Editing WKB approximation (section)

== 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
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|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
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|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>