Homework 8

Problem 1:

Show that all eigenstates of the Hamiltonian of the one-dimensional harmonic oscillator are not degenerate.

Problem 2:

A particle of mass m moves in one dimension in the potential .

Write down the Hamiltonian.

Write down the time-independent Schroedinger equation.

Show that by appropriate choice of a dimension less coordinate z the Schroedinger equation can be written as

 .

Verify that normalized solutions to this equation are

,

and find the corresponding eigenvalues E0 and E1 .

Suppose that at t=0 the system was prepared in such a way that its wavefunction was .

What is the probability that a measurement of the energy at t=0 would yield the value of E0 ?

What is the probability that a measurement of the energy at t=0 would yield the value of E1 ?

Would the system remain in the state for subsequent times ? Explain !

Problem 3:

A system consists on N independent quantum mechanical harmonic oscillators of frequency n. The system is in thermal equilibrium with a reservoir at temperature T.

(a) Calculate the average energy <E> for the system.

(b) Find the heat capacity of the system when kT>>hn.

Problem 4:

Consider a harmonic oscillator with Hamiltonian

.

Assume that at t=0 the system is in the state such that .

(a) Show that <X>(t) behaves as a function of time as the coordinate x of a classical harmonic oscillator with amplitude .

Such a classical harmonic oscillator has energy .
The expectation value of the energy of the quantum-mechanical oscillator is 
.

(b) Assume that .
Show that implies that
and that  implies .

The two conditions define what is called a quasi classical state.

(c) Let  .
For a normalized find the expansion coefficients cn .

(d) Show that  .

Problem 5:

For the two-dimensional harmonic oscillator find the two lowest energy eigenstates and write down the corresponding wave functions.