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

We have already shown that the ground state of the one-dimensional harmonic oscillator is not degenerate.

All eigenstates of H are also eigenstates of N = a

Assume |Φ

H|Φ

Consider a vector |Φ

a|Φ

|Φ

All vectors |Φ

The eigenvalue (n + 1) is therefore not degenerate.

We have shown:

|Φ

If |Φ

This implies that all eigenvalues are not degenerate.

A particle of mass m moves in one dimension in the
potential energy function U(x) = ½mω^{2}x^{2}.

(a) Write down the Hamiltonian abd the time-independent Schroedinger equation.

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

½ħω(-d^{2}/dz^{2} + z^{2})Ψ = EΨ.

Verify that normalized solutions to this equation are

Ψ_{0} = (mω/(πħ))^{1/4}exp(-½z^{2})
and Ψ_{1} = (4mω/(πħ))^{1/4 }
z exp(-½z^{2})

and find the corresponding eigenvalues E_{0} and E_{1}
.

(c) Suppose that at t = 0 the system was prepared in such a
way that its wavefunction was Ψ = (2mω/(πħ))^{1/4}exp(-z^{2}).

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

What is the probability that a measurement of the energy at t = 0
would yield the value of E_{1}?

(d) Would the system remain in the state Ψ = (2mω/(πħ))^{1/4}exp(-z^{2}) for subsequent times? Explain!

Solution:

A
system consists on N independent quantum mechanical harmonic oscillators of
frequency f. 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 >> ħω.

Consider a harmonic oscillator with
Hamiltonian

H = ½P^{2}/m
+ ½mω^{2}X^{2} = ħω(a^{†}a + ½).

Assume that at t = 0 the system is in the state
|Ψ(0)> such that <Ψ(0)|a|Ψ(0)> = a_{0}.

(a) Show that <X>(t) behaves as a function of time as the coordinate x
of a classical harmonic oscillator with amplitude x_{M} = (2ħ/(mω))^{1/2}|a_{0}|.

Such a classical harmonic oscillator has energy E = ½mω^{2}x_{M}^{2}
= ħω|a_{0}|^{2}.

The expectation value of the energy of the quantum-mechanical oscillator is

ħω<Ψ(0)|a^{†}a|Ψ(0)> + ½ħω.

(b) Assume that <a^{†}a>(0) = |a_{0}|^{2}

Show that
<a^{†}a>(0) = |a_{0}|^{2} or <a>(0) = a_{0} implies that
a|Ψ(0)> = a_{0}|Ψ(0)>,

and that a|Ψ(0)> = a_{0}|Ψ(0)> implies
<a^{†}a>(o) = |a_{0}|^{2} or <a>(0) = a_{0}.

The two conditions <a^{†}a>(0) = |a_{0}|^{2} or <a>(0) =
a_{0} define what is called a **quasi
classical state**.

(c) Let
|Ψ(0)> = ∑_{n}c_{n}|Φ_{n}>, H|Φ_{n}>
= (n + ½)ħω|Φ_{n}>.

For a normalized |Ψ(0)> find the expansion
coefficients c_{n} .

(d) Show that ΔXΔP = ħ/2.

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