Solving Stat Phys Probs: Classical & Quantum HO Entropy

  • Thread starter Pacopag
  • Start date
In summary, you are on the right track with your calculation of the partition function for N independent harmonic oscillators in both the classical and quantum cases. Just remember to include both the kinetic and potential energy terms for the classical case, sum over all possible energy levels for the quantum case, and use the Boltzmann factor and natural logarithm in your calculations. Good luck!
  • #1
Pacopag
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Homework Statement


I need to find ntropy S(E) for N independent HO's for both the classical and quantum cases,
using the [tex]\mu[/tex]-canoncal esemble

Homework Equations


For one classical HO
[tex]H={p^2 \over{2m}}+{m\omega^2 q^2 \over 2}[/tex]
For one quantum HO
[tex]H=\hbar \omega (n+{1\over 2})[/tex]

The Attempt at a Solution


I tried to find the partitn function for one HO.
[tex]W_1={1\over h^3}\int d^3q d^3p \delta(H-E)[/tex]
[tex]W_1={1\over h^3} [/tex] times volume of 6-sphere with radius E
[tex]W_1={1\over h^3} {{\pi^3 E^6}\over 6}[/tex]
So for N HOs
[tex]W={W_1^N \over {N!}}={1\over h^{3N}} {{\pi^{3N} E^{6N}}\over 6^N}[/tex]
Then I can use
[tex]S(E)=k ln(W)[/tex]
Can anyone verify if I'm on the right track??
 
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  • #2


Hello! It looks like you are on the right track. However, there are a few things you might want to consider. First, in the classical case, the Hamiltonian is a sum of the kinetic and potential energies, so you would need to include both terms in your calculation of the partition function. Also, in the quantum case, the energy levels are discrete and given by n+1/2, so you would need to sum over all possible energy levels instead of integrating. Additionally, for the quantum case, you would need to use the Boltzmann factor in your calculation of the partition function. Finally, for the entropy, you should use the natural logarithm instead of the base 10 logarithm. Keep up the good work!
 

What is statistical physics?

Statistical physics is a branch of physics that uses statistical methods to study the behavior of large numbers of particles. It is concerned with understanding and predicting the properties and behavior of systems made up of a large number of particles, such as gases, liquids, and solids.

What is a classical harmonic oscillator?

A classical harmonic oscillator is a system that can be described by a simple harmonic motion, where the potential energy is proportional to the square of the displacement from equilibrium. It is used to model many physical systems, such as a mass-spring system or a pendulum.

What is a quantum harmonic oscillator?

A quantum harmonic oscillator is a system that exhibits quantized energy levels, meaning that the energy of the system can only take on certain discrete values. This is in contrast to a classical harmonic oscillator, where the energy can take on any value. It is used to model quantum systems, such as atoms and molecules.

What is entropy?

Entropy is a measure of the disorder or randomness in a system. In statistical physics, it is often used to describe the number of ways in which the particles in a system can be arranged. A system with high entropy has many possible arrangements, while a system with low entropy has fewer possible arrangements.

How is entropy related to classical and quantum harmonic oscillators?

In classical harmonic oscillators, the entropy increases as the system becomes more disordered, meaning that the particles are moving at higher speeds and are more spread out. In quantum harmonic oscillators, the entropy is related to the number of energy levels that are occupied by the particles in the system. As more energy levels are occupied, the entropy increases.

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