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WKB approximation

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</style><div role="note" class="hatnote navigation-not-searchable">"WKB" and "WKBJ" redirect here. For other uses, see <a href="WKB_(disambiguation)" class="mw-disambig" title="WKB (disambiguation)">WKB (disambiguation)</a>. For the television station in Live Oak, Florida, see <a href="WKBJ-LD" class="mw-redirect" title="WKBJ-LD">WKBJ-LD</a>.</div>

In <a href="Mathematical_physics" title="Mathematical physics">mathematical physics</a>, the WKB approximation or WKB method is a technique for finding approximate solutions to <a href="Linear_differential_equation" title="Linear differential equation">linear differential equations</a> with spatially varying coefficients. It is typically used for a <a href="Semiclassical_physics" title="Semiclassical physics">semiclassical</a> calculation in <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> in which the <a href="Wave_function" title="Wave function">wave function</a> is recast as an exponential function, semiclassically expanded, and then either the amplitude or the phase is taken to be changing slowly.

The name is an initialism for Wentzel–Kramers–Brillouin. It is also known as the LG or Liouville–Green method. Other often-used letter combinations include JWKB and WKBJ, where the "J" stands for Jeffreys.

<div class="mw-heading mw-heading2"><h2 id="Brief_history">Brief history</h2></div>

This method is named after physicists <a href="Gregor_Wentzel" title="Gregor Wentzel">Gregor Wentzel</a>, <a href="Hendrik_Anthony_Kramers" class="mw-redirect" title="Hendrik Anthony Kramers">Hendrik Anthony Kramers</a>, and <a href="L%C3%A9on_Brillouin" title="Léon Brillouin">Léon Brillouin</a>, who all developed it in 1926.<a href="#cite_note-Wentzel-1926-1">[1]</a><a href="#cite_note-Kramers-1926-2">[2]</a><a href="#cite_note-Brillouin-1926-3">[3]</a><a href="#cite_note-4">[4]</a> In 1923,<a href="#cite_note-Jefferys-1924-5">[5]</a> mathematician <a href="Harold_Jeffreys" title="Harold Jeffreys">Harold Jeffreys</a> had developed a general method of approximating solutions to linear, second-order differential equations, a class that includes the <a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a>. The Schrödinger equation itself was not developed until two years later, and Wentzel, Kramers, and Brillouin were apparently unaware of this earlier work, so Jeffreys is often neglected credit. Early texts in quantum mechanics contain any number of combinations of their initials, including WBK, BWK, WKBJ, JWKB and BWKJ. An authoritative discussion and critical survey has been given by Robert B. Dingle.<a href="#cite_note-6">[6]</a>

Earlier appearances of essentially equivalent methods are: <a href="Francesco_Carlini" title="Francesco Carlini">Francesco Carlini</a> in 1817,<a href="#cite_note-Carlini-1817-7">[7]</a> <a href="Joseph_Liouville" title="Joseph Liouville">Joseph Liouville</a> in 1837,<a href="#cite_note-Liouville-8">[8]</a> <a href="George_Green_(mathematician)" title="George Green (mathematician)">George Green</a> in 1837,<a href="#cite_note-Green-1837-9">[9]</a> <a href="Lord_Rayleigh" class="mw-redirect" title="Lord Rayleigh">Lord Rayleigh</a> in 1912<a href="#cite_note-Rayleigh-1912-10">[10]</a> and <a href="Richard_Gans" title="Richard Gans">Richard Gans</a> in 1915.<a href="#cite_note-Gans-1915-11">[11]</a> Liouville and Green may be said to have founded the method in 1837, and it is also commonly referred to as the Liouville–Green or LG method.<a href="#cite_note-12">[12]</a><a href="#cite_note-13">[13]</a>

The important contribution of Jeffreys, Wentzel, Kramers, and Brillouin to the method was the inclusion of the treatment of <a href="Stationary_point" title="Stationary point">turning points</a>, connecting the <a href="Evanescent_wave" class="mw-redirect" title="Evanescent wave">evanescent</a> and <a href="Oscillation" title="Oscillation">oscillatory</a> solutions at either side of the turning point. For example, this may occur in the Schrödinger equation, due to a <a href="Potential_energy" title="Potential energy">potential energy</a> hill.

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

Generally, WKB theory is a method for approximating the solution of a differential equation whose highest derivative is multiplied by a small parameter ε. The method of approximation is as follows.

For a differential equation

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon {\frac {d^{n}y}{dx^{n}}}+a(x){\frac {d^{n-1}y}{dx^{n-1}}}+\cdots +k(x){\frac {dy}{dx}}+m(x)y=0,}</annotation>

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assume a solution of the form of an <a href="Asymptotic_series" class="mw-redirect" title="Asymptotic series">asymptotic series</a> expansion

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<annotation encoding="application/x-tex">{\displaystyle y(x)\sim \exp \left[{\frac {1}{\delta }}\sum _{n=0}^{\infty }\delta ^{n}S_{n}(x)\right]}</annotation>

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in the limit δ → 0. The asymptotic scaling of δ in terms of ε will be determined by the equation – see the example below.

Substituting the above <a href="Ansatz" title="Ansatz">ansatz</a> into the differential equation and cancelling out the exponential terms allows one to solve for an arbitrary number of terms Sn(x) in the expansion.

WKB theory is a special case of <a href="Multiple_scale_analysis" class="mw-redirect" title="Multiple scale analysis">multiple scale analysis</a>.<a href="#cite_note-14">[14]</a><a href="#cite_note-15">[15]</a><a href="#cite_note-:0-16">[16]</a>

<div class="mw-heading mw-heading2"><h2 id="An_example">An example</h2></div>

This example comes from the text of <a href="Carl_M._Bender" title="Carl M. Bender">Carl M. Bender</a> and <a href="Steven_Orszag" title="Steven Orszag">Steven Orszag</a>.<a href="#cite_note-:0-16">[16]</a> Consider the second-order homogeneous linear differential equation

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<annotation encoding="application/x-tex">{\displaystyle \epsilon ^{2}{\frac {d^{2}y}{dx^{2}}}=Q(x)y,}</annotation>

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

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

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</math><img src="./0a619b140eb87f5fb745df8780084e6982ecc0ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.238ex; height:2.843ex;" alt="{\displaystyle Q(x)\neq 0}" loading="lazy">. Substituting

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<annotation encoding="application/x-tex">{\displaystyle y(x)=\exp \left[{\frac {1}{\delta }}\sum _{n=0}^{\infty }\delta ^{n}S_{n}(x)\right]}</annotation>

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results in the equation

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<annotation encoding="application/x-tex">{\displaystyle \epsilon ^{2}\left[{\frac {1}{\delta ^{2}}}\left(\sum _{n=0}^{\infty }\delta ^{n}S_{n}^{\prime }\right)^{2}+{\frac {1}{\delta }}\sum _{n=0}^{\infty }\delta ^{n}S_{n}^{\prime \prime }\right]=Q(x).}</annotation>

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To <a href="Leading-order" class="mw-redirect" title="Leading-order">leading order</a> in ϵ (assuming, for the moment, the series will be asymptotically consistent), the above can be approximated as

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\epsilon ^{2}}{\delta ^{2}}}{S_{0}^{\prime }}^{2}+{\frac {2\epsilon ^{2}}{\delta }}S_{0}^{\prime }S_{1}^{\prime }+{\frac {\epsilon ^{2}}{\delta }}S_{0}^{\prime \prime }=Q(x).}</annotation>

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In the limit δ → 0, the <a href="Method_of_dominant_balance" title="Method of dominant balance">dominant balance</a> is given by

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\epsilon ^{2}}{\delta ^{2}}}{S_{0}^{\prime }}^{2}\sim Q(x).}</annotation>

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So δ is proportional to ϵ. Setting them equal and comparing powers yields

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<annotation encoding="application/x-tex">{\displaystyle \epsilon ^{0}:\quad {S_{0}^{\prime }}^{2}=Q(x),}</annotation>

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which can be recognized as the <a href="Eikonal_equation" title="Eikonal equation">eikonal equation</a>, with solution

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<annotation encoding="application/x-tex">{\displaystyle S_{0}(x)=\pm \int _{x_{0}}^{x}{\sqrt {Q(x')}}\,dx'.}</annotation>

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Considering first-order powers of ϵ fixes

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<annotation encoding="application/x-tex">{\displaystyle \epsilon ^{1}:\quad 2S_{0}^{\prime }S_{1}^{\prime }+S_{0}^{\prime \prime }=0.}</annotation>

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This has the solution

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<annotation encoding="application/x-tex">{\displaystyle S_{1}(x)=-{\frac {1}{4}}\ln Q(x)+k_{1},}</annotation>

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where k1 is an arbitrary constant.

We now have a pair of approximations to the system (a pair, because S0 can take two signs); the first-order WKB-approximation will be a linear combination of the two:

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<annotation encoding="application/x-tex">{\displaystyle y(x)\approx c_{1}Q^{-{\frac {1}{4}}}(x)\exp \left[{\frac {1}{\epsilon }}\int _{x_{0}}^{x}{\sqrt {Q(t)}}\,dt\right]+c_{2}Q^{-{\frac {1}{4}}}(x)\exp \left[-{\frac {1}{\epsilon }}\int _{x_{0}}^{x}{\sqrt {Q(t)}}\,dt\right].}</annotation>

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Higher-order terms can be obtained by looking at equations for higher powers of δ. Explicitly,

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<annotation encoding="application/x-tex">{\displaystyle 2S_{0}^{\prime }S_{n}^{\prime }+S_{n-1}^{\prime \prime }+\sum _{j=1}^{n-1}S_{j}^{\prime }S_{n-j}^{\prime }=0}</annotation>

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for n ≥ 2.

<div class="mw-heading mw-heading3"><h3 id="Precision_of_the_asymptotic_series">Precision of the asymptotic series</h3></div>

The asymptotic series for y(x) is usually a <a href="Divergent_series" title="Divergent series">divergent series</a>, whose general term δn Sn(x) starts to increase after a certain value n = nmax. Therefore, the smallest error achieved by the WKB method is at best of the order of the last included term.

For the equation

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<annotation encoding="application/x-tex">{\displaystyle \epsilon ^{2}{\frac {d^{2}y}{dx^{2}}}=Q(x)y,}</annotation>

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with Q(x) <0 an <a href="Analytic_function" title="Analytic function">analytic function</a>, the value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{\max }}">

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</math><img src="./758934b527a77b9f061766d0862104406f8f936f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.686ex; height:2.009ex;" alt="{\displaystyle n_{\max }}" loading="lazy"> and the magnitude of the last term can be estimated as follows:<a href="#cite_note-17">[17]</a>

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<annotation encoding="application/x-tex">{\displaystyle n_{\max }\approx 2\epsilon ^{-1}\left|\int _{x_{0}}^{x_{\ast }}{\sqrt {-Q(z)}}\,dz\right|,}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \delta ^{n_{\max }}S_{n_{\max }}(x_{0})\approx {\sqrt {\frac {2\pi }{n_{\max }}}}\exp[-n_{\max }],}</annotation>

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

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The number nmax can be interpreted as the number of oscillations between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

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If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon ^{-1}Q(x)}">

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<annotation encoding="application/x-tex">{\displaystyle \epsilon ^{-1}Q(x)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \epsilon \left|{\frac {dQ}{dx}}\right|\ll Q^{2},^{{\text{[might be }}Q^{3/2}{\text{?]}}}}</annotation>

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the number nmax will be large, and the minimum error of the asymptotic series will be exponentially small.

<div class="mw-heading mw-heading2"><h2 id="Application_in_non-relativistic_quantum_mechanics">Application in non-relativistic quantum mechanics</h2></div>

The above example may be applied specifically to the one-dimensional, time-independent <a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a>,

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<annotation encoding="application/x-tex">{\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}\Psi (x)+V(x)\Psi (x)=E\Psi (x),}</annotation>

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which can be rewritten as

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}}{dx^{2}}}\Psi (x)={\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)\Psi (x).}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading3"><h3 id="Approximation_away_from_the_turning_points">Approximation away from the turning points</h3></div>

The wavefunction can be rewritten as the exponential of another function S (closely related to the <a href="Action_(physics)" title="Action (physics)">action</a>), which could be complex,

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle \Psi (\mathbf {x} )=e^{iS(\mathbf {x} ) \over \hbar },}</annotation>

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</math>

so that its substitution in Schrödinger's equation gives:

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<annotation encoding="application/x-tex">{\displaystyle i\hbar \nabla ^{2}S(\mathbf {x} )-(\nabla S(\mathbf {x} ))^{2}=2m\left(V(\mathbf {x} )-E\right),}</annotation>

</semantics>

</math>

Next, the semiclassical approximation is used. This means that each function is expanded as a <a href="Power_series" title="Power series">power series</a> in ħ.

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<annotation encoding="application/x-tex">{\displaystyle S=S_{0}+\hbar S_{1}+\hbar ^{2}S_{2}+\cdots }</annotation>

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</math>

Substituting in the equation, and only retaining terms up to first order in ℏ, we get:

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<annotation encoding="application/x-tex">{\displaystyle (\nabla S_{0}+\hbar \nabla S_{1})^{2}-i\hbar (\nabla ^{2}S_{0})=2m(E-V(\mathbf {x} ))}</annotation>

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</math>

which gives the following two relations:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(\nabla S_{0})^{2}=2m(E-V(\mathbf {x} ))=(p(\mathbf {x} ))^{2}\\2\nabla S_{0}\cdot \nabla S_{1}-i\nabla ^{2}S_{0}=0\end{aligned}}}</annotation>

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</math>

which can be solved for 1D systems, first equation resulting in:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{0}(x)=\pm \int {\sqrt {2m\left(E-V(x)\right)}}\,dx=\pm \int p(x)\,dx}">

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<annotation encoding="application/x-tex">{\displaystyle S_{0}(x)=\pm \int {\sqrt {2m\left(E-V(x)\right)}}\,dx=\pm \int p(x)\,dx}</annotation>

</semantics>

</math>and the second equation computed for the possible values of the above, is generally expressed as:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi (x)\approx C_{+}{\frac {e^{+{\frac {i}{\hbar }}\int p(x)\,dx}}{\sqrt {|p(x)|}}}+C_{-}{\frac {e^{-{\frac {i}{\hbar }}\int p(x)\,dx}}{\sqrt {|p(x)|}}}}">

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<annotation encoding="application/x-tex">{\displaystyle \Psi (x)\approx C_{+}{\frac {e^{+{\frac {i}{\hbar }}\int p(x)\,dx}}{\sqrt {|p(x)|}}}+C_{-}{\frac {e^{-{\frac {i}{\hbar }}\int p(x)\,dx}}{\sqrt {|p(x)|}}}}</annotation>

</semantics>

</math>

Thus, the resulting wavefunction in first order WKB approximation is presented as,<a href="#cite_note-18">[18]</a><a href="#cite_note-:1-19">[19]</a>

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<annotation encoding="application/x-tex">{\displaystyle \Psi (x)\approx {\frac {C_{+}e^{+{\frac {i}{\hbar }}\int {\sqrt {2m\left(E-V(x)\right)}}\,dx}+C_{-}e^{-{\frac {i}{\hbar }}\int {\sqrt {2m\left(E-V(x)\right)}}\,dx}}{\sqrt[{4}]{2m\mid E-V(x)\mid }}}}</annotation>

</semantics>

</math><img src="./9a790b467b5680b7ee8ffadaf2063a95232ec713.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:56.765ex; height:8.176ex;" alt="{\displaystyle \Psi (x)\approx {\frac {C_{+}e^{+{\frac {i}{\hbar }}\int {\sqrt {2m\left(E-V(x)\right)}}\,dx}+C_{-}e^{-{\frac {i}{\hbar }}\int {\sqrt {2m\left(E-V(x)\right)}}\,dx}}{\sqrt[{4}]{2m\mid E-V(x)\mid }}}}" loading="lazy">

</div>

In the classically allowed region, namely the region where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(x)<E}">

</mstyle>

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

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</math><img src="./953662117dd97b4a293ca94fb4255181747d441f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.8ex; height:2.843ex;" alt="{\displaystyle V(x)<E}" loading="lazy"> the integrand in the exponent is imaginary and the approximate wave function is oscillatory. In the classically forbidden region <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(x)>E}">

</mstyle>

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

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</math><img src="./9d581f1a5904b320d627b8ac0cda82780e0eb940.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.8ex; height:2.843ex;" alt="{\displaystyle V(x)>E}" loading="lazy">, the solutions are growing or decaying. It is evident in the denominator that both of these approximate solutions become singular near the classical turning points, where E = V(x), and cannot be valid. (The turning points are the points where the classical particle changes direction.)

Hence, when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E>V(x)}">

</mstyle>

</mrow>

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

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</math><img src="./84c685d488512e9cb6f971a21422e9084ed99a19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.8ex; height:2.843ex;" alt="{\displaystyle E>V(x)}" loading="lazy">, the wavefunction can be chosen to be expressed as:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi (x')\approx C{\frac {\cos {({\frac {1}{\hbar }}\int |p(x)|\,dx}+\alpha )}{\sqrt {|p(x)|}}}+D{\frac {\sin {(-{\frac {1}{\hbar }}\int |p(x)|\,dx}+\alpha )}{\sqrt {|p(x)|}}}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \Psi (x')\approx C{\frac {\cos {({\frac {1}{\hbar }}\int |p(x)|\,dx}+\alpha )}{\sqrt {|p(x)|}}}+D{\frac {\sin {(-{\frac {1}{\hbar }}\int |p(x)|\,dx}+\alpha )}{\sqrt {|p(x)|}}}}</annotation>

</semantics>

</math>and for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(x)>E}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./9d581f1a5904b320d627b8ac0cda82780e0eb940.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.8ex; height:2.843ex;" alt="{\displaystyle V(x)>E}" loading="lazy">,<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi (x')\approx {\frac {C_{+}e^{+{\frac {i}{\hbar }}\int |p(x)|\,dx}}{\sqrt {|p(x)|}}}+{\frac {C_{-}e^{-{\frac {i}{\hbar }}\int |p(x)|\,dx}}{\sqrt {|p(x)|}}}.}">

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Psi (x')\approx {\frac {C_{+}e^{+{\frac {i}{\hbar }}\int |p(x)|\,dx}}{\sqrt {|p(x)|}}}+{\frac {C_{-}e^{-{\frac {i}{\hbar }}\int |p(x)|\,dx}}{\sqrt {|p(x)|}}}.}</annotation>

</semantics>

</math>The integration in this solution is computed between the classical turning point and the arbitrary position x'.

<div class="mw-heading mw-heading3"><h3 id="Validity_of_WKB_solutions">Validity of WKB solutions</h3></div>

From the condition:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (S_{0}'(x))^{2}-(p(x))^{2}+\hbar (2S_{0}'(x)S_{1}'(x)-iS_{0}''(x))=0}">

</mrow>

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<msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle (S_{0}'(x))^{2}-(p(x))^{2}+\hbar (2S_{0}'(x)S_{1}'(x)-iS_{0}''(x))=0}</annotation>

</semantics>

</math>

It follows that: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \hbar \mid 2S_{0}'(x)S_{1}'(x)\mid +\hbar \mid iS_{0}''(x)\mid \ll \mid (S_{0}'(x))^{2}\mid +\mid (p(x))^{2}\mid }">

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<annotation encoding="application/x-tex">{\textstyle \hbar \mid 2S_{0}'(x)S_{1}'(x)\mid +\hbar \mid iS_{0}''(x)\mid \ll \mid (S_{0}'(x))^{2}\mid +\mid (p(x))^{2}\mid }</annotation>

</semantics>

</math><img src="./f530b6bb0cadee1e9fdaa2ccd83bbad2c412214e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:56.122ex; height:3.176ex;" alt="{\textstyle \hbar \mid 2S_{0}'(x)S_{1}'(x)\mid +\hbar \mid iS_{0}''(x)\mid \ll \mid (S_{0}'(x))^{2}\mid +\mid (p(x))^{2}\mid }" loading="lazy">

For which the following two inequalities are equivalent since the terms in either side are equivalent, as used in the WKB approximation:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\hbar \mid S_{0}''(x)\mid \ll \mid (S_{0}'(x))^{2}\mid \\2\hbar \mid S_{0}'S_{1}'\mid \ll \mid (p'(x))^{2}\mid \end{aligned}}}</annotation>

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The first inequality can be used to show the following:

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<msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\hbar \mid S_{0}''(x)\mid \ll \mid (p(x))\mid ^{2}\\{\frac {1}{2}}{\frac {\hbar }{|p(x)|}}\left|{\frac {dp^{2}}{dx}}\right|\ll |p(x)|^{2}\\\lambda \left|{\frac {dV}{dx}}\right|\ll {\frac {|p|^{2}}{m}}\\\end{aligned}}}</annotation>

</semantics>

</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle |S_{0}'(x)|=|p(x)|}">

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<annotation encoding="application/x-tex">{\textstyle |S_{0}'(x)|=|p(x)|}</annotation>

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</math><img src="./e80d67b6d1ca0fdb21d46180578fa73473ee0a67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.612ex; height:3.009ex;" alt="{\textstyle |S_{0}'(x)|=|p(x)|}" loading="lazy"> is used and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \lambda (x)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./84bc2f11dc18666e8cdf53b3be52c11daab4776c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.494ex; height:2.843ex;" alt="{\textstyle \lambda (x)}" loading="lazy"> is the local <a href="De_Broglie_waves" class="mw-redirect" title="De Broglie waves">de Broglie wavelength</a> of the wavefunction. The inequality implies that the variation of potential is assumed to be slowly varying.<a href="#cite_note-:1-19">[19]</a><a href="#cite_note-:2-20">[20]</a> This condition can also be restated as the fractional change of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle E-V(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle E-V(x)}</annotation>

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</math><img src="./c5798374c99783fa0ec95afe8bb5497f78f1cdae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.542ex; height:2.843ex;" alt="{\textstyle E-V(x)}" loading="lazy"> or that of the momentum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle p(x)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./416ff23ae3a92d9ccc8ae3ddf87f5f746509d618.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.398ex; height:2.843ex;" alt="{\textstyle p(x)}" loading="lazy">, over the wavelength <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \lambda }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f801b46dadd4b4d0ba4c6592b232053bd79e080b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\textstyle \lambda }" loading="lazy">, being much smaller than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6706df9ed9f240d1a94545fb4e522bda168fe8fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\textstyle 1}" loading="lazy">.<a href="#cite_note-21">[21]</a>

Similarly it can be shown that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \lambda (x)}">

</mstyle>

</mrow>

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

</semantics>

<mrow>

<mfrac>

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle \left|{\frac {d\lambda }{dx}}\right|\ll 1}</annotation>

</semantics>

</math>which implies that the <a href="De_Broglie_waves" class="mw-redirect" title="De Broglie waves">de Broglie wavelength</a> of the particle is slowly varying.<a href="#cite_note-:2-20">[20]</a>

<div class="mw-heading mw-heading3"><h3 id="Behavior_near_the_turning_points">Behavior near the turning points</h3></div>

We now consider the behavior of the wave function near the turning points. For this, we need a different method. Near the first turning points, x1, the term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)}</annotation>

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</math><img src="./263107bf9d8714b3f9709257caa6376ac638f193.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.778ex; height:5.509ex;" alt="{\displaystyle {\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)}" loading="lazy"> can be expanded in a power series,

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)=U_{1}\cdot (x-x_{1})+U_{2}\cdot (x-x_{1})^{2}+\cdots \;.}</annotation>

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</math>

To first order, one finds

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}}{dx^{2}}}\Psi (x)=U_{1}\cdot (x-x_{1})\cdot \Psi (x).}</annotation>

</semantics>

</math>

This differential equation is known as the <a href="Airy_equation" class="mw-redirect" title="Airy equation">Airy equation</a>, and the solution may be written in terms of <a href="Airy_function" title="Airy function">Airy functions</a>,<a href="#cite_note-22">[22]</a>

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<annotation encoding="application/x-tex">{\displaystyle \Psi (x)=C_{A}\operatorname {Ai} \left({\sqrt[{3}]{U_{1}}}\cdot (x-x_{1})\right)+C_{B}\operatorname {Bi} \left({\sqrt[{3}]{U_{1}}}\cdot (x-x_{1})\right)=C_{A}\operatorname {Ai} \left(u\right)+C_{B}\operatorname {Bi} \left(u\right).}</annotation>

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Although for any fixed value of <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">, the wave function is bounded near the turning points, the wave function will be peaked there, as can be seen in the images above. As <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar }">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\hbar }}\int p(x)dx={\sqrt {U_{1}}}\int {\sqrt {x-a}}\,dx={\frac {2}{3}}({\sqrt[{3}]{U_{1}}}(x-a))^{\frac {3}{2}}={\frac {2}{3}}u^{\frac {3}{2}}}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Connection_conditions">Connection conditions</h3></div>

It now remains to construct a global (approximate) solution to the Schrödinger equation. For the wave function to be square-integrable, we must take only the exponentially decaying solution in the two classically forbidden regions. These must then "connect" properly through the turning points to the classically allowed region. For most values of E, this matching procedure will not work: The function obtained by connecting the solution near <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +\infty }">

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

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</math><img src="./bddbb0e4420a7e744cf71bd71216e11b0bf88831.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle +\infty }" loading="lazy"> to the classically allowed region will not agree with the function obtained by connecting the solution near <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }">

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

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</math><img src="./ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }" loading="lazy"> to the classically allowed region. The requirement that the two functions agree imposes a condition on the energy E, which will give an approximation to the exact quantum energy levels.The wavefunction's coefficients can be calculated for a simple problem shown in the figure. Let the first turning point, where the potential is decreasing over x, occur at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=x_{1}}">

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

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</math><img src="./2bdf4f9973d3d0c15948ee465d462b13151563d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.812ex; height:2.009ex;" alt="{\displaystyle x=x_{1}}" loading="lazy"> and the second turning point, where potential is increasing over x, occur at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=x_{2}}">

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

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<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\Psi _{V>E}(x)\approx A{\frac {e^{{\frac {2}{3}}u^{\frac {3}{2}}}}{\sqrt[{4}]{u}}}+B{\frac {e^{-{\frac {2}{3}}u^{\frac {3}{2}}}}{\sqrt[{4}]{u}}}\\\Psi _{E>V}(x)\approx C{\frac {\cos {({\frac {2}{3}}u^{\frac {3}{2}}-\alpha )}}{\sqrt[{4}]{u}}}+D{\frac {\sin {({\frac {2}{3}}u^{\frac {3}{2}}-\alpha )}}{\sqrt[{4}]{u}}}\\\end{aligned}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Psi _{V>E}(x)\approx A{\frac {e^{{\frac {2}{3}}u^{\frac {3}{2}}}}{\sqrt[{4}]{u}}}+B{\frac {e^{-{\frac {2}{3}}u^{\frac {3}{2}}}}{\sqrt[{4}]{u}}}\\\Psi _{E>V}(x)\approx C{\frac {\cos {({\frac {2}{3}}u^{\frac {3}{2}}-\alpha )}}{\sqrt[{4}]{u}}}+D{\frac {\sin {({\frac {2}{3}}u^{\frac {3}{2}}-\alpha )}}{\sqrt[{4}]{u}}}\\\end{aligned}}}</annotation>

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</math>

<div class="mw-heading mw-heading4"><h4 id="First_classical_turning_point">First classical turning point</h4></div>

For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{1}<0}">

<msub>

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

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</math><img src="./96c6c296a50a2ec4165e39cb08afc4eecede2f39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.903ex; height:2.509ex;" alt="{\displaystyle U_{1}<0}" loading="lazy"> ie. decreasing potential condition or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=x_{1}}">

<msub>

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

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</math><img src="./2bdf4f9973d3d0c15948ee465d462b13151563d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.812ex; height:2.009ex;" alt="{\displaystyle x=x_{1}}" loading="lazy"> in the given example shown by the figure, we require the exponential function to decay for negative values of x so that wavefunction for it to go to zero. Considering Bairy functions to be the required connection formula, we get:<a href="#cite_note-:3-23">[23]</a>

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<mtext>where,</mtext>

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<mtext>where,</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {Bi} (u)\rightarrow -{\frac {1}{\sqrt {\pi }}}{\frac {1}{\sqrt[{4}]{u}}}\sin {\left({\frac {2}{3}}|u|^{\frac {3}{2}}-{\frac {\pi }{4}}\right)}\quad {\textrm {where,}}\quad u\rightarrow -\infty \\\operatorname {Bi} (u)\rightarrow {\frac {1}{\sqrt {\pi }}}{\frac {1}{\sqrt[{4}]{u}}}e^{{\frac {2}{3}}u^{\frac {3}{2}}}\quad {\textrm {where,}}\quad u\rightarrow +\infty \\\end{aligned}}}</annotation>

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We cannot use Airy function since it gives growing exponential behaviour for negative x. When compared to WKB solutions and matching their behaviours at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm \infty }">

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

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</math><img src="./c586ae37f8efec026b8a4ea3f6a5253576c2c4e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle \pm \infty }" loading="lazy">, we conclude:

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

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</math><img src="./6ccf48d2cb825855b58234a1534ef3aeacabcae4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.736ex; height:2.343ex;" alt="{\displaystyle A=-D=N}" loading="lazy">,

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

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</math><img src="./d966abd402b0d20318abe297b7c201a657e99034.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.89ex; height:2.176ex;" alt="{\displaystyle B=C=0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ={\frac {\pi }{4}}}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha ={\frac {\pi }{4}}}</annotation>

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</math><img src="./b8f7c5303bf3113490b1bfb240ce9dfd4165e0ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.754ex; height:4.676ex;" alt="{\displaystyle \alpha ={\frac {\pi }{4}}}" loading="lazy">.

Thus, letting some normalization constant be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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

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</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy">, the wavefunction is given for increasing potential (with x) as:<a href="#cite_note-:1-19">[19]</a>

<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{\text{WKB}}(x)={\begin{cases}-{\frac {N}{\sqrt {|p(x)|}}}\exp {(-{\frac {1}{\hbar }}\int _{x}^{x_{1}}|p(x)|dx)}&{\text{if }}x<x_{1}\\{\frac {N}{\sqrt {|p(x)|}}}\sin {({\frac {1}{\hbar }}\int _{x}^{x_{1}}|p(x)|dx-{\frac {\pi }{4}})}&{\text{if }}x_{2}>x>x_{1}\\\end{cases}}}">

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<annotation encoding="application/x-tex">{\displaystyle \Psi _{\text{WKB}}(x)={\begin{cases}-{\frac {N}{\sqrt {|p(x)|}}}\exp {(-{\frac {1}{\hbar }}\int _{x}^{x_{1}}|p(x)|dx)}&{\text{if }}x<x_{1}\\{\frac {N}{\sqrt {|p(x)|}}}\sin {({\frac {1}{\hbar }}\int _{x}^{x_{1}}|p(x)|dx-{\frac {\pi }{4}})}&{\text{if }}x_{2}>x>x_{1}\\\end{cases}}}</annotation>

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</math><img src="./2f03e6fda1d9b23afc82774069c379ae52ad8bb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.773ex; margin-bottom: -0.232ex; width:62.481ex; height:9.176ex;" alt="{\displaystyle \Psi _{\text{WKB}}(x)={\begin{cases}-{\frac {N}{\sqrt {|p(x)|}}}\exp {(-{\frac {1}{\hbar }}\int _{x}^{x_{1}}|p(x)|dx)}&{\text{if }}x<x_{1}\\{\frac {N}{\sqrt {|p(x)|}}}\sin {({\frac {1}{\hbar }}\int _{x}^{x_{1}}|p(x)|dx-{\frac {\pi }{4}})}&{\text{if }}x_{2}>x>x_{1}\\\end{cases}}}" loading="lazy">

<div class="mw-heading mw-heading4"><h4 id="Second_classical_turning_point">Second classical turning point</h4></div>

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

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</math><img src="./f0ae098615c2ca6b5c0b727fd50a89edb62d52e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.903ex; height:2.509ex;" alt="{\displaystyle U_{1}>0}" loading="lazy"> ie. increasing potential condition or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=x_{2}}">

<msub>

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</mstyle>

</mrow>

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

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</math><img src="./bc021716332d474013047a1d6b800bd6cfd9f3a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.812ex; height:2.009ex;" alt="{\displaystyle x=x_{2}}" loading="lazy"> in the given example shown by the figure, we require the exponential function to decay for positive values of x so that wavefunction for it to go to zero. Considering <a href="Airy_function" title="Airy function">Airy functions</a> to be the required connection formula, we get:<a href="#cite_note-:3-23">[23]</a>

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<mtext>where,</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {Ai} (u)\rightarrow {\frac {1}{2{\sqrt {\pi }}}}{\frac {1}{\sqrt[{4}]{u}}}e^{-{\frac {2}{3}}u^{\frac {3}{2}}}\quad {\textrm {where,}}\quad u\rightarrow +\infty \\\operatorname {Ai} (u)\rightarrow {\frac {1}{\sqrt {\pi }}}{\frac {1}{\sqrt[{4}]{u}}}\cos {\left({\frac {2}{3}}|u|^{\frac {3}{2}}-{\frac {\pi }{4}}\right)}\quad {\textrm {where,}}\quad u\rightarrow -\infty \\\end{aligned}}}</annotation>

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</math>

We cannot use Bairy function since it gives growing exponential behaviour for positive x. When compared to WKB solutions and matching their behaviours at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm \infty }">

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

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<annotation encoding="application/x-tex">{\displaystyle 2B=C=N'}</annotation>

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</math><img src="./1f4c67c40c49ca7f26abe7f24f1b215fbc44ea8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.697ex; height:2.509ex;" alt="{\displaystyle 2B=C=N'}" loading="lazy">,

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

</semantics>

</math><img src="./f0c329950ef8562f362cbd73c9d89e811ee43d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.027ex; height:2.176ex;" alt="{\displaystyle D=A=0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ={\frac {\pi }{4}}}">

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \alpha ={\frac {\pi }{4}}}</annotation>

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

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</math><img src="./3b2b7bc9c31faccd4695895c4259bbe5a6b53419.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.808ex; height:2.509ex;" alt="{\displaystyle N'}" loading="lazy">, the wavefunction is given for increasing potential (with x) as:<a href="#cite_note-:1-19">[19]</a>

<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{\text{WKB}}(x)={\begin{cases}{\frac {N'}{\sqrt {|p(x)|}}}\cos {({\frac {1}{\hbar }}\int _{x}^{x_{2}}|p(x)|dx-{\frac {\pi }{4}})}&{\text{if }}x_{1}<x<x_{2}\\{\frac {N'}{2{\sqrt {|p(x)|}}}}\exp {(-{\frac {1}{\hbar }}\int _{x_{2}}^{x}|p(x)|dx)}&{\text{if }}x>x_{2}\\\end{cases}}}">

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<annotation encoding="application/x-tex">{\displaystyle \Psi _{\text{WKB}}(x)={\begin{cases}{\frac {N'}{\sqrt {|p(x)|}}}\cos {({\frac {1}{\hbar }}\int _{x}^{x_{2}}|p(x)|dx-{\frac {\pi }{4}})}&{\text{if }}x_{1}<x<x_{2}\\{\frac {N'}{2{\sqrt {|p(x)|}}}}\exp {(-{\frac {1}{\hbar }}\int _{x_{2}}^{x}|p(x)|dx)}&{\text{if }}x>x_{2}\\\end{cases}}}</annotation>

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</math><img src="./81f6bf78c5bdaa5ecc96f09caa7e19056e7aa11a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:62.736ex; height:9.843ex;" alt="{\displaystyle \Psi _{\text{WKB}}(x)={\begin{cases}{\frac {N'}{\sqrt {|p(x)|}}}\cos {({\frac {1}{\hbar }}\int _{x}^{x_{2}}|p(x)|dx-{\frac {\pi }{4}})}&{\text{if }}x_{1}<x<x_{2}\\{\frac {N'}{2{\sqrt {|p(x)|}}}}\exp {(-{\frac {1}{\hbar }}\int _{x_{2}}^{x}|p(x)|dx)}&{\text{if }}x>x_{2}\\\end{cases}}}" loading="lazy">

<div class="mw-heading mw-heading4"><h4 id="Common_oscillating_wavefunction">Common oscillating wavefunction</h4></div>

Matching the two solutions for region <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}<x<x_{2}}">

<msub>

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

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</math><img src="./5a1322657cfe23659f1369f22472e939fd5d7a4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.294ex; height:2.176ex;" alt="{\displaystyle x_{1}<x<x_{2}}" loading="lazy">, it is required that the difference between the angles in these functions is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi (n+1/2)}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \pi (n+1/2)}</annotation>

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</math><img src="./53e55d2e14b367604990c847c46980c238d5cbd9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.864ex; height:2.843ex;" alt="{\displaystyle \pi (n+1/2)}" loading="lazy"> where the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{2}}}">

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

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</math><img src="./98f98bef5d4981ff6e2aa827d4699e347fb30db2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.168ex; height:4.676ex;" alt="{\displaystyle {\frac {\pi }{2}}}" loading="lazy"> phase difference accounts for changing cosine to sine for the wavefunction and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\pi }">

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

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</math><img src="./6b51e35e4bab58263b808970f2d8231c0ae388e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.727ex; height:1.676ex;" alt="{\displaystyle n\pi }" loading="lazy"> difference since negation of the function can occur by letting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=(-1)^{n}N'}">

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<annotation encoding="application/x-tex">{\displaystyle N=(-1)^{n}N'}</annotation>

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</math><img src="./6f9c5d72ec4f56096ee251cd15689adb84f8b944.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.968ex; height:3.009ex;" alt="{\displaystyle N=(-1)^{n}N'}" loading="lazy">. Thus:

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<annotation encoding="application/x-tex">{\displaystyle \int _{x_{1}}^{x_{2}}{\sqrt {2m\left(E-V(x)\right)}}\,dx=(n+1/2)\pi \hbar ,}</annotation>

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</math>

Where n is a non-negative integer. This condition can also be rewritten as saying that:

<dl><dd><dl><dd>The area enclosed by the classical energy curve is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi \hbar (n+1/2)}">

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<annotation encoding="application/x-tex">{\displaystyle 2\pi \hbar (n+1/2)}</annotation>

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</math><img src="./22ba731db7df1c9b2a5bee4926b80db44b57a047.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.333ex; height:2.843ex;" alt="{\displaystyle 2\pi \hbar (n+1/2)}" loading="lazy">.</dd></dl></dd></dl>

Either way, the condition on the energy is a version of the <a href="Bohr%E2%80%93Sommerfeld_quantization" class="mw-redirect" title="Bohr–Sommerfeld quantization">Bohr–Sommerfeld quantization</a> condition, with a "<a href="Lagrangian_Grassmannian#Maslov_index" title="Lagrangian Grassmannian">Maslov correction</a>" equal to 1/2.<a href="#cite_note-24">[24]</a>

It is possible to show that after piecing together the approximations in the various regions, one obtains a good approximation to the actual <a href="Eigenfunction" title="Eigenfunction">eigenfunction</a>. In particular, the Maslov-corrected Bohr–Sommerfeld energies are good approximations to the actual eigenvalues of the Schrödinger operator.<a href="#cite_note-25">[25]</a> Specifically, the error in the energies is small compared to the typical spacing of the quantum energy levels. Thus, although the "old quantum theory" of Bohr and Sommerfeld was ultimately replaced by the Schrödinger equation, some vestige of that theory remains, as an approximation to the eigenvalues of the appropriate Schrödinger operator.

<div class="mw-heading mw-heading4"><h4 id="General_connection_conditions">General connection conditions</h4></div>

Thus, from the two cases the connection formula is obtained at a classical turning point, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=a}">

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<annotation encoding="application/x-tex">{\displaystyle x=a}</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">:<a href="#cite_note-:2-20">[20]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {N}{\sqrt {|p(x)|}}}\sin {\left({\frac {1}{\hbar }}\int _{x}^{a}|p(x)|dx-{\frac {\pi }{4}}\right)}\Longrightarrow -{\frac {N}{\sqrt {|p(x)|}}}\exp {\left({\frac {1}{\hbar }}\int _{a}^{x}|p(x)|dx\right)}}</annotation>

</semantics>

</math><img src="./a32b318a473f7ec0cffbbc909824c4d706ba4231.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:73.567ex; height:6.843ex;" alt="{\displaystyle {\frac {N}{\sqrt {|p(x)|}}}\sin {\left({\frac {1}{\hbar }}\int _{x}^{a}|p(x)|dx-{\frac {\pi }{4}}\right)}\Longrightarrow -{\frac {N}{\sqrt {|p(x)|}}}\exp {\left({\frac {1}{\hbar }}\int _{a}^{x}|p(x)|dx\right)}}" loading="lazy">

and:

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {N'}{\sqrt {|p(x)|}}}\cos {\left({\frac {1}{\hbar }}\int _{x}^{a}|p(x)|dx-{\frac {\pi }{4}}\right)}\Longleftarrow {\frac {N'}{2{\sqrt {|p(x)|}}}}\exp {\left(-{\frac {1}{\hbar }}\int _{a}^{x}|p(x)|dx\right)}}</annotation>

</semantics>

</math><img src="./d7f541ae6e1617a084eaf475c516754fd18edffb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:74.917ex; height:6.843ex;" alt="{\displaystyle {\frac {N'}{\sqrt {|p(x)|}}}\cos {\left({\frac {1}{\hbar }}\int _{x}^{a}|p(x)|dx-{\frac {\pi }{4}}\right)}\Longleftarrow {\frac {N'}{2{\sqrt {|p(x)|}}}}\exp {\left(-{\frac {1}{\hbar }}\int _{a}^{x}|p(x)|dx\right)}}" loading="lazy">

The WKB wavefunction at the classical turning point away from it is approximated by oscillatory sine or cosine function in the classically allowed region, represented in the left and growing or decaying exponentials in the forbidden region, represented in the right. The implication follows due to the dominance of growing exponential compared to decaying exponential. Thus, the solutions of oscillating or exponential part of wavefunctions can imply the form of wavefunction on the other region of potential as well as at the associated turning point.

<div class="mw-heading mw-heading3"><h3 id="Probability_density">Probability density</h3></div>

One can then compute the probability density associated to the approximate wave function. The probability that the quantum particle will be found in the classically forbidden region is small. In the classically allowed region, meanwhile, the probability the quantum particle will be found in a given interval is approximately the fraction of time the classical particle spends in that interval over one period of motion.<a href="#cite_note-26">[26]</a> Since the classical particle's velocity goes to zero at the turning points, it spends more time near the turning points than in other classically allowed regions. This observation accounts for the peak in the wave function (and its probability density) near the turning points.

Applications of the WKB method to Schrödinger equations with a large variety of potentials and comparison with perturbation methods and path integrals are treated in Müller-Kirsten.<a href="#cite_note-27">[27]</a>

<div class="mw-heading mw-heading2"><h2 id="Examples_in_quantum_mechanics">Examples in quantum mechanics</h2></div>

Although WKB potential only applies to smoothly varying potentials,<a href="#cite_note-:2-20">[20]</a> in the examples where rigid walls produce infinities for potential, the WKB approximation can still be used to approximate wavefunctions in regions of smoothly varying potentials. Since the rigid walls have highly discontinuous potential, the connection condition cannot be used at these points and the results obtained can also differ from that of the above treatment.<a href="#cite_note-:1-19">[19]</a>

<div class="mw-heading mw-heading3"><h3 id="Bound_states_for_1_rigid_wall">Bound states for 1 rigid wall</h3></div>

The potential of such systems can be given in the form:

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<annotation encoding="application/x-tex">{\displaystyle V(x)={\begin{cases}V(x)&{\text{if }}x\geq x_{1}\\\infty &{\text{if }}x<x_{1}\\\end{cases}}}</annotation>

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</math><img src="./49cfef4aff5e503527502bb68143a135a8feff90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.519ex; height:6.176ex;" alt="{\displaystyle V(x)={\begin{cases}V(x)&{\text{if }}x\geq x_{1}\\\infty &{\text{if }}x<x_{1}\\\end{cases}}}" loading="lazy">

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<annotation encoding="application/x-tex">{\textstyle x_{1}<x_{2}}</annotation>

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</math><img src="./78e1a04af37e07bf845956ecbf04f4218a1c2933.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.866ex; height:2.176ex;" alt="{\textstyle x_{1}<x_{2}}" loading="lazy">.

Finding wavefunction in bound region, ie. within classical turning points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x_{1}}">

<msub>

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

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</math><img src="./c849fcfc381b6b79acea37e334a4203ceecb5335.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\textstyle x_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x_{2}}">

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

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</math><img src="./4f6c8d0c84c9c04c36f898bc568d6d8c61dbfdd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\textstyle x_{2}}" loading="lazy">, by considering approximations far from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x_{1}}">

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

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<annotation encoding="application/x-tex">{\displaystyle \Psi _{\text{WKB}}(x)={\frac {A}{\sqrt {|p(x)|}}}\sin {\left({\frac {1}{\hbar }}\int _{x}^{x_{1}}|p(x)|dx+\alpha \right)}}</annotation>

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</math><img src="./f74bf2bd84c5c402a81b17c1a691b8a972f954ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:48.083ex; height:6.843ex;" alt="{\displaystyle \Psi _{\text{WKB}}(x)={\frac {A}{\sqrt {|p(x)|}}}\sin {\left({\frac {1}{\hbar }}\int _{x}^{x_{1}}|p(x)|dx+\alpha \right)}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \Psi _{\text{WKB}}(x)={\frac {B}{\sqrt {|p(x)|}}}\cos {\left({\frac {1}{\hbar }}\int _{x}^{x_{2}}|p(x)|dx+\beta \right)}}</annotation>

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</math><img src="./1db60fd4c6fb9f3a4be37e3923f9f6584857389c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:48.183ex; height:6.843ex;" alt="{\displaystyle \Psi _{\text{WKB}}(x)={\frac {B}{\sqrt {|p(x)|}}}\cos {\left({\frac {1}{\hbar }}\int _{x}^{x_{2}}|p(x)|dx+\beta \right)}}" loading="lazy">

Since wavefunction must vanish near <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x_{1}}">

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

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</math><img src="./c849fcfc381b6b79acea37e334a4203ceecb5335.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\textstyle x_{1}}" loading="lazy">, we conclude <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \alpha =0}">

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

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</math><img src="./d964c9d0cb72ad6ea944fdae0860f5d7a74f2796.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\textstyle \alpha =0}" loading="lazy">. For airy functions near <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x_{2}}">

<msub>

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</mstyle>

</mrow>

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

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</math><img src="./4f6c8d0c84c9c04c36f898bc568d6d8c61dbfdd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\textstyle x_{2}}" loading="lazy">, we require <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \beta =-{\frac {\pi }{4}}}">

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<annotation encoding="application/x-tex">{\textstyle \beta =-{\frac {\pi }{4}}}</annotation>

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</math><img src="./2111bf61c0c3fcf471469bde1f73c10199f5557d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.017ex; height:3.176ex;" alt="{\textstyle \beta =-{\frac {\pi }{4}}}" loading="lazy">. We require that angles within these functions have a phase difference <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi (n+1/2)}">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \pi (n+1/2)}</annotation>

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

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</math><img src="./98f98bef5d4981ff6e2aa827d4699e347fb30db2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.168ex; height:4.676ex;" alt="{\displaystyle {\frac {\pi }{2}}}" loading="lazy"> phase difference accounts for changing sine to cosine and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\pi }">

</mstyle>

</mrow>

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

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</math><img src="./6b51e35e4bab58263b808970f2d8231c0ae388e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.727ex; height:1.676ex;" alt="{\displaystyle n\pi }" loading="lazy"> allowing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=(-1)^{n}A}">

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<annotation encoding="application/x-tex">{\displaystyle B=(-1)^{n}A}</annotation>

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</math><img src="./2b773f17b6a1d2abc75a19c84a981f2799355c14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.604ex; height:2.843ex;" alt="{\displaystyle B=(-1)^{n}A}" loading="lazy">.

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\hbar }}\int _{x_{1}}^{x_{2}}|p(x)|dx=\pi \left(n+{\frac {3}{4}}\right)}</annotation>

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</math>Where n is a non-negative integer.<a href="#cite_note-:1-19">[19]</a> Note that the right hand side of this would instead be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi (n-1/4)}">

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<annotation encoding="application/x-tex">{\displaystyle \pi (n-1/4)}</annotation>

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</math><img src="./e2876ff6f1b30c33a2ed876d2d0a44886df3e1bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.864ex; height:2.843ex;" alt="{\displaystyle \pi (n-1/4)}" loading="lazy"> if n was only allowed to non-zero natural numbers.

Thus we conclude that, for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n=1,2,3,\cdots }">

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<annotation encoding="application/x-tex">{\textstyle n=1,2,3,\cdots }</annotation>

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</math><img src="./395d14595a93bc65e51b5cb262e0ef562975e9b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.806ex; height:2.509ex;" alt="{\textstyle n=1,2,3,\cdots }" loading="lazy"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{x_{1}}^{x_{2}}{\sqrt {2m\left(E-V(x)\right)}}\,dx=\left(n-{\frac {1}{4}}\right)\pi \hbar }">

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<annotation encoding="application/x-tex">{\displaystyle \int _{x_{1}}^{x_{2}}{\sqrt {2m\left(E-V(x)\right)}}\,dx=\left(n-{\frac {1}{4}}\right)\pi \hbar }</annotation>

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</math>In 3 dimensions with spherically symmetry, the same condition holds where the position x is replaced by radial distance r, due to its similarity with this problem.<a href="#cite_note-28">[28]</a>

<div class="mw-heading mw-heading3"><h3 id="Bound_states_within_2_rigid_wall">Bound states within 2 rigid wall</h3></div>

The potential of such systems can be given in the form:

<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(x)={\begin{cases}\infty &{\text{if }}x>x_{2}\\V(x)&{\text{if }}x_{2}\geq x\geq x_{1}\\\infty &{\text{if }}x<x_{1}\\\end{cases}}}">

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<annotation encoding="application/x-tex">{\displaystyle V(x)={\begin{cases}\infty &{\text{if }}x>x_{2}\\V(x)&{\text{if }}x_{2}\geq x\geq x_{1}\\\infty &{\text{if }}x<x_{1}\\\end{cases}}}</annotation>

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</math><img src="./11d6cf586a2614ce93bc986f0debbe8f81092f38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:32.325ex; height:8.509ex;" alt="{\displaystyle V(x)={\begin{cases}\infty &{\text{if }}x>x_{2}\\V(x)&{\text{if }}x_{2}\geq x\geq x_{1}\\\infty &{\text{if }}x<x_{1}\\\end{cases}}}" loading="lazy">

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For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle E\geq V(x)}">

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</math><img src="./422d3a9a3aa4042b4f19fa6a79c5d5e20e13f4b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.8ex; height:2.843ex;" alt="{\textstyle E\geq V(x)}" loading="lazy"> between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x_{1}}">

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</math><img src="./4f6c8d0c84c9c04c36f898bc568d6d8c61dbfdd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\textstyle x_{2}}" loading="lazy"> which are thus the classical turning points, by considering approximations far from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle \Psi _{\text{WKB}}(x)={\frac {A}{\sqrt {|p(x)|}}}\sin {\left({\frac {1}{\hbar }}\int _{x}^{x_{1}}|p(x)|dx\right)}}</annotation>

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</math><img src="./14ec5a8b21da37dec6eda05ddd2920ee02401a95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:43.755ex; height:6.843ex;" alt="{\displaystyle \Psi _{\text{WKB}}(x)={\frac {A}{\sqrt {|p(x)|}}}\sin {\left({\frac {1}{\hbar }}\int _{x}^{x_{1}}|p(x)|dx\right)}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \Psi _{\text{WKB}}(x)={\frac {B}{\sqrt {|p(x)|}}}\sin {\left({\frac {1}{\hbar }}\int _{x}^{x_{2}}|p(x)|dx\right)}}</annotation>

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</math><img src="./f4da3dfd816bc2a398589e4e06e6c55a492906d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:43.755ex; height:6.843ex;" alt="{\displaystyle \Psi _{\text{WKB}}(x)={\frac {B}{\sqrt {|p(x)|}}}\sin {\left({\frac {1}{\hbar }}\int _{x}^{x_{2}}|p(x)|dx\right)}}" loading="lazy">

Since wavefunctions must vanish at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x_{1}}">

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</math><img src="./4f6c8d0c84c9c04c36f898bc568d6d8c61dbfdd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\textstyle x_{2}}" loading="lazy">. Here, the phase difference only needs to account for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\pi }">

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</math><img src="./6b51e35e4bab58263b808970f2d8231c0ae388e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.727ex; height:1.676ex;" alt="{\displaystyle n\pi }" loading="lazy"> which allows <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=(-1)^{n}A}">

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<annotation encoding="application/x-tex">{\displaystyle \int _{x_{1}}^{x_{2}}{\sqrt {2m\left(E-V(x)\right)}}\,dx=n\pi \hbar }</annotation>

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</math>where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n=1,2,3,\cdots }">

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</math><img src="./395d14595a93bc65e51b5cb262e0ef562975e9b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.806ex; height:2.509ex;" alt="{\textstyle n=1,2,3,\cdots }" loading="lazy"> but not equal to zero since it makes the wavefunction zero everywhere.<a href="#cite_note-:1-19">[19]</a>

<div class="mw-heading mw-heading3"><h3 id="Quantum_bouncing_ball">Quantum bouncing ball</h3></div>

Consider the following potential a bouncing ball is subjected to:

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

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</math><img src="./43a2612f08f6dca814b0f813f6e48353060e1ac1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.858ex; height:6.176ex;" alt="{\displaystyle V(x)={\begin{cases}mgx&{\text{if }}x\geq 0\\\infty &{\text{if }}x<0\\\end{cases}}}" loading="lazy">

The wavefunction solutions of the above can be solved using the WKB method by considering only odd parity solutions of the alternative potential <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(x)=mg|x|}">

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</math><img src="./c6d027c5631eb0722e6aef3cff5b9608346f459f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.804ex; height:2.843ex;" alt="{\displaystyle V(x)=mg|x|}" loading="lazy">. The classical turning points are identified <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x_{1}=-{E \over mg}}">

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</math><img src="./479d7f1c448fa6645b0e0f0d0ac542ffb2d1b815.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.55ex; height:3.676ex;" alt="{\textstyle x_{2}={E \over mg}}" loading="lazy">. Thus applying the quantization condition obtained in WKB:

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<annotation encoding="application/x-tex">{\displaystyle \int _{x_{1}}^{x_{2}}{\sqrt {2m\left(E-V(x)\right)}}\,dx=(n_{\text{odd}}+1/2)\pi \hbar }</annotation>

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Letting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n_{\text{odd}}=2n-1}">

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</math><img src="./6293964a979ac31c7381b8210745e57a153f1f43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.935ex; height:2.509ex;" alt="{\textstyle n_{\text{odd}}=2n-1}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n=1,2,3,\cdots }">

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</math><img src="./395d14595a93bc65e51b5cb262e0ef562975e9b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.806ex; height:2.509ex;" alt="{\textstyle n=1,2,3,\cdots }" loading="lazy">, solving for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle E}">

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</math><img src="./6d934f67126f64e9c061b598b8941b8767a8d343.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="{\textstyle E}" loading="lazy"> with given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(x)=mg|x|}">

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</math><img src="./c6d027c5631eb0722e6aef3cff5b9608346f459f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.804ex; height:2.843ex;" alt="{\displaystyle V(x)=mg|x|}" loading="lazy">, we get the quantum mechanical energy of a bouncing ball:<a href="#cite_note-29">[29]</a>

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<annotation encoding="application/x-tex">{\displaystyle E={\left(3\left(n-{\frac {1}{4}}\right)\pi \right)^{\frac {2}{3}} \over 2}(mg^{2}\hbar ^{2})^{\frac {1}{3}}.}</annotation>

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This result is also consistent with the use of equation from <a href="Bound_state" title="Bound state">bound state</a> of one rigid wall without needing to consider an alternative potential.

<div class="mw-heading mw-heading3"><h3 id="Quantum_Tunneling">Quantum Tunneling</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Quantum_tunnelling" title="Quantum tunnelling">Quantum tunnelling</a></div>

The potential of such systems can be given in the form:

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<annotation encoding="application/x-tex">{\displaystyle V(x)={\begin{cases}0&{\text{if }}x<x_{1}\\V(x)&{\text{if }}x_{2}\geq x\geq x_{1}\\0&{\text{if }}x>x_{2}\\\end{cases}}}</annotation>

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Its solutions for an incident wave is given as

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x)={\begin{cases}A\exp({ip_{0}x \over \hbar })+B\exp({-ip_{0}x \over \hbar })&{\text{if }}x<x_{1}\\{\frac {C}{\sqrt {|p(x)|}}}\exp {(-{\frac {1}{\hbar }}\int _{x_{1}}^{x}|p(x)|dx)}&{\text{if }}x_{2}\geq x\geq x_{1}\\D\exp({ip_{0}x \over \hbar })&{\text{if }}x>x_{2}\\\end{cases}}}">

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<annotation encoding="application/x-tex">{\displaystyle \psi (x)={\begin{cases}A\exp({ip_{0}x \over \hbar })+B\exp({-ip_{0}x \over \hbar })&{\text{if }}x<x_{1}\\{\frac {C}{\sqrt {|p(x)|}}}\exp {(-{\frac {1}{\hbar }}\int _{x_{1}}^{x}|p(x)|dx)}&{\text{if }}x_{2}\geq x\geq x_{1}\\D\exp({ip_{0}x \over \hbar })&{\text{if }}x>x_{2}\\\end{cases}}}</annotation>

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where the wavefunction in the classically forbidden region is the WKB approximation but neglecting the growing exponential. This is a fair assumption for wide potential barriers through which the wavefunction is not expected to grow to high magnitudes.

By the requirement of continuity of wavefunction and its derivatives, the following relation can be shown:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {|D|^{2}}{|A|^{2}}}={\frac {4}{(1+{a_{1}^{2}}/{p_{0}^{2}})}}{\frac {a_{1}}{a_{2}}}\exp \left(-{\frac {2}{\hbar }}\int _{x_{1}}^{x_{2}}|p(x')|dx'\right)}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {|D|^{2}}{|A|^{2}}}={\frac {4}{(1+{a_{1}^{2}}/{p_{0}^{2}})}}{\frac {a_{1}}{a_{2}}}\exp \left(-{\frac {2}{\hbar }}\int _{x_{1}}^{x_{2}}|p(x')|dx'\right)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle a_{1}=|p(x_{1})|}</annotation>

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</math><img src="./5db5136fd5c4952f3f18c90c3f49b46578089f1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.039ex; height:2.843ex;" alt="{\displaystyle a_{1}=|p(x_{1})|}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{2}=|p(x_{2})|}">

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<annotation encoding="application/x-tex">{\displaystyle a_{2}=|p(x_{2})|}</annotation>

</semantics>

Using <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {J} (\mathbf {x} ,t)={\frac {i\hbar }{2m}}(\psi ^{*}\nabla \psi -\psi \nabla \psi ^{*})}">

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \mathbf {J} (\mathbf {x} ,t)={\frac {i\hbar }{2m}}(\psi ^{*}\nabla \psi -\psi \nabla \psi ^{*})}</annotation>

</semantics>

</math><img src="./77403126ec4a6c7a78a6537571db46d67e7a36c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:29.357ex; height:3.676ex;" alt="{\textstyle \mathbf {J} (\mathbf {x} ,t)={\frac {i\hbar }{2m}}(\psi ^{*}\nabla \psi -\psi \nabla \psi ^{*})}" loading="lazy"> we express the values without signs as:

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle J_{\text{inc.}}={\frac {\hbar }{2m}}({\frac {2p_{0}}{\hbar }}|A|^{2})}</annotation>

</semantics>

</math><img src="./fbbf3130885ff75673941e6c13403c26b923e3c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:19.497ex; height:4.176ex;" alt="{\textstyle J_{\text{inc.}}={\frac {\hbar }{2m}}({\frac {2p_{0}}{\hbar }}|A|^{2})}" loading="lazy">

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle J_{\text{ref.}}={\frac {\hbar }{2m}}({\frac {2p_{0}}{\hbar }}|B|^{2})}</annotation>

</semantics>

</math><img src="./0c2c9951366b6f5eaf739635b247a8960788bcff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:19.295ex; height:4.176ex;" alt="{\textstyle J_{\text{ref.}}={\frac {\hbar }{2m}}({\frac {2p_{0}}{\hbar }}|B|^{2})}" loading="lazy">

<msub>

<mtext>trans.</mtext>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle J_{\text{trans.}}={\frac {\hbar }{2m}}({\frac {2p_{0}}{\hbar }}|D|^{2})}</annotation>

</semantics>

</math><img src="./cab5d3c9a3bd89984cc65667f7521b7f0d3dff8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:21.245ex; height:4.176ex;" alt="{\textstyle J_{\text{trans.}}={\frac {\hbar }{2m}}({\frac {2p_{0}}{\hbar }}|D|^{2})}" loading="lazy">

Thus, the <a href="Transmission_coefficient" title="Transmission coefficient">transmission coefficient</a> is found to be:

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mrow>

<msup>

</msup>

</mrow>

<msup>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T={\frac {|D|^{2}}{|A|^{2}}}={\frac {4}{(1+{a_{1}^{2}}/{p_{0}^{2}})}}{\frac {a_{1}}{a_{2}}}\exp \left(-{\frac {2}{\hbar }}\int _{x_{1}}^{x_{2}}|p(x')|dx'\right)}</annotation>

</semantics>

</math>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle p(x)={\sqrt {2m(E-V(x))}}}</annotation>

</semantics>

</math><img src="./1d41f254a755cf2866d2bb01e76fb607d84eee75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:24.374ex; height:3.343ex;" alt="{\textstyle p(x)={\sqrt {2m(E-V(x))}}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}=|p(x_{1})|}">

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{1}=|p(x_{1})|}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{2}=|p(x_{2})|}</annotation>

</semantics>

</math><img src="./5d9038851952c01a986ad4f458f880c021393f70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.039ex; height:2.843ex;" alt="{\displaystyle a_{2}=|p(x_{2})|}" loading="lazy">. The result can be stated as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle T\sim ~e^{-2\gamma }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle T\sim ~e^{-2\gamma }}</annotation>

</semantics>

</math><img src="./aa81436ebd2c5bb344c067d344ac058bc174227e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.624ex; height:2.509ex;" alt="{\textstyle T\sim ~e^{-2\gamma }}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \gamma =\int _{x_{1}}^{x_{2}}|p(x')|dx'}">

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mrow>

<msup>

</msup>

</mrow>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \gamma =\int _{x_{1}}^{x_{2}}|p(x')|dx'}</annotation>

</semantics>

</math><img src="./bcea28730f2765ad313f7e36a25f8d5ab035a540.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:17.783ex; height:3.509ex;" alt="{\textstyle \gamma =\int _{x_{1}}^{x_{2}}|p(x')|dx'}" loading="lazy">.<a href="#cite_note-:1-19">[19]</a>

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<ul><li><a href="Airy_function" title="Airy function">Airy function</a></li>

<li><a href="Einstein%E2%80%93Brillouin%E2%80%93Keller_method" title="Einstein–Brillouin–Keller method">Einstein–Brillouin–Keller method</a></li>

<li><a href="Field_electron_emission" title="Field electron emission">Field electron emission</a></li>

<li><a href="Instanton" title="Instanton">Instanton</a></li>

<li><a href="Langer_correction" title="Langer correction">Langer correction</a></li>

<li><a href="Maslov_index" class="mw-redirect" title="Maslov index">Maslov index</a></li>

<li><a href="Method_of_dominant_balance" title="Method of dominant balance">Method of dominant balance</a></li>

<li><a href="Method_of_matched_asymptotic_expansions" title="Method of matched asymptotic expansions">Method of matched asymptotic expansions</a></li>

<li><a href="Method_of_steepest_descent" title="Method of steepest descent">Method of steepest descent</a></li>

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

<li><a href="Perturbation_methods" class="mw-redirect" title="Perturbation methods">Perturbation methods</a></li>

<li><a href="Quantum_tunneling" class="mw-redirect" title="Quantum tunneling">Quantum tunneling</a></li>

<li><a href="Slowly_varying_envelope_approximation" title="Slowly varying envelope approximation">Slowly varying envelope approximation</a></li>

<li><a href="Supersymmetric_WKB_approximation" title="Supersymmetric WKB approximation">Supersymmetric WKB approximation</a></li></ul>

</div>

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

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/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFWentzel,_Gregor1926" class="citation journal cs1"><a href="Gregor_Wentzel" title="Gregor Wentzel">Wentzel, Gregor</a> (1926). "Eine Verallgemeinerung der Quantenbedingungen für die Zwecke der Wellenmechanik". Zeitschrift für Physik. 38 (6–7): 518–529. <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/1926ZPhy...38..518W">1926ZPhy...38..518W</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01397171">10.1007/BF01397171</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:120096571">120096571</a>.</cite>

</li>

<li id="cite_note-Kramers-1926-2"><a href="#cite_ref-Kramers-1926_2-0">^</a> <cite id="CITEREFKramers,_Hendrik_A.1926" class="citation journal cs1"><a href="Hendrik_Anthony_Kramers" class="mw-redirect" title="Hendrik Anthony Kramers">Kramers, Hendrik A.</a> (1926). "Wellenmechanik und halbzahlige Quantisierung". Zeitschrift für Physik. 39 (10–11): 828–840. <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/1926ZPhy...39..828K">1926ZPhy...39..828K</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01451751">10.1007/BF01451751</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:122955156">122955156</a>.</cite>

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<li id="cite_note-Brillouin-1926-3"><a href="#cite_ref-Brillouin-1926_3-0">^</a> <cite id="CITEREFBrillouin,_Léon1926" class="citation journal cs1"><a href="L%C3%A9on_Brillouin" title="Léon Brillouin">Brillouin, Léon</a> (1926). "La mécanique ondulatoire de Schrödinger: une méthode générale de resolution par approximations successives". Comptes Rendus de l'Académie des Sciences. 183: 24–26.</cite>

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<li id="cite_note-4"><a href="#cite_ref-4">^</a> <a href="#CITEREFHall2013">Hall 2013</a> Section 15.1

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<li id="cite_note-Jefferys-1924-5"><a href="#cite_ref-Jefferys-1924_5-0">^</a> <cite id="CITEREFJeffreys,_Harold1924" class="citation journal cs1"><a href="Harold_Jeffreys" title="Harold Jeffreys">Jeffreys, Harold</a> (1924). "On certain approximate solutions of linear differential equations of the second order". Proceedings of the London Mathematical Society. 23: 428–436. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fplms%2Fs2-23.1.428">10.1112/plms/s2-23.1.428</a>.</cite>

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<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFDingle1973" class="citation book cs1">Dingle, Robert Balson (1973). Asymptotic Expansions: Their Derivation and Interpretation. Academic Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-12-216550-0</bdi>.</cite>

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<li id="cite_note-Liouville-8"><a href="#cite_ref-Liouville_8-0">^</a> <cite id="CITEREFLiouville,_Joseph1837" class="citation journal cs1"><a href="Joseph_Liouville" title="Joseph Liouville">Liouville, Joseph</a> (1837). "Sur le développement des fonctions et séries.". Journal de Mathématiques Pures et Appliquées. 1: 16–35.</cite>

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<li id="cite_note-Green-1837-9"><a href="#cite_ref-Green-1837_9-0">^</a> <cite id="CITEREFGreen,_George1837" class="citation journal cs1"><a href="George_Green_(mathematician)" title="George Green (mathematician)">Green, George</a> (1837). "On the motion of waves in a variable canal of small depth and width". Transactions of the Cambridge Philosophical Society. 6: 457–462.</cite>

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<li id="cite_note-Rayleigh-1912-10"><a href="#cite_ref-Rayleigh-1912_10-0">^</a> <cite id="CITEREFRayleigh,_Lord_(John_William_Strutt)1912" class="citation journal cs1"><a href="Lord_Rayleigh" class="mw-redirect" title="Lord Rayleigh">Rayleigh, Lord (John William Strutt)</a> (1912). <a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1912.0014">"On the propagation of waves through a stratified medium, with special reference to the question of reflection"</a>. <a href="Proceedings_of_the_Royal_Society_A" class="mw-redirect" title="Proceedings of the Royal Society A">Proceedings of the Royal Society A</a>. 86 (586): 207–226. <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/1912RSPSA..86..207R">1912RSPSA..86..207R</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1912.0014">10.1098/rspa.1912.0014</a>.</cite>

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<li id="cite_note-Gans-1915-11"><a href="#cite_ref-Gans-1915_11-0">^</a> <cite id="CITEREFGans,_Richard1915" class="citation journal cs1"><a href="Richard_Gans" title="Richard Gans">Gans, Richard</a> (1915). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1447303">"Fortplantzung des Lichts durch ein inhomogenes Medium"</a>. Annalen der Physik. 47 (14): 709–736. <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/1915AnP...352..709G">1915AnP...352..709G</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fandp.19153521402">10.1002/andp.19153521402</a>.</cite>

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<li id="cite_note-12"><a href="#cite_ref-12">^</a> <cite id="CITEREFAdrian_E._Gill1982" class="citation book cs1">Adrian E. Gill (1982). <a rel="nofollow" class="external text" href="https://archive.org/details/atmosphereoceand0000gill">Atmosphere-ocean dynamics</a>. Academic Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/atmosphereoceand0000gill/page/297">297</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-12-283522-3</bdi>. <q>Liouville-Green WKBJ WKB.</q></cite>

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<cite id="CITEREFRenato_SpiglerMarco_Vianello1998" class="citation book cs1">Renato Spigler & Marco Vianello (1998). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=a36iXw5_VzcC&dq=Liouville-Green+WKBJ+WKB+LG&pg=PA567">"A Survey on the Liouville–Green (WKB) approximation for linear difference equations of the second order"</a>. In Saber Elaydi; I. Győri & G. E. Ladas (eds.). Advances in difference equations: proceedings of the Second International Conference on Difference Equations : Veszprém, Hungary, August 7–11, 1995. CRC Press. p. 567. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-90-5699-521-8</bdi>.</cite>

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<li id="cite_note-14"><a href="#cite_ref-14">^</a> <cite id="CITEREFFilippi1999" class="citation book cs1">Filippi, Paul (1999). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xHWiOMp63WsC&q=wkb%20multi-scale&pg=PA171">Acoustics: basic physics, theory and methods</a>. Academic Press. p. 171. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-12-256190-0</bdi>.</cite>

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<cite id="CITEREFHolmes,_M.2013" class="citation book cs1">Holmes, M. (2013). Introduction to Perturbation Methods, 2nd Ed. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4614-5476-2</bdi>.</cite>

</li>

<li id="cite_note-:0-16">^ <a href="#cite_ref-:0_16-0">a</a> <a href="#cite_ref-:0_16-1">b</a> <cite id="CITEREFBenderOrszag1999" class="citation book cs1"><a href="Carl_M._Bender" title="Carl M. Bender">Bender, Carl M.</a>; <a href="Steven_Orszag" title="Steven Orszag">Orszag, Steven A.</a> (1999). Advanced mathematical methods for scientists and engineers. Springer. pp. 549–568. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-98931-5</bdi>.</cite>

</li>

<li id="cite_note-17"><a href="#cite_ref-17">^</a> <cite id="CITEREFWinitzki2005" class="citation journal cs1">Winitzki, S. (2005). "Cosmological particle production and the precision of the WKB approximation". Phys. Rev. D. 72 (10): 104011, 14 pp. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/gr-qc/0510001">gr-qc/0510001</a>. <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/2005PhRvD..72j4011W">2005PhRvD..72j4011W</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%2FPhysRevD.72.104011">10.1103/PhysRevD.72.104011</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:119152049">119152049</a>.</cite>

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> <a href="#CITEREFHall2013">Hall 2013</a> Section 15.4

</li>

<li id="cite_note-:1-19">^ <a href="#cite_ref-:1_19-0">a</a> <a href="#cite_ref-:1_19-1">b</a> <a href="#cite_ref-:1_19-2">c</a> <a href="#cite_ref-:1_19-3">d</a> <a href="#cite_ref-:1_19-4">e</a> <a href="#cite_ref-:1_19-5">f</a> <a href="#cite_ref-:1_19-6">g</a> <a href="#cite_ref-:1_19-7">h</a> <cite id="CITEREFZettili2009" class="citation book cs1">Zettili, Nouredine (2009). Quantum mechanics: concepts and applications (2nd ed.). Chichester: Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-470-02679-3</bdi>.</cite>

</li>

<li id="cite_note-:2-20">^ <a href="#cite_ref-:2_20-0">a</a> <a href="#cite_ref-:2_20-1">b</a> <a href="#cite_ref-:2_20-2">c</a> <a href="#cite_ref-:2_20-3">d</a> <cite id="CITEREFZwiebach" class="citation web cs1">Zwiebach, Barton. <a rel="nofollow" class="external text" href="https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/bf207c35150e1f5d93ef05d4664f406d_MIT8_06S18ch3.pdf">"Semiclassical approximation"</a> (PDF).</cite>

</li>

<li id="cite_note-21"><a href="#cite_ref-21">^</a> <cite id="CITEREFBransdenJoachain2003" class="citation book cs1">Bransden, B. H.; Joachain, Charles Jean (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ST_DwIGZeTQC">Physics of Atoms and Molecules</a>. Prentice Hall. pp. 140–141. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-582-35692-4</bdi>.</cite>

</li>

<li id="cite_note-22"><a href="#cite_ref-22">^</a> <a href="#CITEREFHall2013">Hall 2013</a> Section 15.5

</li>

<li id="cite_note-:3-23">^ <a href="#cite_ref-:3_23-0">a</a> <a href="#cite_ref-:3_23-1">b</a> <cite id="CITEREFRamkarthikPereira2021" class="citation journal cs1">Ramkarthik, M. S.; Pereira, Elizabeth Louis (2021-06-01). <a rel="nofollow" class="external text" href="https://doi.org/10.1007/s12045-021-1179-z">"Airy Functions Demystified — II"</a>. Resonance. 26 (6): 757–789. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs12045-021-1179-z">10.1007/s12045-021-1179-z</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0973-712X">0973-712X</a>.</cite>

</li>

<li id="cite_note-24"><a href="#cite_ref-24">^</a> <a href="#CITEREFHall2013">Hall 2013</a> Section 15.2

</li>

<li id="cite_note-25"><a href="#cite_ref-25">^</a> <a href="#CITEREFHall2013">Hall 2013</a> Theorem 15.8

</li>

<li id="cite_note-26"><a href="#cite_ref-26">^</a> <a href="#CITEREFHall2013">Hall 2013</a> Conclusion 15.5

</li>

<li id="cite_note-27"><a href="#cite_ref-27">^</a> Harald J.W. Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed. (World Scientific, 2012).

</li>

<li id="cite_note-28"><a href="#cite_ref-28">^</a> <cite id="CITEREFWeinberg2015" class="citation book cs1">Weinberg, Steven (2015-09-10). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1017/cbo9781316276105">Lectures on Quantum Mechanics</a> (2nd ed.). Cambridge University Press. p. 204. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2Fcbo9781316276105">10.1017/cbo9781316276105</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-107-11166-0</bdi>.</cite>

</li>

<li id="cite_note-29"><a href="#cite_ref-29">^</a> <cite id="CITEREFSakuraiNapolitano2021" class="citation book cs1">Sakurai, Jun John; Napolitano, Jim (2021). Modern quantum mechanics (3rd ed.). Cambridge: Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-108-47322-4</bdi>.</cite>

</li>

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

<div class="mw-heading mw-heading3"><h3 id="Further_reading">Further reading</h3></div>

<ul><li><cite id="CITEREFChild,_M._S.1991" class="citation book cs1">Child, M. S. (1991). Semiclassical mechanics with molecular applications. Oxford: Clarendon Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-19-855654-3</bdi>.</cite></li>

<li><cite id="CITEREFFrömanFröman1965" class="citation book cs1">Fröman, N.; Fröman, P.-O. (1965). JWKB Approximation: Contributions to the Theory. Amsterdam: North-Holland.</cite></li>

<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="CITEREFHall2013" class="citation cs2">Hall, Brian C. (2013), Quantum Theory for Mathematicians, Graduate Texts in Mathematics, vol. 267, Springer, <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/2013qtm..book.....H">2013qtm..book.....H</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1461471158</bdi></cite></li>

<li><cite id="CITEREFLiboff,_Richard_L.2003" class="citation book cs1">Liboff, Richard L. (2003). Introductory Quantum Mechanics (4th ed.). Addison-Wesley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8053-8714-5</bdi>.</cite></li>

<li><cite id="CITEREFOlver,_Frank_William_John1974" class="citation book cs1"><a href="Frank_William_John_Olver" class="mw-redirect" title="Frank William John Olver">Olver, Frank William John</a> (1974). <a rel="nofollow" class="external text" href="https://archive.org/details/asymptoticsspeci0000olve">Asymptotics and Special Functions</a>. Academic Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-12-525850-X</bdi>.</cite></li>

<li><cite id="CITEREFRazavy,_Mohsen2003" class="citation book cs1">Razavy, Mohsen (2003). <a rel="nofollow" class="external text" href="https://archive.org/details/quantumtheoryoft0000raza">Quantum Theory of Tunneling</a>. World Scientific. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>981-238-019-1</bdi>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite id="CITEREFFitzpatrick2002" class="citation web cs1">Fitzpatrick, Richard (2002). <a rel="nofollow" class="external text" href="http://farside.ph.utexas.edu/teaching/jk1/lectures/node70.html">"The W.K.B. Approximation"</a>.</cite> (An application of the WKB approximation to the scattering of radio waves from the ionosphere.)</li></ul></div><div><div class="zim-footer">

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