Solve Tricky Commutator: Heisenberg Picture, a_k(t)

[IHateMayonnaise]

Homework Statement

Part of a much larger problem dealing with the Heisenberg picture. I am not remembering how to start evaluating the following commutator:

$$\left [ a_k(t),\left(\sum_{k,\ell}a_k^\dagger <k|h|\ell>a_\ell\right)\right]$$

Homework Equations

See (a)

The Attempt at a Solution

Just need some help getting started on this one, after that I'm good. What I do know is that when you do the commutator you cannot just lump the first term ($a_k(t)$) into the sum.. any hints on how to go about breaking this down? Halp!

Thanks

IHateMayonnaise

 

[gabbagabbahey]

To start with, the commutator is distributive (i.e. $[A,B+C]=[A,B]+[A,C]$ ), so you take the sum out front. Also, like any inner product, $\langle k|H|l\rangle$ will be a scalar, and so can be pulled outside the commutator...

 

[IHateMayonnaise]

Thanks for the reply!

So you said that because the commutator is distributive I can take the sum out front. Do you mean I can do this:

$$\left [ a_k(t),\left(\sum_{k,\ell}a_k^\dagger a_\ell\right)\right]=\sum_{k,\ell}<k|h|\ell> \left [ a_k(t),a_k^\dagger a_\ell\right]$$

 

[gabbagabbahey]

Yes, exactly...now keep going...simplify the commutator $\left [ a_k(t),a_k^\dagger a_\ell\right]$

 

[IHateMayonnaise]

Yes, exactly...now keep going...simplify the commutator $\left [ a_k(t),a_k^\dagger a_\ell\right]$

I think I got it from here, thank you so much for your help!

 

[gabbagabbahey]

Wait, what does the index $k$ represent in the $a_k(t)$?...If it is not being summed over you should use a different letter for the dummy index in your sum.

$$\left [ a_k(t),\left(\sum_{n,\ell} a_n^\dagger a_\ell\right)\right]=\sum_{n,\ell} \left [ a_k(t),a_n^\dagger a_\ell\right]$$