Q3 A one-dimensional harmonic oscillator has the quantum Hamiltonian

H =1

2mp2 +

1

2m2x2, [x, p] = i~.

Recall that the normalized energy eigenstates are given by |n = (a)n|0/n!where

a =

m

2~x i

1

2~mp, a =

m

2~x+ i

1

2~mp,

and |0 is the ground state, which obeys a|0 = 0.

a) Compute the commutators [a, a], [H, a] and [H, a], and hence evaluatethe time-dependent operator

a(t)def= eiHt/~ a eiHt/~

in terms of a, m, and ~.

b) Given a complex parameter , we define the corresponding coherentstate | to be

| = e||2/2n=0

nn!|n.

Show that | is a normalized eigenstate of a. What is the associatedeigenvalue?

c) If the system is in the coherent state |, what is the probability thatthe system is in the energy eigenstate |n?

d) If the system is the state | at time t = 0, show that at time t it is inanother coherent state |. How is the parameter related to ?

e) Compute X = |x| and P = |p|. Then, using your results fromparts (a),(b) and (d), evaluate X(t), P (t). Draw a figure showing thetrajectory of the point (X(t), P (t)) in the X,P plane, and relate yourplot to some aspects of the classical harmonic oscillator.