Physics GRE Q: 3D Harmonic Oscillator & Thermal Equilibrium

In summary, the question is about a three-dimensional harmonic oscillator in thermal equilibrium with a temperature reservoir at temperature T. The average total energy of the oscillator can be calculated using the Equipartition Theorem, which states that there is a kt/2 contribution to the energy from each degree of quadratic freedom in the Hamiltonian. The correct answer is D, or 3kT, and the expression for the Hamiltonian for any n-dimensional 1-particle system is <E> = skT.
  • #1
daveyman
88
0
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
 
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  • #2
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?
 
Last edited:
  • #3
I'm really not sure where to start. I guess I'll just post this in a different part of the forum then.
 
  • #4
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
 
  • #5
Well the equipartition theorem says that each Degree Of Freedom = 1/2KT. Can you think of why there would be 6 DOF?
 
  • #6
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...
 
  • #7
Ok let's go back to the 1 dimensional harmonic oscillator... How would you write its total energy? How many DOF is that?
 
  • #8
Total energy for a 1-dimensional harmonic oscillator is (1/2)kx^2, right? And wouldn't that just be one degree of freedom?
 
  • #9
Total energy of a 1D SHO is (1/2)kx^2 + (1/2)mv^2...
 
  • #11
daveyman said:
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 [itex]kt/2[/itex] contribution to the energy from each degree of quadratic freedom in the Hamiltonian. In equation form, the average total energy is [itex]<E> = skT[/itex], where s is the degrees of freedom.

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

Related to Physics GRE Q: 3D Harmonic Oscillator & Thermal Equilibrium

1. What is a 3D harmonic oscillator?

A 3D harmonic oscillator is a physical system in which the restoring force acting on a particle is directly proportional to its displacement from its equilibrium position. In 3D, this means that the particle can move in three different directions, each with its own oscillation frequency.

2. How is a 3D harmonic oscillator different from a 1D or 2D harmonic oscillator?

In a 1D harmonic oscillator, the particle can only move in one direction, while in a 2D harmonic oscillator, the particle can move in two perpendicular directions. In 3D, the particle can move in three perpendicular directions, making it a more complex system.

3. What is thermal equilibrium in the context of a 3D harmonic oscillator?

Thermal equilibrium refers to a state in which the average energy of the particles in a system is constant. In the case of a 3D harmonic oscillator, thermal equilibrium is achieved when the kinetic energy of the particle is equal to its potential energy, resulting in a stable oscillation.

4. How is the energy of a 3D harmonic oscillator related to its oscillation frequency?

The energy of a 3D harmonic oscillator is directly proportional to its oscillation frequency, meaning that the higher the frequency, the higher the energy. This occurs because as the frequency increases, the particle is oscillating at a faster rate and therefore has more kinetic energy.

5. Can a 3D harmonic oscillator ever reach a state of absolute zero temperature?

No, a 3D harmonic oscillator cannot reach absolute zero temperature, as it would require the particle to come to a complete stop, which is not possible in this type of system. However, the energy of the particle can approach zero as the temperature decreases.

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