The Quantum Hamilton Equations

Feed: Dr. Myron Evans
Posted on: Monday, March 07, 2011 2:41 AM
Author: metric345
Subject: The Quantum Hamilton Equations

To Dr. Horst Eckardt:

Thanks as usual for going through this note. I will give the final proof with more detail in the next note. I followed Atkins in the definition of hermiticity, his eqs. (5.2.1) to (5.2.6) of the second edition. The result is right because it is self checking. I will go through the Atkins definition and expand it. Atkins writes that his eq. (5.2.2) is obtained from the basic definition of hermiticity, his eq. (5.2.1), by taking the complex conjugate of each term (page 88 of the second edition of “Molecular Quantum Mechanics”, it should also be in your edition). It is right because he uses it in deriving the time evolution equation of quantum mechanics, his eq. (5.5.2), second edition. The other quantum Hamilton equation is obtained using the momentum representation instead of the position representation, and the tautology:

d<p> / dp = 1

The Hamilton equations are the expectation values of the quantum Hamilton equations. So this result shows that x and p can be observable simultaneously in quantum mechanics and that the whole of Copenhagen is incorrect. This is because x and p are simultaneously observable in the classical Hamilton equations, derivable from the quantum Hamilton equations. I am not quite satisfied with the proof as yet, but the result is right, self checking in at least two ways. The final version of the proof will be sent in the next note.

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