Question

4.5 Generalise the res sults of Section 4.4 to the case in which the level E is g , times degenerate and the level E, is g, times degenerate, and show that the Einstein coefficients satisfy the relations o ba

Solve the 4.5 problm.. The results of section 4.4 are given above

Answer 1

The only change is in equation 4.76, which becomes for degenerate states

$\frac{N_a}{N_b}=\frac{g_ae^{-\beta E_a}}{g_be^{-\beta E_b}}=\frac{g_a}{g_b}e^{-\beta \hbar\omega_{ba}}$

where for simplicity I have used

$\beta=\frac{1}{k_BT}$.

As to why, is

$N\propto g(E)e^{-\beta E}$

is based on the classical statistical mechanics. Think of partition function; the energy E term appears g(E) times.

The equations 4.75 and the modified 4.76 yield,

$\rho(\omega_{ba})=\frac{g_bA_{ab}}{g_aB_{ba}e^{\beta\hbar\omega_{ba}}-g_bB_{ab}}$

OR

$\rho(\omega_{ba})=\frac{A_{ab}/B_{ab}}{\frac{g_aB_{ba}}{g_bB_{ab}}e^{\beta\hbar\omega_{ba}}-1}$

Thus,

4.79b remains the same, but 4.79a becomes

$\frac{g_aB_{ba}}{g_bB_{ab}}=1$