Average energy of a 3D harmonic oscillator in thermal equilibrium

[daveyman]

I'm studying for the Physics GRE and I came across this question. The correct answer is D, but I'm not sure quite how to do it. Any help would be greatly appreciated!

A three-dimensional harmonic oscillator is in thermal equilibrium with a temperature reservoir at temperature T. The average total energy of the oscillator is

(A) (1/2)kT
(B) kT
(C) (3/2)kT
(D) 3kT
(E) 6kT

 

[G01]

You need to show work in order to get help here. What have you tried so far? What concepts, etc. apply? What are your thoughts?

 

[daveyman]

I'm really not sure where to start. I guess I'll just post this in a different part of the forum then.

 

[daveyman]

I'm studying for the Physics GRE and I came across this question. The correct answer is D, but I'm not sure quite how to do it. Any help would be greatly appreciated!

A three-dimensional harmonic oscillator is in thermal equilibrium with a temperature reservoir at temperature T. The average total energy of the oscillator is

(A) (1/2)kT
(B) kT
(C) (3/2)kT
(D) 3kT
(E) 6kT

 

[nicksauce]

Well the equipartition theorem says that each Degree Of Freedom = 1/2KT. Can you think of why there would be 6 DOF?

 

[daveyman]

That's exactly what I was thinking. It seems like the problem basically states that there are 3 degrees of freedom, which would yield an answer of (3/2)kT. However, the answer guide clearly says that the answer is 6kT (this is from an official GRE practice test).

I can't think of why there would be 6 degrees of freedom...

 

[nicksauce]

Ok let's go back to the 1 dimensional harmonic oscillator... How would you write its total energy? How many DOF is that?

 

[daveyman]

Total energy for a 1-dimensional harmonic oscillator is (1/2)kx^2, right? And wouldn't that just be one degree of freedom?

 

[nicksauce]

Total energy of a 1D SHO is (1/2)kx^2 + (1/2)mv^2...

 

[daschaich]

Here's a place to start: http://en.wikipedia.org/wiki/Equipartition_theorem

 

[Reshma]

I'm studying for the Physics GRE and I came across this question. The correct answer is D, but I'm not sure quite how to do it. Any help would be greatly appreciated!

A three-dimensional harmonic oscillator is in thermal equilibrium with a temperature reservoir at temperature T. The average total energy of the oscillator is

(A) (1/2)kT
(B) kT
(C) (3/2)kT
(D) 3kT
(E) 6kT

Acording to the Equipartition Theorem, there is a $kt/2$ contribution to the energy from each degree of quadratic freedom in the Hamiltonian. In equation form, the average total energy is $<E> = skT$, where s is the degrees of freedom.

What is the expression for the Hamiltonian for any n-dimensional 1-particle system?