0}\cosh(0)=v_{0}}</annotation>

</semantics>

</math><img src="./b8addc51731aae250e84b9ee772349478aba6d65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:75.826ex; height:8.176ex;" alt="{\displaystyle \lim _{x\to 1}f(x)=\lim _{x\to 1}{\frac {\sinh(v_{0}{\sqrt {1-x}})}{(1-x)}}=\lim _{x\to 1}{\frac {{\frac {d}{dx}}\sinh(v_{0}{\sqrt {1-x}})}{{\frac {d}{dx}}{\sqrt {1-x}}}}=v_{0}\cosh(0)=v_{0}}" loading="lazy">

also the limit of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(x)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1171c29b4c2b5575f50a4ea9313f90448a2cbe05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.775ex; height:2.843ex;" alt="{\displaystyle T(x)}" loading="lazy"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</mrow>

</munder>

</mrow>

</munder>

<mfrac>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<mfrac>

</mrow>

</mrow>

</msubsup>

</mfrac>

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</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lim _{x\to 1}T(x)=\lim _{x\to 1}{\frac {1}{1+{\frac {f(x)^{2}}{4x}}}}={\frac {1}{1+{\frac {v_{0}^{2}}{4}}}}}</annotation>

</semantics>

</math><img src="./c1a9d6391661ad52864fd1835f5c9708f0012d75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:35.491ex; height:7.843ex;" alt="{\displaystyle \lim _{x\to 1}T(x)=\lim _{x\to 1}{\frac {1}{1+{\frac {f(x)^{2}}{4x}}}}={\frac {1}{1+{\frac {v_{0}^{2}}{4}}}}}" loading="lazy">

By plugging in the above expression for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

<div class="mw-heading mw-heading2"><h2 id="Remarks_and_applications">Remarks and applications</h2></div>

The calculation presented above may at first seem unrealistic and hardly useful. However it has proved to be a suitable model for a variety of real-life systems. One such example are interfaces between two <a href="Electrical_conductivity" class="mw-redirect" title="Electrical conductivity">conducting</a> materials. In the bulk of the materials, the motion of the electrons is quasi-free and can be described by the kinetic term in the above Hamiltonian with an <a href="Effective_mass_(solid-state_physics)" title="Effective mass (solid-state physics)">effective mass</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy">. Often the surfaces of such materials are covered with oxide layers or are not ideal for other reasons. This thin, non-conducting layer may then be modeled by a barrier potential as above. Electrons may then tunnel from one material to the other giving rise to a current.

The operation of a <a href="Scanning_tunneling_microscope" title="Scanning tunneling microscope">scanning tunneling microscope</a> (STM) relies on this tunneling effect. In that case, the barrier is due to the gap between the tip of the STM and the underlying object. Since the tunnel current depends exponentially on the barrier width, this device is extremely sensitive to height variations on the examined sample.

The above model is one-dimensional, while space is three-dimensional. One should solve the Schrödinger equation in three dimensions. On the other hand, many systems only change along one coordinate direction and are translationally invariant along the others; they are <a href="Separation_of_variables" title="Separation of variables">separable</a>. The Schrödinger equation may then be reduced to the case considered here by an ansatz for the wave function of the type: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi (x,y,z)=\psi (x)\phi (y,z)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Psi (x,y,z)=\psi (x)\phi (y,z)}</annotation>

</semantics>

</math><img src="./0701d782899936a26eb15c947f584b6baaeb4635.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.482ex; height:2.843ex;" alt="{\displaystyle \Psi (x,y,z)=\psi (x)\phi (y,z)}" loading="lazy">.

For another, related model of a barrier, see <a href="Delta_potential_barrier_(QM)" class="mw-redirect" title="Delta potential barrier (QM)">Delta potential barrier (QM)</a>, which can be regarded as a special case of the finite potential barrier. All results from this article immediately apply to the delta potential barrier by taking the limits <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{0}\to \infty ,\;a\to 0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V_{0}\to \infty ,\;a\to 0}</annotation>

</semantics>

</math><img src="./540a84d37d43488c6fcd729964977c5aed84ac8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.033ex; height:2.509ex;" alt="{\displaystyle V_{0}\to \infty ,\;a\to 0}" loading="lazy"> while keeping <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{0}a=\lambda }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e35edbe39c21423ffb89236eb2881f25b7b6214f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.093ex; height:2.509ex;" alt="{\displaystyle V_{0}a=\lambda }" loading="lazy"> constant.

<ul><li><a href="Morse/Long-range_potential" title="Morse/Long-range potential">Morse/Long-range potential</a></li>

<li><a href="Step_potential" title="Step potential">Step potential</a></li>

<li><a href="Finite_potential_well" title="Finite potential well">Finite potential well</a></li></ul>

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

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</style><cite id="CITEREFMcQuarrieSimon1997" class="citation book cs1">McQuarrie DA, Simon JD (1997). Physical Chemistry - A molecular Approach (1st ed.). University Science Books. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0935702996</bdi>.</cite>

</li>

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

<ul><li><cite id="CITEREFGriffiths,_David_J.2004" class="citation book cs1">Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-13-111892-7</bdi>.</cite></li>

<li><cite id="CITEREFCohen-TannoudjiDiuLaloë1996" class="citation book cs1">Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck; et al. (1996). <a rel="nofollow" class="external text" href="https://archive.org/details/cohen-tannoudji-diu-and-laloe-quantum-mechanics-vol.-i-ii-and-iii-2nd-ed.">Quantum mechanics</a>. transl. from the French by Susan Reid Hemley. Wiley-Interscience: Wiley. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/quantummechanics00cohe/page/n331">231</a>–233. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-56952-7</bdi>.</cite></li></ul>

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