Hamilton's equation from heisenberg equation of motion

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SUMMARY

The discussion focuses on deriving Hamilton's equations in the form of Poisson brackets from Heisenberg's equation of motion. It establishes that for any two operators \(\hat{A}\) and \(\hat{B}\), the limit as \(\hbar\) approaches zero of the commutation relation leads to the Poisson bracket formulation. This derivation is crucial for understanding the transition between quantum mechanics and classical mechanics, particularly in the context of operator algebra.

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  • Understanding of Heisenberg's equation of motion
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  • Knowledge of Poisson brackets in classical mechanics
  • Basic grasp of the role of \(\hbar\) in quantum mechanics
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jelathome
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I was wondering how to derive hamilton's equation (in the form of poisson brackets) from Heisenberg's equation of motion
 
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change the commutation to anticommutation [with some i and hbar factors], and there you go...
 
jelathome said:
I was wondering how to derive hamilton's equation (in the form of poisson brackets) from Heisenberg's equation of motion

For any two operators \hat{ A } and \hat{ B }, one can show that
\mbox{ Lim }_{ \hbar \rightarrow 0 } \frac{ 1 }{ i \hbar } [ \hat{ A } , \hat{ B } ] = \{ A ( x , p ) , B ( x , p ) \}_{ PB } .
This is proved in

https://www.physicsforums.com/showpost.php?p=1082430&postcount=7

Sam
 

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