Partition Function, Grand Potential

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SUMMARY

The discussion centers on the correct application of the factor ##(2S+1)## in the expression for grand potential, specifically in relation to Eq.(30.3). Participants agree that the factor should not be included both in front of the integral and in the density of states function g(E). The consensus is that the correct formulation should utilize ##(2S+1)^2## to accurately represent the number of states for each value of k without redundancy.

PREREQUISITES
  • Understanding of statistical mechanics and grand canonical ensemble
  • Familiarity with the concept of density of states (g(E))
  • Knowledge of quantum states and their multiplicities
  • Proficiency in mathematical expressions involving integrals
NEXT STEPS
  • Review the derivation of grand potential in statistical mechanics
  • Study the implications of density of states in quantum systems
  • Examine the role of multiplicity factors in thermodynamic equations
  • Learn about common pitfalls in applying statistical mechanics equations
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Researchers in statistical mechanics, physicists working with quantum systems, and students studying thermodynamics will benefit from this discussion.

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It seems like they have missed out a factor of ##(2S+1)## in the final expression for grand potential? I'm thinking it should be ##(2S+1)^2## instead.

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I don't know, didn't they write the (2S+1) factor twice? There are 2S+1 states for each value of k, so in Eq.(30.3) you can put this factor in front of the integral, or include it in g(E), but you should not do both!
 
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Bill_K said:
I don't know, didn't they write the (2S+1) factor twice? There are 2S+1 states for each value of k, so in Eq.(30.3) you can put this factor in front of the integral, or include it in g(E), but you should not do both!

Good point. Thanks a lot
 

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