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...
Okay, plugging X into that equation (get rid of _t for legibility) I get
d/dt <X> = i/h <[H,X]> + <dX/dt>
= i/h <[(P^2/2m - E*q*X_1), X]>+<dX/dt>
= i/h (-E*q) <X,X1>+<dX/dt>
and I'm not sure where to go from here...
Homework Statement
Consider a particle, with mass m, charge q, moving in a uniform e-field with magnitude E and direction X_1.
The Hamiltonian is (where X, P, and X_1 are operators):
The initial expectation of position and momentum are <X(0)> = 0 and <P(0)>=0
Calculate the expectation...
Thank you for the equation and the link!
Hmm I am getting a negative entropy for some values of a
How come you posted it as log 2 but it simply says log, presumably log 10 on the site?
In my notes purity is defined as the square of the trace of a state's density operator, it is between 1/d and 1 where d is the dimension of the Hilbert space. P = 1 for pure states. The states with maximal classical uncertainty (maximally mixed states) have the minimum possible purity P = 1/d...
Homework Statement
Consider the bipartite state:
|q> = a/sqrt (2) (|1_A 1_B> +|0_A 0_B>)+sqrt((1-a^2)/2) (|0_A 1_B>+|1_A 0_B>)
where a is between or equal to 0 and 1
a) compute the state of the subsystem p_b
b) compute the purity of p_b as a function of a
c) for what values of a is the...
Ah, I see. That makes sense.
Thank you for you help.
For the tensor product of the states in a) and b),the matrix representation would just be a 9x9 matrix but how would it be represented in abstract notation?
Ah, now I'm confused. I thought you said that to construct the density matrix I needed to do
P = |phi><phi|
so I did that in the middle step (there shouldn't be an equal sign after 1/9) and then I did <i|P|j> for the terms in the matrix.
So if the original way I put it is already as the...
And then for b)
p = 1/9 = (|0><0|+|1><1|+|2><2|)(<0|0>+<1|1>+<2|2>) = 1/3 (|0><0|+|1><1|+|2><2|)
= 1/3 (1 1 1
1 1 1
1 1 1)
which is the same result as in a)
Is this correct?
Okay, so then I get
p = 1/3(|0><0|+|0><1|+|0><2|+|1><0|+|1><1|+|1><2|+|2><0|+|2><1|+|2><2|)
and then constructing the matrix, it becomes
p = 1/3 (1 1 1
1 1 1
1 1 1)
Huh I looked in my notes and we actually do just use the dyadic version and the 4x4 matrix to find the expectation value
In fact it specifies to use the kronecker product to represent the tensor product.
using this matrix represetnation I get 0 for the expectation value