WKB approximation#Application to the Schr.C3.B6dinger equation

This method is named after physicists Gregor Wentzel, Hendrik Anthony Kramers, and Léon Brillouin, who all developed it in 1926.{{harvnb|Hall|2013}} Section 15.1 In 1923, 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.{{cite book |first=Robert Balson |last=Dingle |title=Asymptotic Expansions: Their Derivation and Interpretation |publisher=Academic Press |year=1973 |isbn=0-12-216550-0 }}

Earlier appearances of essentially equivalent methods are: Francesco Carlini in 1817, Joseph Liouville in 1837, George Green in 1837, Lord Rayleigh in 1912 and Richard Gans in 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.{{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.

}}

{{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

}}

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

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

$$\backslash varepsilon\; \backslash frac\{d^ny\}\{dx^n\}\; +\; a(x)\backslash frac\{d^\{n-1\}y\}\{dx^\{n-1\}\}\; +\; \backslash cdots\; +\; k(x)\backslash frac\{dy\}\{dx\}\; +\; m(x)y=\; 0,$$

assume a solution of the form of an asymptotic series expansion

$$y(x)\; \backslash sim\; \backslash exp\backslash left[\backslash frac\{1\}\{\backslash delta\}\backslash sum\_\{n=0\}^\{\backslash infty\}\; \backslash delta^n\; S\_n(x)\backslash right]$$

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_n(x)}} in the expansion.

WKB theory is a special case of multiple scale analysis.{{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

}}

{{Cite book

| author1=Holmes, M.

| title=Introduction to Perturbation Methods, 2nd Ed

| year=2013

| publisher=Springer

| isbn=978-1-4614-5476-2

}}{{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

}}

This example comes from the text of Carl M. Bender and Steven Orszag. Consider the second-order homogeneous linear differential equation

$$\backslash epsilon^2\; \backslash frac\{d^2\; y\}\{dx^2\}\; =\; Q(x)\; y,$$

where $Q(x)\; \backslash neq\; 0$. Substituting

$$y(x)\; =\; \backslash exp\; \backslash left[\backslash frac\{1\}\{\backslash delta\}\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash delta^n\; S\_\{n\}(x)\backslash right]$$

results in the equation

$$\backslash epsilon^2\backslash left[\backslash frac\{1\}\{\backslash delta^2\}\; \backslash left(\backslash sum\_\{n=0\}^\backslash infty\; \backslash delta^nS\_\{n\}^\{\backslash prime\}\backslash right)^2\; +\; \backslash frac\{1\}\{\backslash delta\}\; \backslash sum\_\{n=0\}^\{\backslash infty\}\backslash delta^n\; S\_\{n\}^\{\backslash prime\backslash prime\}\backslash right]\; =\; Q(x).$$

To leading order in ϵ (assuming, for the moment, the series will be asymptotically consistent), the above can be approximated as

$$\backslash frac\{\backslash epsilon^2\}\{\backslash delta^2\}\; \{S\_\{0\}^\{\backslash prime\}\}^2\; +\; \backslash frac\{2\backslash epsilon^2\}\{\backslash delta\}\; S\_\{0\}^\{\backslash prime\}\; S\_\{1\}^\{\backslash prime\}\; +\; \backslash frac\{\backslash epsilon^2\}\{\backslash delta\}\; S\_\{0\}^\{\backslash prime\backslash prime\}\; =\; Q(x).$$

In the limit {{math|δ → 0}}, the dominant balance is given by

$$\backslash frac\{\backslash epsilon^2\}\{\backslash delta^2\}\; \{S\_\{0\}^\{\backslash prime\}\}^2\; \backslash sim\; Q(x).$$

So {{mvar|δ}} is proportional to ϵ. Setting them equal and comparing powers yields

$$\backslash epsilon^0:\; \backslash quad\; \{S\_\{0\}^\{\backslash prime\}\}^2\; =\; Q(x),$$

which can be recognized as the eikonal equation, with solution

$$S\_\{0\}(x)\; =\; \backslash pm\; \backslash int\_\{x\_0\}^x\; \backslash sqrt\{Q(x\text{'})\}\backslash ,dx\text{'}.$$

Considering first-order powers of {{mvar|ϵ}} fixes

$$\backslash epsilon^1:\; \backslash quad\; 2\; S\_\{0\}^\{\backslash prime\}\; S\_\{1\}^\{\backslash prime\}\; +\; S\_\{0\}^\{\backslash prime\backslash prime\}\; =\; 0.$$

This has the solution

$$S\_\{1\}(x)\; =\; -\backslash frac\{1\}\{4\}\; \backslash ln\; Q(x)\; +\; k\_1,$$

where {{math|k₁}} is an arbitrary constant.

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

$$y(x)\; \backslash approx\; c\_1\; Q^\{-\backslash frac\{1\}\{4\}\}(x)\; \backslash exp\backslash left[\backslash frac\{1\}\{\backslash epsilon\}\; \backslash int\_\{x\_0\}^x\; \backslash sqrt\{Q(t)\}\; \backslash ,\; dt\backslash right]\; +\; c\_2\; Q^\{-\backslash frac\{1\}\{4\}\}(x)\; \backslash exp\backslash left[-\backslash frac\{1\}\{\backslash epsilon\}\; \backslash int\_\{x\_0\}^x\backslash sqrt\{Q(t)\}\; \backslash ,\; dt\backslash right].$$

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

$$2S\_\{0\}^\{\backslash prime\}\; S\_\{n\}^\{\backslash prime\}\; +\; S^\{\backslash prime\backslash prime\}\_\{n-1\}\; +\; \backslash sum\_\{j=1\}^\{n-1\}S^\{\backslash prime\}\_\{j\}\; S^\{\backslash prime\}\_\{n-j\}\; =\; 0$$

for {{math|n ≥ 2}}.

The asymptotic series for {{math|y(x)}} is usually a divergent series, whose general term {{math|δⁿ S_n(x)}} starts to increase after a certain value {{math|1=n = n_max}}. Therefore, the smallest error achieved by the WKB method is at best of the order of the last included term.

For the equation

$$\backslash epsilon^2\; \backslash frac\{d^2\; y\}\{dx^2\}\; =\; Q(x)\; y,$$

with {{math|Q(x) <0}} an analytic function, the value $n\_\backslash max$ and the magnitude of the last term can be estimated as follows:{{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 pp |doi=10.1103/PhysRevD.72.104011 |bibcode = 2005PhRvD..72j4011W |s2cid=119152049 }}

$$n\_\backslash max\; \backslash approx\; 2\backslash epsilon^\{-1\}\; \backslash left|\; \backslash int\_\{x\_0\}^\{x\_\{\backslash ast\}\}\; \backslash sqrt\{-Q(z)\}\backslash ,dz\; \backslash right|\; ,$$

$$\backslash delta^\{n\_\backslash max\}S\_\{n\_\backslash max\}(x\_0)\; \backslash approx\; \backslash sqrt\{\backslash frac\{2\backslash pi\}\{n\_\backslash max\}\}\; \backslash exp[-n\_\backslash max],$$

where $x\_0$ is the point at which $y(x\_0)$ needs to be evaluated and $x\_\{\backslash ast\}$ is the (complex) turning point where $Q(x\_\{\backslash ast\})\; =\; 0$, closest to $x\; =\; x\_0$.

The number {{math|n_max}} can be interpreted as the number of oscillations between $x\_0$ and the closest turning point.

If $\backslash epsilon^\{-1\}Q(x)$ is a slowly changing function,

$$\backslash epsilon\backslash left|\; \backslash frac\{dQ\}\{dx\}\; \backslash right|\; \backslash ll\; Q^2\; ,\; ^\{\backslash text\{[might\; be\; \}Q^\{3/2\}\backslash text\{?]\}\}$$

the number {{math|n_max}} will be large, and the minimum error of the asymptotic series will be exponentially small.

File:WKB approximation example.svg

File:WKB approximation to probability density.svg

The above example may be applied specifically to the one-dimensional, time-independent Schrödinger equation,

$$-\backslash frac\{\backslash hbar^2\}\{2m\}\; \backslash frac\{d^2\}\{dx^2\}\; \backslash Psi(x)\; +\; V(x)\; \backslash Psi(x)\; =\; E\; \backslash Psi(x),$$

which can be rewritten as

$$\backslash frac\{d^2\}\{dx^2\}\; \backslash Psi(x)\; =\; \backslash frac\{2m\}\{\backslash hbar^2\}\; \backslash left(\; V(x)\; -\; E\; \backslash right)\; \backslash Psi(x).$$

The wavefunction can be rewritten as the exponential of another function {{math|S}} (closely related to the action), which could be complex,

$$\backslash Psi(\backslash mathbf\; x)\; =\; e^\{i\; S(\backslash mathbf\{x\})\; \backslash over\; \backslash hbar\},$$

so that its substitution in Schrödinger's equation gives:

$$i\backslash hbar\; \backslash nabla^2\; S(\backslash mathbf\; x)\; -\; (\backslash nabla\; S(\backslash mathbf\; x))^2\; =\; 2m\; \backslash left(\; V(\backslash mathbf\; x)\; -\; E\; \backslash right),$$

Next, the semiclassical approximation is used. This means that each function is expanded as a power series in {{mvar|ħ}}.

$$S\; =\; S\_0\; +\; \backslash hbar\; S\_1\; +\; \backslash hbar^2\; S\_2\; +\; \backslash cdots$$

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

$$(\backslash nabla\; S\_0+\backslash hbar\; \backslash nabla\; S\_1)^2-i\backslash hbar(\backslash nabla^2\; S\_0)\; =\; 2m(E-V(\backslash mathbf\; x))$$

which gives the following two relations:

$$\backslash begin\{align\}$$

(\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}

which can be solved for 1D systems, first equation resulting in:$$S\_0(x)\; =\; \backslash pm\; \backslash int\; \backslash sqrt\{\; 2m\; \backslash left(\; E\; -\; V(x)\backslash right)\; \}\; \backslash ,dx=\backslash pm\backslash int\; p(x)\; \backslash ,dx$$and the second equation computed for the possible values of the above, is generally expressed as:$$\backslash Psi(x)\; \backslash approx\; C\_+\; \backslash frac\{\; e^\{+\; \backslash frac\; i\; \backslash hbar\; \backslash int\; p(x)\backslash ,dx\}\; \}\{\backslash sqrt$$

From the condition:$$(S\_0\text{'}(x))^2-(p(x))^2\; +\; \backslash hbar\; (2\; S\_0\text{'}(x)S\_1\text{'}(x)-iS\_0\text{'}\text{'}(x))\; =\; 0$$

It follows that: $\backslash hbar\backslash mid\; 2\; S\_0\text{'}(x)S\_1\text{'}(x)\backslash mid+\backslash hbar\; \backslash mid\; i\; S\_0\text{'}\text{'}(x)\backslash mid\; \backslash ll\; \backslash mid(S\_0\text{'}(x))^2\backslash mid\; +\backslash mid\; (p(x))^2\backslash mid$

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

$$\backslash begin\{align\}$$

\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}

The first inequality can be used to show the following:

$$\backslash begin\{align\}$$

\hbar \mid S_0''(x)\mid \ll \mid(p(x))\mid^2\\

\frac{1}{2}\frac{\hbar}

\lambda \left|\frac{dV}{dx}\right| \ll \frac{|p|^2}{m}\\

\end{align}

where $|S\_0\text{'}(x)|=\; |p(x)|$ is used and $\backslash lambda(x)$ is the local de Broglie wavelength of the wavefunction. The inequality implies that the variation of potential is assumed to be slowly varying.{{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}} This condition can also be restated as the fractional change of $E-V(x)$ or that of the momentum $p(x)$, over the wavelength $\backslash lambda$, being much smaller than $1$.{{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}}

Similarly it can be shown that $\backslash lambda(x)$ also has restrictions based on underlying assumptions for the WKB approximation that:$$\backslash left|\backslash frac\{d\backslash lambda\}\{dx\}\backslash right|\; \backslash ll\; 1$$which implies that the de Broglie wavelength of the particle is slowly varying.

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|x₁}}, the term $\backslash frac\{2m\}\{\backslash hbar^2\}\backslash left(V(x)-E\backslash right)$ can be expanded in a power series,

$$\backslash frac\{2m\}\{\backslash hbar^2\}\backslash left(V(x)-E\backslash right)\; =\; U\_1\; \backslash cdot\; (x\; -\; x\_1)\; +\; U\_2\; \backslash cdot\; (x\; -\; x\_1)^2\; +\; \backslash cdots\backslash ;.$$

To first order, one finds

$$\backslash frac\{d^2\}\{dx^2\}\; \backslash Psi(x)\; =\; U\_1\; \backslash cdot\; (x\; -\; x\_1)\; \backslash cdot\; \backslash Psi(x).$$

This differential equation is known as the Airy equation, and the solution may be written in terms of Airy functions,{{harvnb|Hall|2013}} Section 15.5

$$\backslash Psi(x)\; =\; C\_A\; \backslash operatorname\{Ai\}\backslash left(\; \backslash sqrt[3]\{U\_1\}\; \backslash cdot\; (x\; -\; x\_1)\; \backslash right)\; +\; C\_B\; \backslash operatorname\{Bi\}\backslash left(\; \backslash sqrt[3]\{U\_1\}\; \backslash cdot\; (x\; -\; x\_1)\; \backslash right)=\; C\_A\; \backslash operatorname\{Ai\}\backslash left(\; u\; \backslash right)\; +\; C\_B\; \backslash operatorname\{Bi\}\backslash left(\; u\; \backslash right).$$

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

$$\backslash frac\{1\}\{\backslash hbar\}\backslash int\; p(x)\; dx\; =\; \backslash sqrt\{U\_1\}\; \backslash int\; \backslash sqrt\{x-a\}\backslash ,\; dx\; =\; \backslash frac\; 2\; 3\; (\backslash sqrt[3]\{U\_1\}\; (x-a))^\{\backslash frac\; 3\; 2\}\; =\; \backslash frac\; 2\; 3\; u^\{\backslash frac\; 3\; 2\}$$

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 $+\backslash infty$ to the classically allowed region will not agree with the function obtained by connecting the solution near $-\backslash infty$ 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 at $x=x\_1$

and $U\_1\; >\; 0$ at $x=x\_2$

.]]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 $x=x\_1$

and the second turning point, where potential is increasing over x, occur at $x=x\_2$

. 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.

$$\backslash begin\{align\}$$

\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}

For ie. decreasing potential condition or $x=x\_1$

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:{{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}}

$$\backslash begin\{align\}$$

\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}

We cannot use Airy function since it gives growing exponential behaviour for negative x. When compared to WKB solutions and matching their behaviours at $\backslash pm\; \backslash infty$, we conclude:

$A=-D=N$,

$B=C=0$ and $\backslash alpha\; =\; \backslash frac\; \backslash pi\; 4$.

Thus, letting some normalization constant be $N$, the wavefunction is given for increasing potential (with x) as:

$\backslash Psi\_\{\backslash text\{WKB\}\}(x)\; =\; \backslash begin\{cases\}$

-\frac{N}{\sqrt

\frac{N}{\sqrt

\end{cases}

For $U\_1\; >\; 0$ ie. increasing potential condition or $x=x\_2$

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 functions to be the required connection formula, we get:

$$\backslash begin\{align\}$$

\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}

We cannot use Bairy function since it gives growing exponential behaviour for positive x. When compared to WKB solutions and matching their behaviours at $\backslash pm\; \backslash infty$, we conclude:

$2B=C=N\text{'}$,

$D=A=0$ and $\backslash alpha\; =\; \backslash frac\; \backslash pi\; 4$.

Thus, letting some normalization constant be $N\text{'}$, the wavefunction is given for increasing potential (with x) as:

$\backslash Psi\_\{\backslash text\{WKB\}\}(x)\; =\; \backslash begin\{cases\}$

\frac{N'}{\sqrt

\frac{N'}{2\sqrt

\end{cases}

Matching the two solutions for region

$$\backslash int\_\{x\_1\}^\{x\_2\}\; \backslash sqrt\{2m\; \backslash left(\; E-V(x)\backslash right)\}\backslash ,dx\; =\; (n+1/2)\backslash pi\; \backslash hbar\; ,$$

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 $2\backslash pi\backslash hbar(n+1/2)$.

Either way, the condition on the energy is a version of the Bohr–Sommerfeld quantization condition, with a "Maslov correction" equal to 1/2.{{harvnb|Hall|2013}} Section 15.2

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.{{harvnb|Hall|2013}} Theorem 15.8 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.

Thus, from the two cases the connection formula is obtained at a classical turning point, $x=a$

:

$\backslash frac\{N\}\{\backslash sqrt$

and:

$\backslash frac\{N\text{'}\}\{\backslash sqrt$

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.

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

Although WKB potential only applies to smoothly varying potentials, 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.

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

$V(x)\; =\; \backslash begin\{cases\}$

V(x) & \text{if } x \geq x_1\\

\infty & \text{if } x < x_1 \\

\end{cases}

where .

Finding wavefunction in bound region, ie. within classical turning points $x\_1$ and $x\_2$, by considering approximations far from $x\_1$ and $x\_2$ respectively we have two solutions:

$\backslash Psi\_\{\backslash text\{WKB\}\}(x)\; =$

\frac{A}{\sqrt

$\backslash Psi\_\{\backslash text\{WKB\}\}(x)\; =$

\frac{B}{\sqrt

Since wavefunction must vanish near $x\_1$, we conclude $\backslash alpha\; =\; 0$. For airy functions near $x\_2$, we require $\backslash beta\; =\; -\; \backslash frac\; \backslash pi\; 4$. We require that angles within these functions have a phase difference $\backslash pi(n+1/2)$ where the $\backslash frac\; \backslash pi\; 2$ phase difference accounts for changing sine to cosine and $n\; \backslash pi$ allowing $B=\; (-1)^n\; A$.

$$\backslash frac\; 1\; \backslash hbar\; \backslash int\_\{x\_1\}^\{x\_2\}\; |p(x)|\; dx\; =\; \backslash pi\; \backslash left(n\; +\; \backslash frac\; 3\; 4\backslash right)$$Where n is a non-negative integer. Note that the right hand side of this would instead be $\backslash pi(n-1/4)$ if n was only allowed to non-zero natural numbers.

Thus we conclude that, for $n\; =\; 1,2,3,\backslash cdots$$$\backslash int\_\{x\_1\}^\{x\_2\}\; \backslash sqrt\{2m\; \backslash left(\; E-V(x)\backslash right)\}\backslash ,dx\; =\; \backslash left(n-\backslash frac\; 1\; 4\backslash right)\backslash pi\; \backslash hbar$$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.{{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 }}

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

$V(x)\; =\; \backslash begin\{cases\}$

\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}

where .

For $E\; \backslash geq\; V(x)$ between $x\_1$ and $x\_2$ which are thus the classical turning points, by considering approximations far from $x\_1$ and $x\_2$ respectively we have two solutions:

$\backslash Psi\_\{\backslash text\{WKB\}\}(x)\; =$

\frac{A}{\sqrt

$\backslash Psi\_\{\backslash text\{WKB\}\}(x)\; =$

\frac{B}{\sqrt

Since wavefunctions must vanish at $x\_1$ and $x\_2$. Here, the phase difference only needs to account for $n\; \backslash pi$ which allows $B=\; (-1)^n\; A$. Hence the condition becomes:

$$\backslash int\_\{x\_1\}^\{x\_2\}\; \backslash sqrt\{2m\; \backslash left(\; E-V(x)\backslash right)\}\backslash ,dx\; =\; n\backslash pi\; \backslash hbar$$where $n\; =\; 1,2,3,\backslash cdots$ but not equal to zero since it makes the wavefunction zero everywhere.

Consider the following potential a bouncing ball is subjected to:

$V(x)\; =\; \backslash begin\{cases\}$

mgx & \text{if } x \geq 0\\

\infty & \text{if } x < 0 \\

\end{cases}

The wavefunction solutions of the above can be solved using the WKB method by considering only odd parity solutions of the alternative potential $V(x)\; =\; mg|x|$. The classical turning points are identified $x\_1\; =\; -\; \{E\; \backslash over\; mg\}$ and $x\_2\; =\; \{E\; \backslash over\; mg\}$. Thus applying the quantization condition obtained in WKB:

$$\backslash int\_\{x\_1\}^\{x\_2\}\; \backslash sqrt\{2m\; \backslash left(\; E-V(x)\backslash right)\}\backslash ,dx\; =\; (n\_\{\backslash text\{odd\}\}+1/2)\backslash pi\; \backslash hbar$$

Letting $n\_\{\backslash text\{odd\}\}=2n-1$ where $n\; =\; 1,2,3,\backslash cdots$, solving for $E$ with given $V(x)\; =\; mg|x|$, we get the quantum mechanical energy of a bouncing ball:{{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}}

$$E\; =\; \{\backslash left(3\backslash left(n-\backslash frac\; 1\; 4\backslash right)\backslash pi\backslash right)^\{\backslash frac\; 2\; 3\}\; \backslash over\; 2\}(mg^2\backslash hbar^2)^\{\backslash frac\; 1\; 3\}.$$

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

{{Main|Quantum tunnelling}}

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

$$V(x)\; =\; \backslash begin\{cases\}$$

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}

where .

Its solutions for an incident wave is given as

$$\backslash psi(x)\; =\; \backslash begin\{cases\}$$

A \exp({ i p_0 x \over \hbar} ) + B \exp({- i p_0 x \over \hbar}) & \text{if } x < x_1 \\

\frac{C}{\sqrt

D \exp({ i p_0 x \over \hbar} ) & \text{if } x > x_2 \\

\end{cases}

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:$$\backslash frac\; \{|D|^2\}\; \{|A|^2\}\; =\; \backslash frac\{4\}\{(1+\{a\_1^2\}/\{p\_0^2\}\; )\}\; \backslash frac\{a\_1\}\{a\_2\}\backslash exp\backslash left(-\backslash frac\; 2\; \backslash hbar\; \backslash int\_\{x\_1\}^\{x\_2\}\; |p(x\text{'})|\; dx\text{'}\backslash right)$$

where $a\_1\; =\; |p(x\_1)|$ and $a\_2\; =\; |p(x\_2)|$.

Using $\backslash mathbf\; J(\backslash mathbf\; x,t)\; =\; \backslash frac\{i\backslash hbar\}\{2m\}(\backslash psi^*\; \backslash nabla\backslash psi-\backslash psi\backslash nabla\backslash psi^*)$ we express the values without signs as:

$J\_\{\backslash text\{inc.\}\}\; =\; \backslash frac\{\backslash hbar\}\{2m\}(\backslash frac\{2p\_0\}\{\backslash hbar\}|A|^2)$

$J\_\{\backslash text\{ref.\}\}\; =\; \backslash frac\{\backslash hbar\}\{2m\}(\backslash frac\{2p\_0\}\{\backslash hbar\}|B|^2)$

$J\_\{\backslash text\{trans.\}\}\; =\; \backslash frac\{\backslash hbar\}\{2m\}(\backslash frac\{2p\_0\}\{\backslash hbar\}|D|^2)$

Thus, the transmission coefficient is found to be:

$$T\; =\; \backslash frac\; \{|D|^2\}\; \{|A|^2\}\; =\; \backslash frac\{4\}\{(1+\{a\_1^2\}/\{p\_0^2\}\; )\}\; \backslash frac\{a\_1\}\{a\_2\}\backslash exp\backslash left(-\backslash frac\; 2\; \backslash hbar\; \backslash int\_\{x\_1\}^\{x\_2\}\; |p(x\text{'})|\; dx\text{'}\backslash right)$$

where $p(x)\; =\; \backslash sqrt\; \{2m(\; E\; -\; V(x))\}$, $a\_1\; =\; |p(x\_1)|$ and $a\_2\; =\; |p(x\_2)|$. The result can be stated as $T\; \backslash sim\; ~\; e^\{-2\backslash gamma\}$ where $\backslash gamma\; =\; \backslash int\_\{x\_1\}^\{x\_2\}\; |p(x\text{'})|\; dx\text{'}$.

{{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}}

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{{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 }}

{{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 }}

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• {{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