WKB in Spherical Coordinates

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Quantum Mechanics A
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The most fundamental equation of quantum mechanics; given a Hamiltonian \mathcal{H}, it describes how a state |\Psi\rangle evolves in time.
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Summary of One-Dimensional Systems
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Charged Particles in an Electromagnetic Field
WKB Approximation
The Heisenberg Picture: Equations of Motion for Operators
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The Virial Theorem
Commutation Relations
Angular Momentum as a Generator of Rotations in 3D
Spherical Coordinates
Eigenvalue Quantization
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Free Particle in Spherical Coordinates
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Hydrogen Atom
WKB in Spherical Coordinates
Feynman Path Integrals
The Free-Particle Propagator
Propagator for the Harmonic Oscillator
Differential Cross Section and the Green's Function Formulation of Scattering
Central Potential Scattering and Phase Shifts
Coulomb Potential Scattering

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

Recall that


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,



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

Hint: Use the relation,




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}.


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