3-D Harmonic Oscillator

Consider a particle subject to a central force F=-kr directed towards the origin and proportional to the distance away from the origin. Then

with and .

The Hamiltonian is

The state space E can be written as a tensor product space, E=E_xE_yE_z .

H_x acts in E_x , H_y acts in E_y , and H_z acts in E_z . We know the eigenfunctions if H_i in E_i .

is an orthonormal basis for E_i .

is an orthonormal basis for E .

We have

The energy levels of the three-dimensional harmonic oscillator are denoted by , with n a non-negative integer, n=n_x+n_y+n_z. All energies except E₀are degenerate. is not degenerate.

Problems:

For the three-dimensional harmonic oscillator the energy eigenvalues are

, with n=n₁+n₂+n₃ , where n₁, n₂, n₃ are the numbers of quanta associated with oscillations along the Cartesian axes. Derive a formula for the degeneracy of the quantum state n, for spinless particles confined in this potential.

Solution:
We have n=n₁+n₂+n₃ , with n_i = 0,1,2, ... .
For a given n choose a particular n₁. Then n₂+n₃=n-n₁.
There are n-n₁+1 possible pairs {n₂,n₃}. n₂ can take on the values 0 to n-1, and for each n₂ the value of n₃ is fixed. The degree of degeneracy therefore is

In one dimension, consider two particles of mass m, coordinates x₁ and x₂, momenta p₁ and p₂, and potential energy

. Find the eigenvalues and eigenfunctions of the Hamiltonian H of the system.

Solution:
We have

Let

Then

H is the Hamiltonian of two non interacting fictitious particle of mass m in harmonic oscillator potentials with frequency respectively.
The state space E is the tensor product space E=E_GE_R .
The eigenfunctions of H are tensor product functions
.

are the eigenvalues of H.

With and the corresponding eigenfunctions are

Consider the one-dimensional problem in which an electron is placed in a harmonic oscillator potential

and at t=0 an electric field

is turned on.

(a) Write down the Hamiltonian H for t>0.
(b) Find the eigenvalues of H.
(c) Find the eigenfunctions f(x) of H.
(d) Find <x> for all eigenstates of H.

Solution:

	(a) .
	(b) , . Let us try to complete the square. . Let . Then . This is the equation for a harmonic oscillator in the absence of an electric field. .
	(c) The eigenfunctions associated with E_n are .
	(d) <x’>=0 for all eigenstates, <x>= for all eigenstates.

The one-dimensional harmonic oscillator in thermodynamic equilibrium

An important case of a statistical mixture is that of a system in thermodynamic equilibrium with a heat reservoir at temperature T. The various possible dynamical states are the eigenstates of the Hamiltonian H. The statistical weight of a given eigenstate depends upon the corresponding eigenvalue of H. It is proportional to the Boltzmann factor , in which E is the eigenvalue of H and k is the Boltzmann constant. (k=1.38´10^-23J/K)

A system in thermodynamic equilibrium is represented by the density operator

Let be the eigenstates of H.

Then , where N is a normalization constant to make the total probability equal to one.

We need

N^-1 is called the partition function Z, and we write .

Let us calculate the partition function for the harmonic oscillator.

Therefore

and

Let us calculate the average energy.

since

since.gif (1852 bytes) .

Using we find a simpler expression for .

We now have

This is Planck’s formula (to within a constant ) for the average energy of a quantized oscillator.

The energy of a classical one dimensional oscillator is . The mean energy of such an oscillator in thermodynamic equilibrium at temperature T is

Temperature	QM oscillator <H>	Classical oscillator <E>
		0
		kT
		kT

Note: For the three-dimensional harmonic oscillator in thermodynamic equilibrium the mean energy <H> is three times that of a one-dimensional oscillator with the same frequency.