WKB in Spherical Coordinates

Quantum Mechanics A

Schrödinger Equation The most fundamental equation of quantum mechanics; given a Hamiltonian ${\mathcal {H}}$, it describes how a state $|\Psi \rangle$ evolves in time.

Physical Basis of Quantum Mechanics

Basic Concepts and Theory of Motion UV Catastrophe (Black-Body Radiation) Photoelectric Effect Stability of Matter Double Slit Experiment Stern-Gerlach Experiment The Principle of Complementarity The Correspondence Principle The Philosophy of Quantum Theory

Schrödinger Equation

Brief Derivation of Schrödinger Equation Relation Between the Wave Function and Probability Density Stationary States Heisenberg Uncertainty Principle Some Consequences of the Uncertainty Principle

Operators, Eigenfunctions, and Symmetry

Linear Vector Spaces and Operators Commutation Relations and Simultaneous Eigenvalues The Schrödinger Equation in Dirac Notation Transformations of Operators and Symmetry Time Evolution of Expectation Values and Ehrenfest's Theorem

Motion in One Dimension

One-Dimensional Bound States Oscillation Theorem The Dirac Delta Function Potential Scattering States, Transmission and Reflection Motion in a Periodic Potential Summary of One-Dimensional Systems

Discrete Eigenvalues and Bound States

Harmonic Oscillator Spectrum and Eigenstates Analytical Method for Solving the Simple Harmonic Oscillator Coherent States Charged Particles in an Electromagnetic Field WKB Approximation

Time Evolution and the Pictures of Quantum Mechanics

The Heisenberg Picture: Equations of Motion for Operators The Interaction Picture The Virial Theorem

Angular Momentum

Commutation Relations Angular Momentum as a Generator of Rotations in 3D Spherical Coordinates Eigenvalue Quantization Orbital Angular Momentum Eigenfunctions

Central Forces

General Formalism Free Particle in Spherical Coordinates Spherical Well Isotropic Harmonic Oscillator Hydrogen Atom WKB in Spherical Coordinates

The Path Integral Formulation of Quantum Mechanics

Feynman Path Integrals The Free-Particle Propagator Propagator for the Harmonic Oscillator

Continuous Eigenvalues and Collision Theory

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: $\ u(r)=rR(r)$, $\left[-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial r^{2}}}+{\frac {\hbar ^{2}}{2m}}{\frac {l(l+1)}{r^{2}}}+V(r)-E\right]u(r)=0$

Now apply the transformations: $\ r=e^{s};$ $\ u(r)=W(s)e^{{\frac {1}{2}}s}$

Note that for $\ r$ varying from 0 to infinity, $\ s$ will vary from minus infinity to plus infinity.

The radial equation then transforms into:

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

In this case the Bohr-Sommerfeld quantization rule becomes:

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

?

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

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 integral

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

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.

Calculation of Gamow factor using WKB Aprroximation Method