Problem 1:

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

#### Solution: 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†a. Assume |Φn> is not degenerate. H|Φn> = (n+ ½)ħω|Φn>,  N|Φn> = n|Φn>. Consider a vector |Φn+1> with  N|Φn+1> = (n + 1)|Φn+1>. a|Φn+1> = c |Φn>,  a†a|Φn+1> = ca†|Φn>  = (n + 1)|Φn+1>. |Φn+1> =  (c/(n + 1))a†|Φn> for any |Φn+1> for which N|Φn+1> = (n + 1)|Φn+1>. All vectors |Φn+1> are proportional to a unique vector a†|Φn>. The eigenvalue (n + 1) is therefore not degenerate. We have shown: |Φ0> is not degenerate. If |Φn> is not degenerate then |Φn=1> is not degenerate. This implies that all eigenvalues are not degenerate.

Problem 2:

A particle of mass m moves in one dimension in the potential energy function U(x) = ½mω2x2.
(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(-d2/dz2 + z2)Ψ = EΨ.
Verify that normalized solutions to this equation are
Ψ0 =  (mω/(πħ))1/4exp(-½z2)    and    Ψ1 =  (4mω/(πħ))1/4 z exp(-½z2)
and find the corresponding eigenvalues E0 and E1 .

(c)  Suppose that at t = 0 the system was prepared in such a way that its wavefunction was Ψ =  (2mω/(πħ))1/4exp(-z2).
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?
(d)  Would the system remain in the state Ψ =  (2mω/(πħ))1/4exp(-z2) for subsequent times?  Explain!

Solution:

Problem 3:

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.

#### Solution:

Problem 4:

Consider a harmonic oscillator with Hamiltonian
H = ½P2/m + ½mω2X2 = ħω(aa + ½).
Assume that at t = 0 the system is in the state |Ψ(0)> such that <Ψ(0)|a|Ψ(0)> = a0.
(a) Show that <X>(t) behaves as a function of time as the coordinate x of a classical harmonic oscillator with amplitude xM = (2ħ/(mω))1/2|a0|.
Such a classical harmonic oscillator has energy E = ½mω2xM2 = ħω|a0|2.
The expectation value of the energy of the quantum-mechanical oscillator is
ħω<Ψ(0)|aa|Ψ(0)> + ½ħω.

(b) Assume that <aa>(0) = |a0|2
Show that <aa>(0) = |a0|2 or <a>(0) = a0 implies that a|Ψ(0)> = a0|Ψ(0)>,
and that  a|Ψ(0)> = a0|Ψ(0)> implies <aa>(o) = |a0|2 or <a>(0) = a0.
The two conditions <aa>(0) = |a0|2 or <a>(0) = a0 define what is called a quasi classical state.

(c) Let  |Ψ(0)> = ∑ncnn>, H|Φn> = (n + ½)ħω|Φn>.
For a normalized |Ψ(0)> find the expansion coefficients cn .

(d) Show that  ΔXΔP = ħ/2.

#### Sulution:

Problem 5:

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