Alpha Institute for Advanced Studies

Thanks for going through the calculations. As can be seen from page 2 of note 175(14), left hand column, I followed Peter Atkins, “Molecular Quantum Mechanics”, in doing this calculation, applying the Leibniz Theorem to the product of wavefunctions psi * psi, and not to the operator. The tautology comes from the fact that:

<x> = x; x hat psi = x psi

in the position representation. In the momentum representation:

<p> = p, p hat psi = p psi

In fact one does not need the integrals to derive the quantum Hamilton equations. They follow immediately from the tautologies:

d<x> / dx = 1 ; d<p> / dp = 1.

and
[x hat, p hat] psi = i h bar psi

<[x hat, p hat]> = i h bar

So

d<x> / dx = i h bar / (i h bar) = 1 = <[x hat, p hat]> / (i h bar)

Q.E.D. Finally generalize x hat to any operator A hat to get the new equations of motion.