- #1

bananabandana

- 113

- 5

## Homework Statement

[/B]For a general operator ## \hat{O}##, let ##\hat{O}_{mn}(t)## be defined as:

$$ \hat{O_{mn}}(t) = \int u^{*}_{m}(x,t) \hat{O} u_{n}(x,t) $$

and

$$ \hat{O_{mn}} = \int u^{*}_{m}(x) \hat{O} u_{n}(x) $$

##u_{m}## and ##u_{n}## are energy eigenstates with corresponding energy ##E_{n}## and ##E_{m}##.

From this it can be shown that: (sticking in ## \hat{O} = [\hat{A},\hat{H}] ##

$$ d\frac{[\hat{A},\hat{H}]}{dt} =(E_{n}-E_{m}) exp\bigg[-i\frac{(E_{n}-E_{m})t}{\hbar}\bigg] \bigg( -\frac{i \hat{A_{mn}} (E_{n}-E_{m})}{\hbar} + \frac{d \hat{A_{mn}}}{dt}\bigg) $$

But simultaneously, it is also possible to prove that: (as we'd expect from Ehrenfest anyhow)

$$ \frac{d\hat{A_{mn}}}{dt} = -\frac{i}{\hbar} [\hat{A_{mn}},\hat{H}] $$

Where ## \hat{H}## is the Hamiltonian, and ##\hat{A}## some general Hermitian operator.

This is an issue, since if ## [\hat{A},\hat{H}] =0 ## we have simultaneously:

$$ \frac{d \hat{A_{mn}}}{dt} = 0 $$

and

$$ \frac{d\hat{A_{mn}}}{dt} = i \frac{\hat{A_{mn}}}{\hbar} (E_{n}-E_{m}) $$

## Homework Equations

## The Attempt at a Solution

This is a problem I made up and got stuck with! I think the resolution should be fairly trivial, but I can't see it - I'd bet that there is probably a simpler way of looking at this using Dirac notation, but I don't really know that very well...

Thanks!