P316
Fall 2006
Homework #3
1. Consider a 1-dimensional quantum simple harmonic oscillator (QSHO)
with a particle of mass m and angular frequency . :
Write down the Hamiltonian in terms of raising and lowering operators
â . â .
At t=0
( x,0) 
1
2
 0 ( x)   2 ( x)
3
3
where  n (x) are the
stationary states of the QSHO.
What is  ( x, t ) ? What is the average value of energy of this state. Does it
change with time?
2. Show that

 aˆ n dx  (n  1),
2


 aˆ 

2
n
dx  (n).

3. Calculate <x>, <x2>, <p>, <p2> for the stationary states  0  1 of an
QSHO.
4. Calculate kinetic energy expectation value <K> and potential energy
expectation value <V>, < for the stationary states  0  1 of an
QSHO.
5. Find an operator N which counts the number of quanta in a given
stationary state of the QSHO in terms of raising and lowering
operators â . â .