2. Use the creation (¹) and destruction (â) operators that we used for simple harmonic oscillator

in the energy basis to solve the following problems:

a) (4 points) Prove that the state vector Iz), defined as:

|z) = exp[zâ¹] 10).

is an eigenstate of the destruction operator â.

In the above equation, z is an arbitrary complex number. [0) is the usual ground state of the

simple harmonic oscillator, where â 10) = 0.

Hint: You need to expand the exponential into a Taylor-series:

exp(x)=1+x+x 3!

b) (2 points) Does the creation operator at also have eigenstates? If so, write one down. If not,

explain why not.

c) (3 points) Evaluate (2₁122). Use the result to normalize the state [z). Now evaluate the

expectation value of the number operator :

(z|N|z)

(N)

d) (3 points) The simple harmonic oscillator Hamiltonian can be written as: A = hwa¹a = hwÑ

if, for simplicity, we drop the additive constant which represents the contribution of zero-point

energy. Show that the time-evolution of the state (2) is given by:

0(t)|z) = |ze-lot)

Hint: Show e-t|n) = e-n first, where [n) is an eigenstate of the simple harmonic

oscillator.