I now show some derivations regarding quantum commutators,leading to some inconsistencies. Can someone tell what went wrong? What causes the inconsistencies? and what is the correct way of understanding/handling the concepts?(adsbygoogle = window.adsbygoogle || []).push({});

Issue # 1 - Hamiltonian and commutation with time

(1) The Ehrenfest Theorem: [itex]\frac{d<A>}{dt}[/itex] = [itex]\frac{1}{i\hbar}[/itex]<[A, H] + <[itex]\frac{∂A}{∂t}[/itex]>

(2) The Heisenberg Equation: [itex]\frac{d<A(t)>}{dt}[/itex] = [itex]\frac{1}{i\hbar}[/itex]<[A(t), H] + <[itex]\frac{∂A(0)}{∂t}[/itex]>

Now, whenAorA(0) does not depend on timetexplicitly, we have the usual form [A,H] =i[itex]\hbar[/itex][itex]\frac{dA}{dt}[/itex].

Question #1:let's letA=t, therefore, [itex]\frac{dA}{dt}[/itex]=[itex]\frac{∂A}{∂t}[/itex] = 1. Plugging them in the two formulas above, we have [t,H] = 0, instead of the correct form [t,H] =i[itex]\hbar[/itex]. What's wrong?

Issue # 2 - Momemtum and position commutators

[itex]\frac{d<A>}{dx}[/itex] = ([itex]\frac{∂<|}{∂x}[/itex])A|> + <[itex]\frac{∂A}{∂x}[/itex]> + <|A([itex]\frac{∂|>}{∂x}[/itex]) = <|-[itex]\frac{i}{\hbar}p[/itex]A|> + <[itex]\frac{∂A}{∂x}[/itex]> + <|A[itex]\frac{i}{\hbar}p[/itex]|> = [itex]\frac{1}{i\hbar}[p, A][/itex] + <[itex]\frac{∂A}{∂x}[/itex]>

The correct (or usual) form is [itex][p, A][/itex] = [itex]-i\hbar[/itex][itex]\frac{∂A}{∂x}[/itex]. Getting this answer leads to my questions below.

Question #2:Does this mean that we have to force [itex]\frac{d<A>}{dx}[/itex] = 0?!

Question # 3:LetA=x, similar to issue #1, we would have [itex][p, x][/itex] = 0?! because [itex]\frac{dx}{dx}[/itex] = [itex]\frac{∂x}{∂x}[/itex] = 0.

Again, what is wrong here? Thanks for any hint or discussion.

Question #4 - a minor question:Why do we normally have H = [itex]i\hbar\frac{d}{dx}[/itex], while [itex]p = i\hbar \frac{∂}{∂x}[/itex]? In other words, why one is full derivative while the other partial? I think the answer to this question is closely related to the derivations I gave above.

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# Quantum commutator problems

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