I have been studying the theoretical framework of quantum mechanics in an attempt to have a working understanding of the subject, if not a comprehensive one, and I have hit upon the following stumbling block.(adsbygoogle = window.adsbygoogle || []).push({});

Now, given that the orthogonality of states is preserved with time, it is easily shown that any time evolution operator has to be unitary, and with a clever choice of notation, we can in effect, derive the abstract form of the Schrodinger equation, where

i*hbar*(d/dt)Psi = H*Psi

and H is some hermitian operator.

My question then, is how to show that this observable H is in fact the quantum mechanical analogue of the classical Hamiltonian.

There are compelling connections to classical mechanics:

(d/dt)F = {F,H}, where {F,H} is the Poisson bracket of F(x,p) with the classical Hamiltonian H(x.p)

(d/dt)<F> = -(i/hbar)<[F,H]>, where <[F,H]> is the expectation value of the commutator of the quantum mechanical analogue F of F(x,p) with the observable H.

Though these equations are very similar, they don't actually show that H is the quantum mechanical analogue of the Hamiltonian.

Any help with this would be greatly appreciated.

-James

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# Relating the time evoution operator for the state vector to the classical hamiltonian

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