Solve Tricky Commutator: Heisenberg Picture, a_k(t)

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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
 
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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...
 
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:

<br /> \left [ a_k(t),\left(\sum_{k,\ell}a_k^\dagger a_\ell\right)\right]=\sum_{k,\ell}&lt;k|h|\ell&gt; \left [ a_k(t),a_k^\dagger a_\ell\right]<br />
 
Yes, exactly...now keep going...simplify the commutator \left [ a_k(t),a_k^\dagger a_\ell\right]
 
gabbagabbahey said:
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!
 
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]
 
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To solve this, I first used the units to work out that a= m* a/m, i.e. t=z/λ. This would allow you to determine the time duration within an interval section by section and then add this to the previous ones to obtain the age of the respective layer. However, this would require a constant thickness per year for each interval. However, since this is most likely not the case, my next consideration was that the age must be the integral of a 1/λ(z) function, which I cannot model.
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