## WKB approximation

I have a confession to make: I’m scared of approximations.

Scandalous, isn’t it? For there are few things in physics that can be solved exactly, and so if we want to ever be able to do any calculations, rather than just sitting around and writing down one PDE after another that nobody will ever solve, we simply can’t do without approximations. What kind of physics major doesn’t like approximations?

And yet, they’ve never come easily to me. In contrast to theorems and exact solutions, which make a lot of sense to me, approximations always confuse me and feel like they’re something I just have to memorize, and I’m not very good at memorizing things. This started in high school, when we were discussing double-slit diffraction patterns. In order to get an expression for the approximate spacing between the bands, you’ve got to argue that because the screen is far away, you have nearly a right triangle, and $\sin \theta \approx \tan \theta \approx \theta$, as shown here. In the end, I just memorized the formula and felt dirty about it.

In my second year of college, I took an intro quantum mechanics course. It began with a discussion of the wave-like nature of matter, the photoelectric effect, Compton scattering, bremsstrahlung, hydrogen atom energy levels, all that good stuff. Then we did the Schrödinger equation for a particle in a box, a free particle, and plenty of super annoying problems where you have a potential jump and you have to match coefficients at the discontinuity and compute transmission and reflection coefficients. At the very end of term, we were introduced to the WKB approximation for the first time. Now, the prof for this course is notoriously bad (fortunately, he no longer teaches it), so I could barely understand what was going on; in the end he just derived an approximate formula for the tunneling amplitude through a potential barrier, and said WKB wouldn’t be on the final exam. I was relieved, and hoped I’d never come across it again.

Fast forward to present. I’m taking a course called Applications of Quantum Mechanics, and it’s a thinly veiled physics course about time-dependent QM which happens to be classified as CHM (luckily for me, because I didn’t want to take any more organic courses). Naturally, the WKB approximation shows up. There’s a lengthy discussion in the text about assuming an exponential form, expanding in a power series, and then plugging it into the Schrödinger equation. It was terribly dry, so I ended up just looking at the formula. That’s when it finally made sense to me.

The WKB approximation gives the following expression for the wave function of a scattering momentum eigenstate (i.e., $E > V$) subject to a spatially varying potential $V(x)$:

$\displaystyle \psi(x) \approx \frac{A}{\sqrt{p}} e^{i \int p/\hbar \, dx} + \frac{B}{\sqrt{p}} e^{-i \int p /\hbar \, dx}$

where $A$ and $B$ are constants, and the momentum function $p$ is defined as you would expect: $p(x) = \sqrt{2m(E-V(x))}$.

In order to see why this formula makes sense, compare it to the case where $V = 0$ and we have a free particle. Here we have an exact solution for the momentum eigenstates:

$\displaystyle \psi(x) = C e^{ipx/\hbar} + D e^{-ipx/\hbar}$

From the free-particle solution we can see that as the wave travels, it picks up a phase shift proportional to its momentum. If it travels a distance $dx$, then it incurs a phase shift of $p/\hbar \, dx$.

The WKB approximation is nothing more than the extension of this to a spatially varying potential. Here $p$ is a function of $x$, so the total phase shift up to a particular point isn’t just $px/\hbar$, but has to be replaced by an integral, $\int p/\hbar \, dx$.

There’s a twist, however; in order to conserve probability, the amplitude has to vary spatially. Because we have a stationary state, the probability current has to be constant. Now, the probability current transported by the wave $A' e^{ipx/\hbar}$ is proportional to $A'^2 p$. For this to remain constant, we must have $A' = A/\sqrt{p}$.

And really, that’s all there is to it: WKB is what you get when you assume that a free particle propagating through a potential maintains basically the same form; it just doesn’t accumulate a phase shift at a constant rate now since the potential isn’t constant, and its amplitude varies in order to conserve probability (just like how a classical wave’s amplitude decreases when it passes to a denser medium). There wasn’t anything to be scared of!

(In regions where $E < V$, we instead have exponential decay of the wave function. The formula given above is still correct, but a factor of $i$ from the square root cancels the factor of $i$ already in the exponential, and you get a real argument. The factor of $(-1)^{1/4}$ in the denominator can be absorbed into the constant.)