Non-Interacting Particles interact with a Potential

[Lyons_63]

Homework Statement

Non Interacting Particles Interact with a external harmonic Potential. What are the energy levels of the system, and the partition functions when assuming the particles are (b) Bosons and (c) Fermions

Homework Equations

Energy of the system
E=(ρ1)^2/2m + (ρ2)^2/2m+ mω^2/2 (x1+x2)

ρ= momentum
ω=angular frequency of the system

The Attempt at a Solution

The energy levels for a single oscillator are given by E=hbar ω (n + 1/2)
I am not sure to go from here and how to incorperate the fact that there are two particles in the system

Any help would be great!

 

[anathema]

Thank you for your question. In order to determine the energy levels of the system, we need to consider the energy of each individual particle as well as the interaction with the external harmonic potential. The energy of a single particle in the system can be written as:

E1 = ρ1^2/2m + mω^2/2 x1

Similarly, the energy of the second particle can be written as:

E2 = ρ2^2/2m + mω^2/2 x2

To find the total energy of the system, we simply add these two energies together:

E = E1 + E2 = (ρ1^2 + ρ2^2)/2m + mω^2/2 (x1 + x2)

Since we are assuming that the particles are non-interacting, we can treat them as independent systems. Therefore, the energy levels of the system will be the sum of the energy levels of each individual particle. This means that the energy levels for the two-particle system will be:

E = E1 + E2 = E1 + E1 = 2E1

Substituting the energy of a single oscillator (E1 = hbar ω (n + 1/2)) into this equation, we get:

E = 2hbar ω (n + 1/2)

This is the total energy of the system, which is now in terms of the energy levels of a single oscillator. To find the partition function, we need to consider the possible energy states of the system. For bosons, the energy states are:

E = 0, 2hbar ω, 4hbar ω, ...

Therefore, the partition function for bosons can be written as:

Z = Σe^(-βE) = 1 + e^(-2βhbar ω) + e^(-4βhbar ω) + ...

Similarly, for fermions, the energy states are:

E = hbar ω, 3hbar ω, 5hbar ω, ...

Thus, the partition function for fermions can be written as:

Z = Σe^(-βE) = e^(-βhbar ω) + e^(-3βhbar ω) + e^(-5βhbar ω) + ...

I hope