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Klein-Gordon Hamiltonian commutator

  • #1
25
3

Homework Statement


Consider the quantum mechanical Hamiltonian ##H##. Using the commutation relations of the fields and conjugate momenta , show that if ##F## is a polynomial of the fields##\Phi## and ##\Pi## then
##[H,F]-i \partial_0 F##

Homework Equations


For KG we have:
##H=\frac{1}{2} \int d^3\mathbf{x}(\Pi^2 + (\nabla \Phi)^2 + m^2 \Phi^2)##
##\Phi(x) = \int \frac{d^3 \mathbf{p}}{(2\pi)^3 \sqrt{2E_{\mathbf{p}}}}(a_{\mathbf{p}} e^{-ipx} + a_{\mathbf{p}}^\dagger e^{ipx} )##
##\Pi(x) = \int \frac{-i d^3 \mathbf{p}}{(2\pi)^3 }\sqrt{\frac{2E_{\mathbf{p}}}{2}}(a_{\mathbf{p}} e^{-ipx} + a_{\mathbf{p}}^\dagger e^{ipx} )##
##[\Phi(t,\mathbf{x}),\Pi(t,\mathbf{y})]=i\delta^{(3)}(\mathbf{x}-\mathbf{y})##​

The Attempt at a Solution


[/B]
I have written ##F## as a generic two variable polynomial in ##\Phi,\;\Pi## but I don't know how to tackle that commutator, any help or hint is appreciated.
 

Answers and Replies

  • #2
DEvens
Education Advisor
Gold Member
1,154
397
Start by looking at your problem statement. It sure looks like you have it typed in incorrectly, since there is no equals sign in it. What is it you are meant to be showing? The text ends with the word "then" so one would expect to have to show something equals something else.

Heh. It's funny that the forum's spell checker does not know the word commutator. It keeps trying to change it to commenter.

So you have the commutator of phi with pi. And you have the Hamiltonian in terms of phi and pi. So you should be able to work out [H,F]. You just remember your rules for integrals and "integration by parts" and things of that nature.

And then the only difficult part is relating that commutator to the time derivative you have. Assuming there is an equals sign in there some place. For that you are going to need just a little bit more information. For example, what is [H,phi]? And what is [H,pi]? Or, to put it another way, how can you get the time derivative of phi and pi?
 
  • #3
25
3
Yes you're totally right I missed an equals sign! It should be
##[H,F]=-i \partial_o F##​
 
Last edited:
  • #4
Avodyne
Science Advisor
1,396
85
First, can you compute ##[H,\Phi(\vec y,t)]##?
 
  • #5
25
3
Yes indeed I can Avodyne, actually after some work using both commutators with the Hamiltonian I managed to prove what the original question asked. Thanks for the hints guys!
 

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