# Basic calculation problem with commutators

1. Jul 26, 2009

### Unkraut

1. The problem statement, all variables and given/known data
A is a Hermitian operator which commutes with the Hamiltonian: $\left[A,H\right]=AH-HA=0$

To be shown: $\frac{d}{dt}A=0$

2. Relevant equations

Schrödinger equation: $i\hbar\frac{\partial}{\partial t}\psi=H\psi$ with the Hamilton operator H.

3. The attempt at a solution
I have seen this solution on many sites:

$\frac{d}{dt}<A>=\frac{d}{dt}<\psi|A|\psi>=<\psi|\frac{\partial A}{\partial t}|\psi>+<\frac{d\psi}{dt}|A|\psi>+<\psi|A|\frac{d\psi}{dt}>=<\frac{\partial A}{\partial t}>+\frac{1}{i\hbar}<\left[ A, H\right] >=0$

I have a problem with this: $<\frac{d\psi}{dt}|A|\psi>+<\psi|A|\frac{d\psi}{dt}>=\frac{1}{i\hbar}<\left[ A, H\right] >$

Okay, obviously we have from the Schrödinger equation:
$H=i\hbar\frac{\partial}{\partial t}$
and thus
$\frac{\partial}{\partial t}=\frac{1}{i\hbar}H$
and thus
$<\frac{d\psi}{dt}|A|\psi>+<\psi|A|\frac{d\psi}{dt}>=\frac{1}{i\hbar}(<H\psi|A|\psi>+<\psi|A|H\psi>)=\frac{1}{i\hbar}<\psi|HA+AH|\psi>$

But this is not the commutator but the anti-commutator. It is plus and not minus! What did I do wrong here?
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Jul 26, 2009

### Dick

Remember <a|ib>=i*<a|b> but <ia|b>=(-i)*<a|b>.

3. Jul 26, 2009

### Unkraut

Oh, of course! :yuck:
Thank you.