Editing WKB approximation

{{short description|Solution method for linear differential equations}}
{{Redirect2|WKB|WKBJ|other uses|WKB (disambiguation)|the television station in Live Oak, Florida|WKBJ-LD}}
In [[mathematical physics]], the '''WKB approximation''' or '''WKB method''' is a technique for finding approximate solutions to [[Linear differential equation|linear differential equations]] with spatially varying coefficients.  It is typically used for a [[Semiclassical physics|semiclassical]] calculation in [[quantum mechanics]] in which the [[wave function]] 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.

== Brief history ==
This method is named after physicists [[Gregor Wentzel]], [[Hendrik Anthony Kramers]], and [[Léon Brillouin]], who all developed it in 1926.<ref name=Wentzel-1926/><ref name=Kramers-1926/><ref name=Brillouin-1926/><ref>{{harvnb|Hall|2013}} Section 15.1 </ref> In 1923,<ref name=Jefferys-1924/> mathematician [[Harold Jeffreys]] had developed a general method of approximating solutions to linear, second-order differential equations, a class that includes the [[Schrödinger equation]].  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.<ref>{{cite book |first=Robert Balson |last=Dingle |title=Asymptotic Expansions: Their Derivation and Interpretation |publisher=Academic Press |year=1973 |isbn=0-12-216550-0 }}</ref>

Earlier appearances of essentially equivalent methods are: [[Francesco Carlini]] in 1817,<ref name=Carlini-1817/> [[Joseph Liouville]] in 1837,<ref name=Liouville/> [[George Green (mathematician)|George Green]] in 1837,<ref name=Green-1837/> [[Lord Rayleigh]] in 1912<ref name=Rayleigh-1912/> and [[Richard Gans]] in 1915.<ref name=Gans-1915/> 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.<ref>{{cite book
 | title = Atmosphere-ocean dynamics
 | author = Adrian E. Gill
 | publisher = Academic Press
 | year = 1982
 | isbn = 978-0-12-283522-3
 | page = [https://archive.org/details/atmosphereoceand0000gill/page/297 297]
 | url = https://archive.org/details/atmosphereoceand0000gill
 | url-access = registration
 | quote = Liouville-Green WKBJ WKB.
 }}</ref><ref>
{{cite book
 | chapter = A Survey on the Liouville–Green (WKB) approximation for linear difference equations of the second order
 |author1=Renato Spigler  |author2=Marco Vianello
  |name-list-style=amp | title = Advances in difference equations: proceedings of the Second International Conference on Difference Equations : Veszprém, Hungary, August 7–11, 1995     
 |editor1=Saber Elaydi |editor2=I. Győri |editor3=G. E. Ladas | publisher = CRC Press
 | year = 1998
 | isbn = 978-90-5699-521-8
 | page = 567
 | chapter-url = https://books.google.com/books?id=a36iXw5_VzcC&dq=Liouville-Green+WKBJ+WKB+LG&pg=PA567
 }}</ref>

The important contribution of Jeffreys, Wentzel, Kramers, and Brillouin to the method was the inclusion of the treatment of [[stationary point|turning points]], connecting the [[evanescent wave|evanescent]] and [[oscillation|oscillatory]] solutions at either side of the turning point. For example, this may occur in the Schrödinger equation, due to a [[potential energy]] hill.

==Formulation==

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

For a differential equation
<math display="block"> \varepsilon \frac{d^ny}{dx^n} + a(x)\frac{d^{n-1}y}{dx^{n-1}} + \cdots + k(x)\frac{dy}{dx} + m(x)y= 0,</math>
assume a solution of the form of an [[asymptotic series]] expansion
<math display="block"> y(x) \sim \exp\left[\frac{1}{\delta}\sum_{n=0}^{\infty} \delta^n S_n(x)\right]</math>
in the limit {{math|''δ'' → 0}}. The asymptotic scaling of {{mvar|δ}} in terms of {{mvar|ε}} will be determined by the equation – see the example below.

Substituting the above [[ansatz]] into the differential equation and cancelling out the exponential terms allows one to solve for an arbitrary number of terms {{math|''S''<sub>''n''</sub>(''x'')}} in the expansion.

WKB theory is a special case of [[multiple scale analysis]].<ref>{{cite book
 | title = Acoustics: basic physics, theory and methods
 | first = Paul 
 | last = Filippi
 | publisher = Academic Press
 | year = 1999
 | isbn = 978-0-12-256190-0
 | page = 171
 | url = https://books.google.com/books?id=xHWiOMp63WsC&q=wkb%20multi-scale&pg=PA171
 }}</ref><ref>
{{Cite book 
 | author1=Holmes, M. 
 | title=Introduction to Perturbation Methods, 2nd Ed 
 | year=2013 
 | publisher=Springer 
 | isbn=978-1-4614-5476-2 
 }}</ref><ref name=":0">{{cite book
 | first1=Carl M.
 | last1=Bender
 | author-link1=Carl M. Bender
 | first2=Steven A.
 | last2=Orszag
 | author-link2=Steven Orszag
 | title=Advanced mathematical methods for scientists and engineers
 | publisher=Springer
 | year=1999
 | isbn=0-387-98931-5
 | pages=549–568
 }}</ref>

==An example==
This example comes from the text of [[Carl M. Bender]] and [[Steven Orszag]].<ref name=":0" /> Consider the second-order homogeneous linear differential equation
<math display="block"> \epsilon^2 \frac{d^2 y}{dx^2} = Q(x) y, </math>
where <math>Q(x) \neq 0</math>. Substituting
<math display="block">y(x) = \exp \left[\frac{1}{\delta} \sum_{n=0}^\infty \delta^n S_{n}(x)\right]</math>
results in the equation
<math display="block">\epsilon^2\left[\frac{1}{\delta^2} \left(\sum_{n=0}^\infty \delta^nS_{n}^{\prime}\right)^2 + \frac{1}{\delta} \sum_{n=0}^{\infty}\delta^n S_{n}^{\prime\prime}\right] = Q(x).</math>

To [[leading-order|leading order]] in  ''ϵ'' (assuming, for the moment, the series will be asymptotically consistent), the above can be approximated as
<math display="block">\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).</math>

In the limit {{math|''δ'' → 0}}, the [[Method of dominant balance|dominant balance]] is given by
<math display="block">\frac{\epsilon^2}{\delta^2} {S_{0}^{\prime}}^2 \sim Q(x).</math>

So {{mvar|δ}} is proportional to ''ϵ''. Setting them equal and comparing powers yields
<math display="block">\epsilon^0: \quad {S_{0}^{\prime}}^2 = Q(x),</math>
which can be recognized as the [[eikonal equation]], with solution
<math display="block">S_{0}(x) = \pm \int_{x_0}^x \sqrt{Q(x')}\,dx'.</math>

Considering first-order powers of  {{mvar|ϵ}} fixes
<math display="block">\epsilon^1: \quad 2 S_{0}^{\prime} S_{1}^{\prime} + S_{0}^{\prime\prime} = 0.</math>
This has the solution
<math display="block">S_{1}(x) = -\frac{1}{4} \ln Q(x) + k_1,</math>
where {{math|''k''<sub>1</sub>}} is an arbitrary constant.

We now have a pair of approximations to the system (a pair, because   {{math|''S''<sub>0</sub>}} can take two signs); the first-order WKB-approximation will be a linear combination of the two:
<math display="block">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].</math>

Higher-order terms can be obtained by looking at equations for higher powers of {{mvar|δ}}.  Explicitly,
<math display="block"> 2S_{0}^{\prime} S_{n}^{\prime} + S^{\prime\prime}_{n-1} + \sum_{j=1}^{n-1}S^{\prime}_{j} S^{\prime}_{n-j} = 0</math>
for {{math|''n'' ≥ 2}}.

=== Precision of the asymptotic series ===
The asymptotic series for {{math|''y''(''x'')}} is usually a [[divergent series]], whose general term {{math|''δ''<sup>''n''</sup> ''S''<sub>''n''</sub>(''x'')}} starts to increase after a certain value {{math|1=''n'' = ''n''<sub>max</sub>}}. Therefore, the smallest error achieved by the WKB method is at best of the order of the last included term.

For the equation
<math display="block"> \epsilon^2 \frac{d^2 y}{dx^2} = Q(x) y, </math>
with {{math|''Q''(''x'') <0}} an analytic function, the value <math>n_\max</math> and the magnitude of the last term can be estimated as follows:<ref>{{cite journal| last=Winitzki |first=S. |year=2005 |arxiv=gr-qc/0510001 |title=Cosmological particle production and the precision of the WKB approximation |journal=Phys. Rev. D |volume=72 |issue=10 |pages=104011, 14&nbsp;pp |doi=10.1103/PhysRevD.72.104011 |bibcode = 2005PhRvD..72j4011W |s2cid=119152049 }}</ref>
<math display="block">n_\max \approx 2\epsilon^{-1} \left| \int_{x_0}^{x_{\ast}} \sqrt{-Q(z)}\,dz \right| , </math>
<math display="block">\delta^{n_\max}S_{n_\max}(x_0) \approx \sqrt{\frac{2\pi}{n_\max}} \exp[-n_\max], </math>
where <math>x_0</math> is the point at which <math>y(x_0)</math> needs to be evaluated and <math>x_{\ast}</math> is the (complex) turning point where <math>Q(x_{\ast}) = 0</math>, closest to <math>x = x_0</math>.

The number {{math|''n''<sub>max</sub>}} can be interpreted as the number of oscillations between <math>x_0</math> and the closest turning point.

If <math>\epsilon^{-1}Q(x)</math> is a slowly changing function,
<math display="block">\epsilon\left| \frac{dQ}{dx} \right| \ll Q^2 , ^{\text{[might be }Q^{3/2}\text{?]}}</math>
the number {{math|''n''<sub>max</sub>}}  will be large, and the minimum error of the asymptotic series will be exponentially small.

== Application in non relativistic quantum mechanics ==
[[File:WKB approximation example.svg|thumb|WKB approximation to the indicated potential. Vertical lines show the turning points]]
[[File:WKB approximation to probability density.svg|thumb|Probability density for the approximate wave function. Vertical lines show the turning points]]
The above example may be applied specifically to the one-dimensional, time-independent [[Schrödinger equation]],
<math display="block">-\frac{\hbar^2}{2m} \frac{d^2}{dx^2} \Psi(x) + V(x) \Psi(x) = E \Psi(x),</math>
which can be rewritten as
<math display="block">\frac{d^2}{dx^2} \Psi(x) = \frac{2m}{\hbar^2} \left( V(x) - E \right) \Psi(x).</math>

===Approximation away from the turning points===
The wavefunction can be rewritten as the exponential of another function {{math|S}} (closely related to the [[Action (physics)|action]]), which could be complex,
<math display="block">\Psi(\mathbf x) = e^{i S(\mathbf{x}) \over \hbar},    </math>
so that its substitution in Schrödinger's equation gives:

<math display="block">i\hbar \nabla^2 S(\mathbf x) - (\nabla S(\mathbf x))^2 = 2m \left( V(\mathbf x) - E \right),</math>

Next, the semiclassical approximation is used. This means that each function is expanded as a [[power series]] in {{mvar|ħ}}.
<math display="block">S = S_0 + \hbar S_1 + \hbar^2 S_2 + \cdots </math>
Substituting in the equation, and only retaining terms up to first order in {{math|ℏ}}, we get:
<math display="block">(\nabla S_0+\hbar \nabla S_1)^2-i\hbar(\nabla^2 S_0) = 2m(E-V(\mathbf x)) </math>
which gives the following two relations:
<math display="block">\begin{align}
(\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{align}</math>
which can be solved for 1D systems, first equation resulting in:<math display="block">S_0(x) = \pm \int \sqrt{ 2m \left( E - V(x)\right) } \,dx=\pm\int p(x) \,dx     </math>and the second equation computed for the possible values of the above, is generally expressed as:<math display="block">\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)| }} </math>


Thus, the resulting wavefunction in first order WKB approximation is presented as,<ref>{{harvnb|Hall|2013}} Section 15.4</ref><ref name=":1">{{Cite book |last=Zettili |first=Nouredine |title=Quantum mechanics: concepts and applications |date=2009 |publisher=Wiley |isbn=978-0-470-02679-3 |edition=2nd |location=Chichester}}</ref>
{{Equation box 1
|indent =:
|equation =  <math>\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{2 m \left( E - V(x)  \right)}\,dx} }{  \sqrt[4]{2m \mid E - V(x) \mid} }  </math>
|cellpadding= 6
|border
|border colour = #0073CF
|bgcolor=#F9FFF7}}


In the classically allowed region, namely the region where <math>V(x) < E</math> the integrand in the exponent is imaginary and the approximate wave function is oscillatory. In the classically forbidden region <math>V(x) > E</math>, 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 {{math|1=''E'' = ''V''(''x'')}}, and cannot be valid. (The turning points are the points where the classical particle changes direction.)


Hence, when <math>E > V(x)</math>, the wavefunction can be chosen to be expressed as:<math display="block">\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)| }}  </math>and for <math>V(x) > E</math>,<math display="block">\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)|} } . </math>The integration in this solution is computed between the classical turning point and the arbitrary position x'.

=== Validity of WKB solutions ===
From the condition:<math display="block">(S_0'(x))^2-(p(x))^2 + \hbar (2 S_0'(x)S_1'(x)-iS_0''(x)) = 0 </math>

It follows that: <math display="inline">\hbar\mid 2 S_0'(x)S_1'(x)\mid+\hbar \mid i S_0''(x)\mid \ll \mid(S_0'(x))^2\mid +\mid (p(x))^2\mid </math>


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

<math display="block">\begin{align}
\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{align}    </math>

The first inequality can be used to show the following:

<math display="block">\begin{align}
\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{align}    </math>

where <math display="inline">|S_0'(x)|= |p(x)|    </math> is used and <math display="inline">\lambda(x) </math> is the local [[De Broglie waves|de Broglie wavelength]] of the wavefunction. The inequality implies that the variation of potential is assumed to be slowly varying.<ref name=":1" /><ref name=":2">{{Cite web |last=Zwiebach |first=Barton |title=Semiclassical approximation |url=https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/bf207c35150e1f5d93ef05d4664f406d_MIT8_06S18ch3.pdf}}</ref> This condition can also be restated as the fractional change of <math display="inline">E-V(x) </math> or that of the momentum <math display="inline">p(x) </math>, over the wavelength <math display="inline">\lambda </math>, being much smaller than <math display="inline">1 </math>.<ref>{{Cite book |last1=Bransden |first1=B. H. |url=https://books.google.com/books?id=ST_DwIGZeTQC |title=Physics of Atoms and Molecules |last2=Joachain |first2=Charles Jean |date=2003 |publisher=Prentice Hall |isbn=978-0-582-35692-4 |pages=140–141 |language=en}}</ref>



Similarly it can be shown that <math display="inline">\lambda(x) </math> also has restrictions based on underlying assumptions for the WKB approximation that:<math display="block">\left|\frac{d\lambda}{dx}\right| \ll 1 </math>which implies that the [[De Broglie waves|de Broglie wavelength]] of the particle is slowly varying.<ref name=":2" />

=== Behavior near the turning points ===
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, {{math|''x''<sub>1</sub>}}, the term <math>\frac{2m}{\hbar^2}\left(V(x)-E\right)</math> can be expanded in a power series,
<math display="block">\frac{2m}{\hbar^2}\left(V(x)-E\right) = U_1 \cdot (x - x_1) + U_2 \cdot (x - x_1)^2 + \cdots\;.</math>

To first order, one finds
<math display="block">\frac{d^2}{dx^2} \Psi(x) = U_1 \cdot (x - x_1) \cdot \Psi(x).</math>
This differential equation is known as the [[Airy equation]], and the solution may be written in terms of [[Airy function]]s,<ref>{{harvnb|Hall|2013}} Section 15.5</ref>
<math display="block">\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).</math>

Although for any fixed value of <math>\hbar</math>, 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>\hbar</math> gets smaller, the height of the wave function at the turning points grows. It also follows from this approximation that:

<math display="block">\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}</math>

===Connection conditions===
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 {{math|''E''}}, this matching procedure will not work: The function obtained by connecting the solution near <math>+\infty</math> to the classically allowed region will not agree with the function obtained by connecting the solution near <math>-\infty</math> to the classically allowed region. The requirement that the two functions agree imposes a condition on the energy {{math|''E''}}, which will give an approximation to the exact quantum energy levels.[[File:WKB approximation example.svg|thumb|WKB approximation to the indicated potential. Vertical lines show the energy level and its intersection with potential shows the turning points with dotted lines. The problem has two classical turning points with <math>U_1 < 0</math> at <math>x=x_1 
</math> and <math>U_1 > 0</math> at <math>x=x_2
</math>.]]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>x=x_1 
</math> and the second turning point, where potential is increasing over x, occur at <math>x=x_2
</math>. Given that we expect wavefunctions to be of the following form, we can calculate their coefficients by connecting the different regions using Airy and Bairy functions. 

<math display="block">\begin{align} 
\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{align} </math>

==== First classical turning point ====
For <math>U_1 < 0</math> ie. decreasing potential condition or <math>x=x_1 
</math> 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:<ref name=":3">{{Cite journal |last1=Ramkarthik |first1=M. S. |last2=Pereira |first2=Elizabeth Louis |date=2021-06-01 |title=Airy Functions Demystified — II |url=https://doi.org/10.1007/s12045-021-1179-z |journal=Resonance |language=en |volume=26 |issue=6 |pages=757–789 |doi=10.1007/s12045-021-1179-z |issn=0973-712X|url-access=subscription }}</ref>

<math display="block">\begin{align}
\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{align} </math>

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>\pm \infty </math>, we conclude:

<math>A=-D=N </math>,
<math>B=C=0 </math> and <math>\alpha = \frac \pi 4 </math>.  

Thus, letting some normalization constant be <math>N </math>, the wavefunction is given for increasing potential (with x) as:<ref name=":1" />


<math>\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}    </math>

==== Second classical turning point ====
For <math>U_1 > 0</math> ie. increasing potential condition or <math>x=x_2
</math> 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 [[Airy function|Airy functions]] to be the required connection formula, we get:<ref name=":3" />

<math display="block">\begin{align}
\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{align} </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>\pm \infty </math>, we conclude:

<math>2B=C=N'   </math>,
<math>D=A=0 </math> and <math>\alpha = \frac \pi 4 </math>.  

Thus, letting some normalization constant be <math>N' </math>, the wavefunction is given for increasing potential (with x) as:<ref name=":1" />


<math>\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}</math>

==== Common oscillating wavefunction ====
Matching the two solutions for region <math>x_1<x<x_2  </math>, it is required that the difference between the angles in these functions is <math>\pi(n+1/2)</math> where the <math>\frac \pi 2</math> phase difference accounts for changing cosine to sine for the wavefunction and <math>n \pi</math> difference since negation of the function can occur by letting <math>N= (-1)^n N'  </math>. Thus:
<math display="block">\int_{x_1}^{x_2} \sqrt{2m \left( E-V(x)\right)}\,dx = (n+1/2)\pi \hbar ,</math>
Where ''n'' is a non-negative integer. This condition can also be rewritten as saying that:
::The area enclosed by the classical energy curve is <math>2\pi\hbar(n+1/2)</math>.
Either way, the condition on the energy is a version of the [[Bohr–Sommerfeld quantization]] condition, with a "[[Lagrangian Grassmannian#Maslov index|Maslov correction]]" equal to 1/2.<ref>{{harvnb|Hall|2013}} Section 15.2</ref>

It is possible to show that after piecing together the approximations in the various regions, one obtains a good approximation to the actual eigenfunction. In particular, the Maslov-corrected Bohr–Sommerfeld energies are good approximations to the actual eigenvalues of the Schrödinger operator.<ref>{{harvnb|Hall|2013}} Theorem 15.8</ref> 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.

==== General connection conditions ====
Thus, from the two cases the connection formula is obtained at a classical turning point, <math>x=a
</math>:<ref name=":2" />

<math>  \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)}   </math>

and:

<math>  \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)}  </math>

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.

===Probability density===
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.<ref>{{harvnb|Hall|2013}} Conclusion 15.5</ref> 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.<ref>Harald J.W. Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed. (World Scientific, 2012).</ref>

== Examples in quantum mechanics ==
Although WKB potential only applies to smoothly varying potentials,<ref name=":2" /> 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.<ref name=":1" />

=== Bound states for 1 rigid wall ===
The potential of such systems can be given in the form:

<math>V(x) = \begin{cases}
V(x) & \text{if }  x \geq x_1\\
  \infty & \text{if } x < x_1 \\
 \end{cases}</math>

where <math display="inline">x_1 < x_2 </math>.


Finding wavefunction in bound region, ie. within classical turning points <math display="inline">x_1 </math> and <math display="inline">x_2 </math>, by considering approximations far from <math display="inline">x_1 </math> and <math display="inline">x_2 </math> respectively we have two solutions:

<math>\Psi_{\text{WKB}}(x) = 
\frac{A}{\sqrt{|p(x)|}}\sin{\left(\frac 1 \hbar \int_{x}^{x_1} |p(x)| dx +\alpha \right)}   </math>

<math>\Psi_{\text{WKB}}(x) = 
\frac{B}{\sqrt{|p(x)|}}\cos{\left(\frac 1 \hbar \int_{x}^{x_2} |p(x)| dx +\beta \right)}     </math>

Since wavefunction must vanish near <math display="inline">x_1 </math>, we conclude <math display="inline">\alpha = 0 </math>. For airy functions near <math display="inline">x_2 </math>, we require <math display="inline">\beta = - \frac \pi 4    </math>. We require that angles within these functions have a phase difference <math>\pi(n+1/2)</math> where the <math>\frac \pi 2</math> phase difference accounts for changing sine to cosine and <math>n \pi</math> allowing <math>B= (-1)^n A </math>.

<math display="block">\frac 1 \hbar \int_{x_1}^{x_2} |p(x)| dx = \pi \left(n + \frac 3 4\right)  </math>Where ''n'' is a non-negative integer.<ref name=":1" /> Note that the right hand side of this would instead be <math>\pi(n-1/4)</math> if n was only allowed to non-zero natural numbers. 


Thus we conclude that, for <math display="inline">n = 1,2,3,\cdots      </math><math display="block">\int_{x_1}^{x_2} \sqrt{2m \left( E-V(x)\right)}\,dx = \left(n-\frac 1 4\right)\pi \hbar </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.<ref>{{Cite book |last=Weinberg |first=Steven |url=http://dx.doi.org/10.1017/cbo9781316276105 |title=Lectures on Quantum Mechanics |date=2015-09-10 |publisher=Cambridge University Press |isbn=978-1-107-11166-0 |edition=2nd |pages=204|doi=10.1017/cbo9781316276105 }}</ref>

=== Bound states within 2 rigid wall ===
The potential of such systems can be given in the form:

<math>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} </math>

where <math display="inline">x_1 < x_2 </math>.


For <math display="inline">E \geq V(x)          </math> between <math display="inline">x_1 </math> and <math display="inline">x_2 </math> which are thus the classical turning points, by considering approximations far from <math display="inline">x_1 </math> and <math display="inline">x_2 </math> respectively we have two solutions:

<math>\Psi_{\text{WKB}}(x) = 
\frac{A}{\sqrt{|p(x)|}}\sin{\left(\frac 1 \hbar \int_{x}^{x_1} |p(x)| dx \right)}    </math>

<math>\Psi_{\text{WKB}}(x) = 
\frac{B}{\sqrt{|p(x)|}}\sin{\left(\frac 1 \hbar \int_{x}^{x_2} |p(x)| dx \right)}     </math>

Since wavefunctions must vanish at <math display="inline">x_1 </math> and <math display="inline">x_2 </math>. Here, the phase difference only needs to account for <math>n \pi</math> which allows <math>B= (-1)^n A </math>. Hence the condition becomes:

<math display="block">\int_{x_1}^{x_2} \sqrt{2m \left( E-V(x)\right)}\,dx = n\pi \hbar  </math>where <math display="inline">n = 1,2,3,\cdots      </math> but not equal to zero since it makes the wavefunction zero everywhere.<ref name=":1" />

=== Quantum bouncing ball ===
Consider the following potential a bouncing ball is subjected to:

<math>V(x) = \begin{cases}
mgx & \text{if }  x \geq 0\\
  \infty & \text{if } x < 0 \\
 \end{cases}</math>

The wavefunction solutions of the above can be solved using the WKB method by considering only odd parity solutions of the alternative potential <math>V(x) = mg|x|</math>. The classical turning points are identified <math display="inline">x_1 = - {E \over mg}   </math> and <math display="inline">x_2 = {E \over mg}    </math>. Thus applying the quantization condition obtained in WKB:

<math display="block">\int_{x_1}^{x_2} \sqrt{2m \left( E-V(x)\right)}\,dx = (n_{\text{odd}}+1/2)\pi \hbar</math>

Letting <math display="inline">n_{\text{odd}}=2n-1 </math> where <math display="inline">n = 1,2,3,\cdots      </math>, solving for <math display="inline">E </math> with given <math>V(x) = mg|x|</math>, we get the quantum mechanical energy of a bouncing ball:<ref>{{Cite book |last1=Sakurai |first1=Jun John |title=Modern quantum mechanics |last2=Napolitano |first2=Jim |date=2021 |publisher=Cambridge University Press |isbn=978-1-108-47322-4 |edition=3rd |location=Cambridge}}</ref>

<math display="block">E = {\left(3\left(n-\frac 1 4\right)\pi\right)^{\frac 2 3} \over 2}(mg^2\hbar^2)^{\frac 1 3}. </math>

This result is also consistent with the use of equation from bound state of one rigid wall without needing to consider an alternative potential.

=== Quantum Tunneling ===
{{Main|Quantum tunnelling}}
The potential of such systems can be given in the form:

<math display="block">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} </math>

where <math display="inline">x_1 < x_2 </math>. 


Its solutions for an incident wave is given as

<math display="block">\psi(x) = \begin{cases}
A \exp({ i p_0 x \over \hbar} ) + B \exp({- i p_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({ i p_0 x \over \hbar} )  & \text{if } x > x_2 \\

 \end{cases}  </math>

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">\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)    </math>

where <math>a_1 = |p(x_1)|</math> and <math>a_2 = |p(x_2)|    </math>.


Using <math display="inline">\mathbf J(\mathbf x,t) = \frac{i\hbar}{2m}(\psi^* \nabla\psi-\psi\nabla\psi^*) </math> we express the values without signs as:

<math display="inline">J_{\text{inc.}} = \frac{\hbar}{2m}(\frac{2p_0}{\hbar}|A|^2) </math>

<math display="inline">J_{\text{ref.}} = \frac{\hbar}{2m}(\frac{2p_0}{\hbar}|B|^2)  </math>

<math display="inline">J_{\text{trans.}} = \frac{\hbar}{2m}(\frac{2p_0}{\hbar}|D|^2)  </math>


Thus, the [[transmission coefficient]] is found to be:

<math display="block">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)    </math>

where <math display="inline">p(x) = \sqrt {2m( E - V(x))}        </math>, <math>a_1 = |p(x_1)|</math> and <math>a_2 = |p(x_2)|    </math>. The result can be stated as <math display="inline">T \sim ~ e^{-2\gamma} </math> where <math display="inline">\gamma = \int_{x_1}^{x_2} |p(x')| dx' </math>.<ref name=":1" />

==See also==
{{Portal|Mathematics|Physics}}
{{Div col|colwidth=20em}}
* [[Airy function]]
* [[Einstein–Brillouin–Keller method]]
* [[Field electron emission]]
* [[Instanton]]
* [[Langer correction]]
* [[Maslov index]]
* [[Method of dominant balance]]
* [[Method of matched asymptotic expansions]]
* [[Method of steepest descent]]
* [[Old quantum theory]]
* [[Perturbation methods]]
* [[Quantum tunneling]]
* [[Slowly varying envelope approximation]]
* [[Supersymmetric WKB approximation]]
{{div col end}}

==References==
{{Reflist|refs=
<ref name=Carlini-1817>{{cite book | author=Carlini, Francesco | year=1817 | title=Ricerche sulla convergenza della serie che serva alla soluzione del problema di Keplero | publisher=Milano | author-link=Francesco Carlini }}</ref>
<ref name=Liouville>{{cite journal | author=Liouville, Joseph | year=1837 | title=Sur le développement des fonctions et séries..| journal=Journal de Mathématiques Pures et Appliquées | volume=1 | pages=16–35 | author-link=Joseph Liouville }}</ref>
<ref name=Green-1837>{{cite journal | author=Green, George | year=1837 | title=On the motion of waves in a variable canal of small depth and width | journal=Transactions of the Cambridge Philosophical Society | volume=6 | pages=457–462 | author-link=George Green (mathematician) }}</ref>
<ref name=Rayleigh-1912>{{cite journal | author=Rayleigh, Lord (John William Strutt) | year=1912 | title=On the propagation of waves through a stratified medium, with special reference to the question of reflection | journal=[[Proceedings of the Royal Society A]] | volume=86 | pages=207–226 | doi=10.1098/rspa.1912.0014 |bibcode = 1912RSPSA..86..207R | issue=586 | author-link=Lord Rayleigh | doi-access=free }}</ref>
<ref name=Gans-1915>{{cite journal | author=Gans, Richard | year=1915 | title=Fortplantzung des Lichts durch ein inhomogenes Medium | journal=Annalen der Physik | volume=47 | issue=14 | pages=709–736 | doi = 10.1002/andp.19153521402 |bibcode = 1915AnP...352..709G | url=https://zenodo.org/record/1447303 | author-link=Richard Gans }}</ref>
<ref name=Jefferys-1924>{{cite journal | author=Jeffreys, Harold | year=1924 | title=On certain approximate solutions of linear differential equations of the second order | journal=Proceedings of the London Mathematical Society | volume=23 | pages=428–436 | doi=10.1112/plms/s2-23.1.428 | author-link=Harold Jeffreys }}</ref>
<ref name=Brillouin-1926>{{cite journal | author=Brillouin, Léon | year=1926 | title=La mécanique ondulatoire de Schrödinger: une méthode générale de resolution par approximations successives | journal=Comptes Rendus de l'Académie des Sciences | volume=183 | pages=24–26 | author-link=Léon Brillouin }}</ref>
<ref name=Kramers-1926>{{cite journal | author=Kramers, Hendrik A. | year=1926 | title=Wellenmechanik und halbzahlige Quantisierung | journal=Zeitschrift für Physik | volume=39 |pages=828–840 | doi=10.1007/BF01451751 |bibcode = 1926ZPhy...39..828K | issue=10–11 | s2cid=122955156 | author-link=Hendrik Anthony Kramers }}</ref>
<ref name=Wentzel-1926>{{cite journal | author=Wentzel, Gregor | year=1926 | title=Eine Verallgemeinerung der Quantenbedingungen für die Zwecke der Wellenmechanik | journal=Zeitschrift für Physik | volume=38 | pages=518–529 | doi=10.1007/BF01397171 |bibcode = 1926ZPhy...38..518W | issue=6–7 | s2cid=120096571 | author-link=Gregor Wentzel }}</ref>
}}

===Further reading===
*{{cite book | author=Child, M. S. | title=Semiclassical mechanics with molecular applications | year=1991 | publisher = Clarendon Press | location=Oxford | isbn=0-19-855654-3}}
* {{cite book| last1=Fröman|first1=N. |last2= Fröman |first2= P.-O.| title=JWKB Approximation: Contributions to the Theory | publisher=North-Holland |location = Amsterdam | year=1965}}
*{{cite book | author=Griffiths, David J. | title=Introduction to Quantum Mechanics | edition = 2nd | publisher=Prentice Hall |year=2004 |isbn=0-13-111892-7}}
*{{citation|first=Brian C.|last=Hall | title=Quantum Theory for Mathematicians | series=Graduate Texts in Mathematics| volume=267 | publisher=Springer|year=2013|bibcode=2013qtm..book.....H | isbn=978-1461471158}}
*{{cite book | author=Liboff, Richard L. | title=Introductory Quantum Mechanics | edition = 4th | publisher=Addison-Wesley | year=2003 |isbn=0-8053-8714-5| author-link=Liboff, Richard L }}
*{{cite book | author=Olver, Frank William John |author-link=Frank William John Olver | title=Asymptotics and Special Functions | url=https://archive.org/details/asymptoticsspeci0000olve | url-access=registration | publisher=Academic Press | year=1974 | isbn=0-12-525850-X}}
*{{cite book | author=Razavy, Mohsen | title=Quantum Theory of Tunneling | url=https://archive.org/details/quantumtheoryoft0000raza | url-access=registration | publisher=World Scientific | year=2003 | isbn=981-238-019-1}}

==External links==
* {{cite web| first=Richard |last=Fitzpatrick |url=http://farside.ph.utexas.edu/teaching/jk1/lectures/node70.html |title= The W.K.B. Approximation|year=2002}} (An application of the WKB approximation to the scattering of radio waves from the ionosphere.)

[[Category:Approximations]]
[[Category:Asymptotic analysis]]
[[Category:Mathematical physics]]