# Spontaneous emission rates for two-level system

1. Apr 23, 2013

### QuantumIsHard

1. The problem statement, all variables and given/known data

Our quantum system has two levels:

ψ1 = (1/$\pi$)1/4 * e-x2/2

ψ2 = (4/$\pi$)1/4 * x * e-x2/2

The energy of the ground state ψ1 is 0; the energy of the excited state ψ2 is $\hbar$ * $\omega$0. What is the spontaneous emission rate for this system when it is in the excited state?

2. Relevant equations

I believe the equation I want to use is

A = $\frac{\omega_0^3 * |p|^2}{3*\pi*\epsilon_0*\hbar*c}$,

where the above p is drawn with a strange forward tail of sorts.

3. The attempt at a solution

The only equations for that strange p I can come up with is

p = q * $\sqrt{\frac{n * \hbar}{2*m*w}} * \deltan',n-1$ and

p = q * <n|x|n'>.

The latter seems likely right, using the two given wavelengths as n and n', but then where do the energy levels come in?

(Please excuse my Latex rustiness - I'll try to edit any lines that don't work)

Last edited: Apr 23, 2013