WKB Approximation

WKB method (Wentzel-Kramers-Brillouin method) is a technique for finding approximations to certain differential equations, including the one dimensional Schrodinger equation. It was developed in 1926 by Wenzel, Kramers, and Brillouin, for whom it was named. The logic is that as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar\rightarrow 0\!} , the wavelength, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda=2\pi\hbar/ p\!} , tends to zero where the potential is smooth and slowly varying. Therefore, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda\!} can be thought of as a local quantity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda(x)\!} . This is a quasi-classical method of solving the Schrodinger equation.

In WKB, for a certain turning point, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda\!} is infinite, we cannot say that the potential changes slowly. Here, the whole theory is bound to fail, a proper handling of the turning points is the most difficult aspect of the WKB approximation. The potential at the turning point is approximated as linear and slowly varying (almost constant).

Through this way,

1. We can at least solve the Schrodinger equation and obtain the wave function in this "region". We can use this wave function to connect the WKB wave functions at the two sides of the turning point.

2. We also need to know the region where the WKB wave functions are valid.

For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(x) = 0\!} (or constant) the solutions to the Schrodinger equation are simply plane waves of the form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{\pm ikx} } . If the potential varies smoothly, and the energy of the particle is fixed, the wave function can be described locally by writing its plane wave form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{iu(x)} \!} .

The WKB solutions to the Schrodinger equation for a particle in a smoothly varying potential are given by:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(x)\approx \frac{C_\pm}{\sqrt{p(x)}}\exp\left[\pm \frac{i}{\hbar}\int_{x_0}^x p(x')dx'\right]}

for a classically accessible region where E > V(x) and p(x) is real,

and written:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(x)\approx \frac{C_+}{\sqrt{|p(x)|}}\exp\left[\frac{1}{\hbar}\int_{x_0}^x |p(x')|dx'\right] + \frac{C_-}{\sqrt{|p(x)|}}\exp\left[- \frac{1}{\hbar}\int_{x_0}^x |p(x')|dx'\right]}

for a classically inaccessible region where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E < V(x)\!} .

In both cases Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x)\!} is the classical formula for the momentum of a particle with total energy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E\!} and potential energy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(x)\!} given by:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x)=\sqrt{2m(E-V(x))}\!}

This is an exact solution if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(x)\!} is constant, otherwise it's a local solution for a locally defined wavelength. There must be a condition on the region in space where the wavelength is locally defined to be sure the wavelength does not vary too much and the locally defined wave function is a valid approximation. This condition is that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta \lambda (x) \ll 1} , which is equivalent to writing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| \frac{d\lambda}{dx}\right| = \left|\frac{d}{dx}\left(\frac{\hbar}{p(x)}\right)\right| \ll 1 } .

For example, suppose there is a point, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = a \!} , which is a classical turning point at a given value of E and thereby separates the regions where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E > V(x)\!} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E < V(x)\!} . Let the classically inaccessible region be to the right of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \!} . In the region appropriately close to the turning point, the wave functions can be written as:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(x)\approx \frac{A}{\sqrt{|p(x)|}}\exp\left[- \frac{1}{\hbar}\int_{a}^x |p(x')|dx'\right]+ \frac{B}{\sqrt{|p(x)|}}\exp\left[\frac{1}{\hbar}\int_{a}^x |p(x')|dx'\right] }

for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \gg a \! } , and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(x)\approx \frac{C}{\sqrt{p(x)}}\exp\left[- \frac{i}{\hbar}\int_{a}^x p(x')dx'\right]+ \frac{D}{\sqrt{p(x)}}\exp\left[\frac{i}{\hbar}\int_{a}^x p(x')dx'\right] }

for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \ll a \!} .

Note that at the classical turning point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(a) = 0} and the WKB solution diverges, which means it is no longer a valid approximation because the true wave function will not exhibit such divergent behavior at the turning points. Thus, around each turning point we need to splice the two WKB solutions on either side of the turning point with a "patching" function that will straddle each turning point. Because we only need a solution for this function in the vicinity of the turning points, we can approximate the potential as being linear. If we center the turning point at the origin (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = 0 \!} ) we have:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(x)\approx E+V'(0)x\!}

Solving the Schrodinger equation with our now linearized potential leads to the Airy equation whose solutions are Airy functions. Our patching wave function is then:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_p(x)= C_1 \mathrm{Ai}\left(\alpha x\right)+ C_2 \mathrm{Bi}\left(\alpha x\right)\!}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_1, C_2\!} are c-number coefficients and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha=\left(\frac{2m}{\hbar^2}V'(0)\right)^{\frac{1}{3}}}

The key to patching the wavefunction in the region of the turning point is to asymptotically match the patching wavefunction to the wavefunctions outside the region of the classical turning point. In the vicinity of the classical turning point,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p^2=2m(-V'(0)x)\Rightarrow 2p\frac{dp}{dx}=-2mV'(0)\Rightarrow \frac{dp}{dx}=-\frac{m}{p}V'(0) }

Since the region of applicability of the WKB approximation is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1\gg\frac{1}{2\pi}\left|\frac{d\lambda}{dx}\right|}

near the turning point

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |p|^3 \gg \hbar m|V'(0)|\Rightarrow |x|\gg \frac{\hbar^{\frac{2}{3}}}{2}|mV'(0)|^{-\frac{1}{3}} }

This implies that the width of the region around the classical turning point vanishes as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar^{\frac{2}{3}}} . Thus, we can come as close to the turning point as we wish with the WKB approximations by taking a limit as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar\!} approaches zero, as long as the distance from the classical turning point is much less than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar^{\frac{2}{3}}} . Thus, by extending the patching function towards singularity in the direction of the WKB approximated wavefunction, while simultaneously extending the WKB approximated wavefunction toward the classical turning point, it is possible to match the asymptotic forms of the wavefunctions from the two regions, which are then used to patch the wavefunctions together.

This means that it would be useful to have a form of the Airy functions as they approach positive or negative infinity:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z\rightarrow \infty: \mathrm{Ai}(z)\rightarrow \frac{1}{2\sqrt{\pi}}z^{-\frac{1}{4}}e^{-\frac{2}{3}|z|^{\frac{3}{2}}}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z\rightarrow \infty: \mathrm{Bi}(z)\rightarrow \frac{1}{\sqrt{\pi}}z^{-\frac{1}{4}}e^{\frac{2}{3}|z|^{\frac{3}{2}}}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z\rightarrow -\infty: \mathrm{Ai}(z)\rightarrow \frac{1}{\sqrt{\pi}}z^{-\frac{1}{4}}\cos\left(\frac{2}{3}|z|^{\frac{3}{2}}-\frac{\pi}{4}\right)}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z\rightarrow -\infty: \mathrm{Bi}(z)\rightarrow -\frac{1}{\sqrt{\pi}}z^{-\frac{1}{4}}\sin\left(\frac{2}{3}|z|^{\frac{3}{2}}-\frac{\pi}{4}\right)}

And noticing that in the vicinity of the turning point (for negative Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\!} ):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{\hbar}\int_{x}^0p(x')dx'=\sqrt{\frac{2mV'(0)}{\hbar^2}} \int_x^0\sqrt{-x'}dx'=\frac{2}{3}\sqrt{\frac{2mV'(0)}{\hbar^2}}|x|^{\frac{3}{2}}=\frac{2}{3}|\alpha x|^{\frac{3}{2}} }

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{\sqrt{p(x)}}=\left(2mV'(0)\right)^{-\frac{1}{4}}|x|^{-\frac{1}{4}} }

it becomes apparent that our WKB approximation of the wavefunction is the same as the patching function in the asymptotic limit. This must be the case, since as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar\rightarrow 0} the region of invalidity of the semiclassical wavefunction in the vicinity of the turning point shrinks, while the solution of the linarized potential problem depends only on the accuracy of the linearity of the potential, and not on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar} . The two regions must therefore overlap.

For example, one can take Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\approx \hbar^{\frac{1}{3}}} and then take the limit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar\rightarrow 0} . The semiclassical solution must hold as we are always in the region of its validity and so must the solution of the linearized potential problem. Note that the argument of the Airy functions at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\approx \hbar^{\frac{1}{3}}} goes to $\pm \infty$, which is why we need their asymptotic expansion.

For a classical turning point x=a, which separate the classical inaccessible region x>a from the classical accessible region x<a, by comparing the WKB wave functions with the asymptotic expressions of the Airy functions, we find that the WKB wave functions on the two sides of the turning point x=a are connected as:

${\frac {2A}{\sqrt {k(x)}}}\cos(\int _{x}^{a}k(x)dx-{\frac {\pi }{4}})-{\frac {B}{\sqrt {k(x)}}}\sin(\int _{x}^{a}k(x)dx-{\frac {\pi }{4}})\longleftrightarrow {\frac {A}{\sqrt {|k(x)|}}}exp(-\int _{a}^{x}|k(x)|dx)+{\frac {B}{\sqrt {|k(x)|}}}exp(\int _{a}^{x}|k(x)|dx)$

On the other hand, for a classical turning point x=b, which separate the classical accessible region x>b from the classical inaccessible region x<b, by comparing the WKB wave functions with the asymptotic expressions of the Airy functions, we find that the WKB wave functions on the two sides of the turning point x=b are connected as:

${\frac {2A}{\sqrt {k(x)}}}\cos(\int _{b}^{x}k(x)dx-{\frac {\pi }{4}})-{\frac {B}{\sqrt {k(x)}}}\sin(\int _{b}^{x}k(x)dx-{\frac {\pi }{4}})\longleftrightarrow {\frac {A}{\sqrt {|k(x)|}}}exp(-\int _{x}^{b}|k(x)|dx)+{\frac {B}{\sqrt {|k(x)|}}}exp(\int _{x}^{b}|k(x)|dx)$

Bohr-Sommerfeld Quantization Rule

The quantized energy levels of a bound state can be approximated by the WKB method with an expression known as the Bohr-Sommerfeld quantization rule. A particle in a potential well is subject to bound states. This common example of the WKB method can be found in most undergraduate level quanum texts.

For a potential with no rigid walls the Bohn-Sommerfeld Quantization rule is:

$\oint p(x)dx=2\int _{x_{1}}^{x_{2}}{\sqrt {2m(E_{n}-V(x))}}dx=\left(n+{\frac {1}{2}}\right)2\pi \hbar$

For a potential with one rigid wall:

$\int _{x_{1}}^{x_{2}}{\sqrt {2m(E_{n}-V(x))}}dx=\left(n+{\frac {3}{4}}\right)\pi \hbar$

For a potential with two rigid walls:

$\int _{x_{1}}^{x_{2}}{\sqrt {2m(E_{n}-V(x))}}dx=n\pi \hbar$

For a central potential:

$p_{r}^{2}=E-V(r)-{\frac {\hbar ^{2}}{2m}}{\frac {\ell (\ell +1)}{r^{2}}}$

${\begin{aligned}\int _{r_{1}}^{r_{2}}p_{r}(r)dr&=\int _{0}^{\infty }{\sqrt {2m\left(E_{n}-V(r)-{\frac {\hbar ^{2}}{2m}}{\frac {\ell (\ell +1)}{r^{2}}}\right)}}dr\\&=\left(n+{\frac {1}{2}}\right)\pi \hbar \end{aligned}}$

[ Worked Problem - WKB energy spectrum]

sample problem Worked by team

WKB method for the Coulomb Potential

For the coulomb potential, the potential is given by:

$V(r)=-{\frac {-Ze^{2}}{r}}$

Since the electron is bound to the nucleus, it can be veiwed as moving between two rigid walls at $r=0\!$ and $r=a\!$ with energy $E=V(a),a=-{\frac {-Ze^{2}}{E}}\!$. Thus, the energy of the electron is negative.

The energies of the s-state ($\ell =0\!$) can be obtained from:

$\int _{0}^{a}{\sqrt {2m\left(E+{\frac {Ze^{2}}{r}}\right)}}dr=n\pi \hbar$

Using the change of variable: $x={\frac {a}{r}}$

${\begin{aligned}\int _{0}^{a}{\sqrt {2m\left(E+{\frac {Ze^{2}}{r}}\right)}}dr&={\sqrt {-2mE}}\int _{0}^{a}dr{\sqrt {{\frac {a}{r}}-1}}\\&=a{\sqrt {-2mE}}\int _{0}^{1}dx{\sqrt {{\frac {1}{x}}-1}}\\&={\frac {\pi }{2}}a{\sqrt {-2mE}}\\&=-Ze^{2}\pi {\sqrt {-{\frac {2m}{E}}}}\end{aligned}}$

Where I have used the integal

$\int _{0}^{1}{\sqrt {{\frac {1}{x}}-1}}={\frac {\pi }{2}}$

Thus we have the expression:

$-Ze^{2}\pi {\sqrt {-{\frac {2m}{E}}}}=n\pi \hbar$

$\Rightarrow E_{n}=-{\frac {mZ^{2}e^{4}}{\hbar ^{2}}}=-{\frac {Z^{2}e^{2}}{2a_{0}}}$

Where $a_{0}\!$ is the Bohr radius. Notice that this is the correct expression for the energy levels of a Coulomb potential.

of Gamow factor using WKB Aprroximation Method