WKB in Spherical Coordinates

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Spherical Coordinates
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WKB in Spherical Coordinates
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It is possible to apply the WKB approximation to the radial equation using a method by R. E. Langer (1937).

Recall that

u(r)=rR(r),\!

and that u(r)\! satisfies the effective one-dimensional Schrödinger equation,

\left[ -\frac{\hbar^2}{2m}\frac{d^2}{dr^2}+\frac{\hbar^2}{2m}\frac{l(l+1)}{r^2}+V(r)-E\right] u(r)=0.

We now perform the following transformations:

\ r = e^{s}; \ u(r) = W(s)e^{\frac{1}{2}s}

Note that, for 0<r<\infty,\! -\infty<s<\infty.\! The radial equation becomes

\ \frac{d^{2}W}{ds^{2}}+\frac{2m}{\hbar^{2}}\left[E-V(e^{s})-\frac{\hbar^2}{2m}\left(l+\tfrac{1}{2}\right)^{2}e^{-2s}\right]e^{2s}=0.

In this case, the Bohr-Sommerfeld quantization rule is as in the purely one-dimensional case, but with an effective potential,

V_{\text{eff}}(r)=V(r)+\frac{\hbar^2(\ell+\frac{1}{2})^2}{2mr^2}.

Problem

Use the WKB approximation to estimate the energy spectrum for a Hydrogen atom.

Hint: Use the relation,

r^{2}-Vr+T=(r_{1}-r)(r_{2}-r),\!

where

T=-\frac{\hbar^2(l+\tfrac{1}{2})^2}{2mE},\,V=\frac{e^2}{E},

and r_1\! and r_2\! are the classical turning points of the (effective) potential appearing in the WKB approximation for this problem, and the integral,

\int_{r_1}^{r_2}\sqrt{{\frac{(x-r_1)(x-r_2)}{x^{2}}}}\,dx=\frac{\pi }{2}(\sqrt {r_2} -\sqrt {r_1} )^{2}.

Solution

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