Solving Statistical Mechanics PS6: Consider N Oscillators

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SUMMARY

The discussion focuses on solving problem 2(a) from the Statistical Mechanics problem set concerning N independent quantum oscillators. The key challenge is calculating the partition function for multiple oscillators, which involves summing the energy levels of each oscillator, represented as (n + 1/2)ħω. Participants emphasize that the partition function for N oscillators can be derived similarly to that of a single harmonic oscillator by using the additive property of energy and the exponential function.

PREREQUISITES
  • Understanding of quantum harmonic oscillators and their energy levels
  • Familiarity with statistical mechanics concepts, particularly partition functions
  • Knowledge of thermodynamic quantities derivation from partition functions
  • Basic proficiency in mathematical summation and exponential functions
NEXT STEPS
  • Study the derivation of the partition function for N quantum harmonic oscillators
  • Explore the relationship between energy levels and partition functions in statistical mechanics
  • Learn about the implications of the partition function on thermodynamic properties
  • Review the mathematical properties of exponential functions in the context of statistical mechanics
USEFUL FOR

Students and researchers in physics, particularly those studying statistical mechanics, thermodynamics, and quantum mechanics, will benefit from this discussion.

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Homework Statement



Trying to solve 2(a) on this problem set:

http://docs.google.com/viewer?a=v&q...3AiZT6&sig=AHIEtbSSU6Yh_bICouU_44M_Vg-XlMBYTQ

Homework Equations





The Attempt at a Solution



The thing is..I\'m not sure how to calculate the partition function.

For a single SHO the energy levels are given by (n+1/2)hbar w so we sum over n = 0 to infinty and get the partition fn from which we can derive all thermodynamic quantities..what do we sum over here? how to do we get the partition fn?
Thanks!
 
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You do the exact same thing you do for a single harmonic oscillator, but now there are N oscillators. Knowing that E = E1+E2+...+EN and e^{A+B}=e^A e^B, you can solve this problem very similarly to a single oscillator. I suggest first writing out the partition function, and you will get more help with the math.
 

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