Prove Van Leeuwen's Theorem: Diamagnetism Does Not Exist in Classical Physics

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SUMMARY

Van Leeuwen's theorem establishes that diamagnetism does not exist in classical physics. The discussion focuses on the transformation of the Hamiltonian, where the Hamiltonian Ha changes to Ha(p1-(e/c)A1, p2-(e/c)A2, ..., pN-(e/c)AN, q1, q2, ..., qN) under an external magnetic field H. The induced magnetization M is derived from the equation M = kT*d(log QN)/dH, where QN represents the partition function. The challenge lies in calculating QN when the Hamiltonian is altered, which can be addressed by changing integration variables and simplifying the integral.

PREREQUISITES
  • Understanding of classical statistical mechanics
  • Familiarity with Hamiltonian mechanics
  • Knowledge of partition functions in thermodynamics
  • Basic concepts of magnetism and induced magnetization
NEXT STEPS
  • Study the derivation of the partition function QN in the context of magnetic fields
  • Explore Hamiltonian dynamics and its applications in statistical mechanics
  • Learn about the implications of Van Leeuwen's theorem on classical and quantum systems
  • Investigate techniques for changing integration variables in complex integrals
USEFUL FOR

This discussion is beneficial for physics students, particularly those studying statistical mechanics, as well as researchers exploring the foundations of magnetism in classical physics.

Henk
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For a statistical mechanics course we have to prove Van Leeuwen's theorem: Diamagnetism does not exist in classical physics.

I know that in an external magnetic field H the Hamiltonian Ha goes from Ha(p1,p2,--------,pN ,q1,q2,-------qN) to Ha(p1-(e/c)A1, p2-(e/c)A2, ------- pN-(e/c)AN, q1,q2,------.qN)

I also know that the induced magnetization M = kT*d(log QN)/dH

So the problem is finding QN. I know how to calculate it for a perfect gas without the magnetic field but I can't seem to solve the integral when Ha changes.
 
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Try shifting integration variables.
 
Ah I found bij changing p - (e/c)A to p' and then integrating. Thanks a lot.
 

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