2.4 WKB approximation

The WKB approximation is named for G. Wentzel, H.A. Kramers, and L. Brillouin, who independently developed the method in 1926. There are pre-quantum antecedents due to Jeffreys and Raleigh, though.

We can always write the one-dimensional Schrödinger equation as

where $k\left(x\right)\equiv \sqrt{2m\left(E-V\left(x\right)\right)}∕\hslash$. We could think of the quantity $k\left(x\right)$ as a spatially-varying wavenumber ($k=2\pi ∕\lambda$), though we anticipate that this can only make sense if it doesn’t change too quickly with position - else we can’t identify a wavelength at all.

Let’s see under what conditions a solution of the form

might be a good approximate solution. Plugging this into the SE above, the LHS reads $-\left(k^2\mp i{k}^{\prime }\right)\psi$. (Here and hereafter, primes denote differentiation wrt x — except when they indicate an integration variable.) So provided $\left|k^{\prime }∕k\right|\ll \left|k\right|$, or $\left|{\lambda }^{\prime }\right|\ll 1$, this is indeed a good solution as the second term can be ignored. And $\left|{\lambda }^{\prime }\right|\ll 1$ does indeed mean that the wavelength is slowly varying. (One sometimes reads that what is needed is that the potential is slowly varying. But that is not a well defined statement, because $\text{d}V∕\text{d}x$ is not dimensionless. For any smooth potential, at high-enough energy we will have $\left|{\lambda }^{\prime }\right|\ll 1$. What is required is that the lengthscale of variation of $\lambda$, or $k$, or $V$ (the scales are all approximately equal) is large compared with the de Broglie wavelength of the particle.

An obvious problem with this form is that the current isn’t constant: if we calculate it we get ${\left|A\right|}^2\hslash k\left(x\right)∕m$. A better approximation is

which gives a constant flux. (Classically, the probability of finding a particle in a small region is inversely proportional to the speed with which it passes through that region.) Furthermore one can show that if the error in the first approximation is $𝒪\left|{\lambda }^{\prime }\right|$, the residual error with the second approximation is $𝒪{\left|{\lambda }^{\prime }\right|}^2$. At first glance there is a problem with the second form when $k\left(x\right)=0$, ie when $E=V\left(x\right)$. But near these points - the classical turning points - the whole approximation scheme is invalid, because $\lambda \to \infty$ and so the potential cannot be “slowly varying” on the scale of $\lambda$.

For a region of constant potential, of course, there is no difference between the two approximations and both reduce to a plain wave, since ${\int \; <!--nolimits-->}^{x}k\left(x^{\prime }\right)\text{d}{x}^{\prime }=kx$.

For regions where $E<V\left(x\right)$, $k\left(x\right)$ will be imaginary and there is no wavelength as such. But defining $\lambda =2\pi ∕k$ still, the WKB approximation will continue to be valid if $\left|{\lambda }^{\prime }\right|\ll 1$.

Tunnelling and bound-state problems inevitably include regions where $E\approx V\left(x\right)$ and the WKB approximation isn’t valid. This would seem to be a major problem. However if such regions are short the requirement that the wave function and its derivative be continuous can help us to “bridge the gap”.

• Gasiorowicz Supplement 4 A
• Shankar ch 16.2