Operator commutator (Heisenberg picture)

[AntiStrange]

Homework Statement

It's a part of a bigger problem, but what I need help with is finding the commutator between x(t) and x(t) at a different time. So basically I need [x(t1),x(t2)]

Homework Equations

$$A(t)=e^{iHt/\hbar}Ae^{-iHt/\hbar}$$

The Attempt at a Solution

The farthest I can get is writing out the commutator using the above equation:
[x(t1),x(t2)] =
$$e^{iHt1/\hbar}xe^{-iHt1/\hbar}e^{iHt2/\hbar}xe^{-iHt2/\hbar}-e^{iHt2/\hbar}xe^{-iHt2/\hbar}e^{iHt1/\hbar}xe^{-iHt1/\hbar}$$

But that's just a big mess, and I don't know how to move forward from there. Can anyone help me out?

 

[Jhero]

Hi there! I understand your struggle with finding the commutator between x(t) and x(t) at different times. The key to solving this problem is to use the Heisenberg equation of motion:

dx(t)/dt = (1/iℏ)[H,x(t)]

We can use this equation to rewrite the commutator as:

[x(t1),x(t2)] = (1/iℏ)[x(t1),x(t2)] = (1/iℏ)(dx(t1)/dt)(dx(t2)/dt)

Now, we can use the definition of the commutator to expand this further:

[x(t1),x(t2)] = (1/iℏ)(dx(t1)/dt)(dx(t2)/dt) = (1/iℏ)([x(t1),x(t2)]-[x(t2),x(t1)])

We can then substitute this back into our original equation and simplify to get:

[x(t1),x(t2)] = (1/iℏ)([x(t1),x(t2)]-[x(t2),x(t1)]) = (1/iℏ)([x(t1),x(t2)]-[x(t2),x(t1)])

This is as far as we can go without knowing the specific form of your Hamiltonian. I hope this helps you move forward with your problem!