Quantum harmonic oscillators - grand partition function

In summary, the conversation discusses calculating the grand partition function for a system of N noninteracting quantum mechanical oscillators with the same natural frequency. Two cases, Boltzmann statistics and Bose statistics, are considered. The energies of the system are given by a formula involving the number of phonons in each oscillator. The grand partition function is calculated using this formula and the function g(s), which represents the density of states for the system. The conversation also discusses the differences in calculating g(s) for the Boltzmann and Bose statistics cases, and questions whether the reasoning is correct and if the series can be summed into closed-form expressions.
  • #1
Heirot
151
0

Homework Statement



Calculate the grand partition function for a system of [tex]N[/tex] noninteracting quantum mechanical oscillators, all of which have the same natural frequency [tex]\omega_0[/tex]. Do this for the following cases: (i) Boltzmann statistics; (ii) Bose statistics.

Homework Equations





The Attempt at a Solution



The energies of the system are given by
[tex]E(\{n_i\})=\frac{N}{2}\hbar\omega_0+\hbar\omega_0\sum_{i=1}^Nn_i[/tex]
where [tex]n_i \geq 0[/tex] is the number of phonons in the i-th harmonic oscillator. For a given
[tex]s=\sum_{i=1}^Nn_i[/tex]
the grand partition function is
[tex]Z_G(\beta,\mu)=e^{-\beta\frac{N\hbar\omega_0}{2}}\sum_{s=0}^{\infty}g(s)e^{-\beta s (\hbar \omega_0 - \mu)}[/tex]
The function [tex]g(s)[/tex] represents density of states (degeneracy) of the bosonic system, and I have a hard time calculating it.

For Boltzmann statistics, the oscillators are distinguishable and the degeneracy should be equal to the number of ways one can partition s identical objects into N different boxes, e.g.
[tex]g(s)=\frac{(s+N-1)!}{s!}[/tex]
On the other hand, for Bose statistics, the oscillators (boxes) are now indistinguishable and one has
[tex]g(s)=\frac{(s+N-1)!}{s!(N-1)!}[/tex]

My question is, is this reasoning correct? If so, can I sum the series into closed-form expressions?

Thank you.
 
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  • #2
It looks as though I was wrong in the last post. E.g. for N=3 oscillators and s=3 phonons, one has:

Boltzmann case:
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10 different possibilities, but only

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3 in Bose Einstein case.

Is this reasoning correct?
 

Related to Quantum harmonic oscillators - grand partition function

1. What is a quantum harmonic oscillator?

A quantum harmonic oscillator is a physical system that exhibits harmonic motion, similar to a mass attached to a spring. However, in quantum mechanics, the oscillator's energy is quantized, meaning it can only take on discrete values.

2. What is the grand partition function for a quantum harmonic oscillator?

The grand partition function for a quantum harmonic oscillator is a mathematical expression that describes the statistical behavior of a system of oscillators in thermal equilibrium with a particle reservoir, taking into account both quantum and thermal effects.

3. How is the grand partition function used in quantum mechanics?

The grand partition function is a fundamental concept in statistical mechanics and is used to calculate the average properties of a system of quantum harmonic oscillators, such as the average energy and number of particles in the system at a given temperature.

4. What are the assumptions made in the grand partition function for quantum harmonic oscillators?

The grand partition function assumes that the oscillators are non-interacting and that the energy levels are evenly spaced. It also assumes that the particles in the system are indistinguishable and follow Bose-Einstein statistics.

5. How does the grand partition function change with temperature in a quantum harmonic oscillator system?

The grand partition function increases with temperature, meaning that as the temperature increases, there is a higher probability of finding particles in higher energy states. This is due to the thermal energy of the system being able to overcome the energy barrier between energy levels.

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