I feel like there is a way to manipulate the terms so as to put them back into a known relationship but I am not seeing it
I know that
[X_aH (t), X_bH (0)] = [U_d (t,0) X(t)_aH U (t,0),U_d (0,0) X (0)_bH U (0,0)] = U_d (t,0) X(t)_aH U (t,0) X (0)_b - X (0)_b U (t,0)_d X (t)_aH U (t,0)
Where...
Ah of course, silly mistake
d/dt <x> = (Eq t i_hat + C)/m
<x> = (E q t^2 i_hat )/(2m) + C_1
and then C_1 is zero from the IC so
<x> = (E q t^2 i_hat )/(2m)
But the whole thing has been done with the whole X anyway and the only reason the X_1 is being used is because it is in the Hamiltonian so isn't it fine?
Then
d/dt <p> = Eq i_hat
<p> = Eq t i_hat + C
using the first equation then:
d/dt <x> = (Eq t i_hat + C)/m
<x> = t(Eq t i_hat + C)/m + C_1
As ih is a constant wouldn't <ih> just be equal to ih since the exponentials of the U term would become 1 and the phi(0) would as well.
And then I could sub in the first equation to get the value for <x>?
So then for the heisenberg picture, because e^(0) = 1 and thus for X_bH (0), U would equal 1. I would get (where U_d represents U dagger)
U_d X_aH U X_b - X_b U_d X_aH U
then I don't know what to do
Ah, thank you.
So then
<p> = m (d/dt <X>)
and
d/dt <P>= i(-E*q)/h <[P,X_1]>
Is that the final answer?
The second portion of the question is to, with the same Hamiltonian, knowing that in the Heisenberg Picture , X_H = (X_1H,X_2H,X_3H), find the commutator
[X_aH (t), X_bH (0)]
where all...
If I did the same procedure for P then it would be
d/dt <P> = i/h <[H,P]> + <dP/dt>
= i/h <[(P^2/2m - E*q*X_1), P]>+<dP/dt>
= i/h <[P*-E*q*X_1 -E*q*X_1*P]>+<dP/dt>
= i(-E*q)/h <[P,X_1]>+<dP/dt>
= i/(2mh) <i*h>+<dP/dt>
But then if [P,X_1] = ih then this doesn't really make sense...
Ah ok so then, assuming that the commutator of x and x_1 is 0 I would get:
d/dt <X> = i/h <[H,X]> + <dX/dt>
= i/h <[(P^2/2m - E*q*X_1), X]>+<dX/dt>
= i/h <[X*P^2/2m -X*P^2/2m]>+<dX/dt>
= i/(2mh) <[X,P^2]>+<dX/dt>
= i/(2mh) <2*i*h*p>+<dX/dt>
= i/(2mh)*(2*i*h) <p>+<dX/dt>
= -1/m<p> +<dX/dt>
==>...
What if I just did:
U (t,0) = exp (-i*t*H/hbar)
= exp (-i*t*(P^2/2m - E*q*X_1)/hbar)
|phi (t)> = exp (-i*t*(P^2/2m - E*q*X_1)/hbar) |phi(0)>
<phi (t)|X|phi(t)> = <phi(0) exp (i*t*(P^2/2m - E*q*X_1/hbar) | X | exp (-i*t*(P^2/2m - E*q*X_1)/hbar) |phi(0)>
but I don't know where I...