Canonical partition function for two coupled oscillators

[mv_]

Homework Statement

A one-dimensional quantum harmonic oscillator has discrete energy levels, and its Hamiltonian can be expressed as
H(n) = (n + 1/2 )ℏω,
where n = 0, 1, 2, . . . is its quantum number and ω > 0 its characteristic frequency. Now consider the case where you have two coupled oscillators. They can interact with each other as follows:
• If their quantum numbers are different (n1 different from n2), the total energy of the
pair is Utot = U1 + U2, where U1 and U2 are the energies of each oscillator.
• If their quantum numbers are the same, (n1 = n2), Utot = U1 + U2 + ∆,
where ∆ > 0 is a constant
Consider an ensemble made by N distinguishable pairs of quantum oscillators
(the pairs are independent) in contact with a heat bath at temperature T .
1. Find the canonical partition function Z(T, N ) for this system.
2. Calculate the probability that the two oscillators in a pair have the same
quantum number

Relevant Equations

Can someone please help me with the canonical partition function?

I got the hamiltonians for n1 and n2 as (n1+1/2)hw and (n2+1/2)hw. Since the an ensemble is made by N distinguishable pairs of quantum oscillators, the general canonical partition function for the system is 1/N!((sum(-BH(n1))sum(-BH(n2))), where B is the thermodynamic beta.

I got to the step above, but not sure the work is right or not since 2 cases (n1 different from n2 and n1=n2) make me a bit confused. Can someone please check my work and let me know if I'm doing something wrong? Thanks so much

 

[TSny]

Welcome to PF!

Since the an ensemble is made by N distinguishable pairs of quantum oscillators, the general canonical partition function for the system is 1/N!((sum(-BH(n1))sum(-BH(n2))), where B is the thermodynamic beta.

This doesn't look correct.

Let the word "molecule" refer to one pair of coupled oscillators. So, the system consists of N distinguishable, noninteracting molecules.

For distinguishable molecules, should you include the factor 1/N! ?

The rest of your expression is ((sum(-BH(n1))sum(-BH(n2)). Did you intend to include some exponential functions? $$\sum_{n_1=0}^\infty e^{-\beta H(n_1)} \sum_{n_2=0} ^\infty e^{-\beta H(n_2)}$$

This expression appears to apply to a single molecule rather than to the system of N molecules. Also, how does this expression include the interaction energy ##\Delta## between the two oscillators in a molecule?

Since the molecules are noninteracting, the partition function for the system of N molecules can be expressed easily in terms of the partition function of a single molecule. So, I recommend that you first find the partition function for a single molecule.

Since the two oscillators within the molecule are coupled (i.e., they interact with each other), the partition function for the molecule will not be the product of two separate sums as in the expression above.

Start with the general definition of the partition function ##Z## for a system with discrete energies and apply it to a single molecule.

This homework problem involves detailed calculations. I believe that the only way we are going to be able to follow your work is if you type in your mathematical expressions using Latex. See Latex Guide.