Principles of Quantum Mechanics · Problem 18.8

Time-Dependent Perturbation Theory — R. Shankar

Physics — Quantum Mechanics · time-dependent-perturbation-theory

Problem

The half-life of the

2p

state of hydrogen is

t_{1/2} = \ln 2 / A_{21}

where

A_{21}

is the Einstein

A

coefficient. (a) Find an expression for

A_{21}

in terms of the transition frequency

\omega_0

and the matrix element

|\langle 1 | \hat{r} | 2 \rangle|^2

. (b) It is sometimes convenient to characterize a transition by the oscillator strength:

f_{ij} \equiv \frac{2m_e \omega_{ij}}{3\hbar} |\langle j | \mathbf{r} | i \rangle|^2.

Show that the Thomas–Reiche–Kuhn sum rule holds:

\sum_j f_{ij} = 1.

[Hint: See Problem 3.39.]