Physics 742 – Graduate Quantum Mechanics 2
Solutions to Chapter 18
1. [10] An electron is trapped in a 3D harmonic oscillator potential, H  P 2 2m  12 m02 R 2 .
It is in the quantum state nx , ny , nz  2,1, 0 .
(a) [5] Calculate every non-vanishing matrix element of the form nx , ny , nz R 2,1, 0
where the final state is lower in energy than the initial state.
The three coordinate operators can be written in the form Ri   2m0  ai  ai†  , so
R 210 


 xˆ  ax  ax†   yˆ  a y  a †y   zˆ  az  az†   210

2m0 
 
xˆ 2 110  xˆ 3 310  yˆ 200  yˆ 2 220  zˆ 211  .
2m0 
We are interested in decay, which implies we want a lower energy state, so there are only two
possible final states that work
110 R 210  xˆ

m0
and
200 R 210  yˆ

2m0
(b) [5] Calculate the decay rate   210  nx , ny , nz  for this decay in the dipole
approximation for every possible final state, and find the corresponding branching
ratios.
The decay rate is given by   I  F   43 IF3 rFI
2
c 2 . In each case, the frequency
difference is simply 0 , so we have
  2,1, 0  1,1, 0  
403 
4 02
403 
2 02
,
2,1,
0
2,
0,
0
.







3c 2 m0
3mc 2
3c 2 2m0
3mc 2
2 02
. The branching ratios are
The total decay rate is  
mc 2
BR  2,1, 0  1,1, 0  
4 02 mc 2
2 02 mc 2
2
1
,
2,1,
0
2,
0,
0





 .
BR


2
2
2
2
3mc 2 0 3
3mc 2 0 3