B.11 Chapter 23
Problem 1:
For the coherent state|α〉, compute
〈α|Q|α〉and
〈α|P|α〉
Show that coherent states are not eigenstates of the number operatorN=
a†aand compute
〈α|N|α〉
Problem 2:
Show that the propagator 23.12 for the harmonic oscillator satisfies the
Schr ̈odinger equation for the harmonic oscillator Hamiltonian.
B.12 Chapters 24 to 26
Problem 1:
Consider the harmonic oscillator in two dimensions, with the Hamiltonian
H=
1
2 m(P 12 +P 22 ) +
1
2
mω^2 (Q^21 +Q^22 )There are two differentU(1) =SO(2) groups acting on the phase space of
this system as symmetries, with corresponding operators:
- The rotation action on position space, with a simultaneous rotation ac-
tion on momentum space. The operator here will be thed= 2 angular
momentum operatorQ 1 P 2 −Q 2 P 1. - Simultaneous rotations in theq 1 ,p 1 andq 2 ,p 2 planes. The operator here
will be the Hamiltonian.
For each case, the state spaceH=F 2 will be a representation of the group
U(1) =SO(2). For each energy eigenspace, which irreducible representations
(weights) occur? What are the corresponding joint eigenfunctions of the two
operators?
Problem 2:
Consider the harmonic oscillator in three dimensions, with the Hamiltonian
H=
1
2 m(P 12 +P 22 +P 32 ) +
1
2
mω^2 (Q^21 +Q^22 +Q^23 )