# Search results

1. ### Expectation of position and momentum at time t, pictures

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...
2. ### Expectation of position and momentum at time t, pictures

Thank you for your help! Onto the Heisenberg picture portion of the question :)
3. ### Expectation of position and momentum at time t, pictures

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)
4. ### Expectation of position and momentum at time t, pictures

From the initial conditions <p(0)> = 0 = C = 0 <x(0) = 0 = C_1 = 0 so <p> = Eqt i_hat <x> = t (Eq t i_hat)/m = <p> *t/m
5. ### Expectation of position and momentum at time t, pictures

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
6. ### Expectation of position and momentum at time t, pictures

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>?
7. ### Expectation of position and momentum at time t, pictures

Yes so then it would just be [P,X_1] = ih ? Then the equation would be, as I put in post 9 d/t <P> = = i/(2mh) <i*h>
8. ### Expectation of position and momentum at time t, pictures

It doesn't specify but I would assume it is a vector
9. ### Expectation of position and momentum at time t, pictures

ah, okay. Plus I still need to find <[P,X_1]> right. Could you help me with that commutator? Thanks
10. ### Expectation of position and momentum at time t, pictures

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
11. ### Expectation of position and momentum at time t, pictures

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...
12. ### Expectation of position and momentum at time t, pictures

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...
13. ### Expectation of position and momentum at time t, pictures

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> ==>...
14. ### Expectation of position and momentum at time t, pictures

I know that the commutator of x and p is i*hbar and that in the Schrödinger picture the operators are time independent
15. ### Expectation of position and momentum at time t, pictures

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...
16. ### Expectation of position and momentum at time t, pictures

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...
17. ### Expectation of position and momentum at time t, pictures

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...
18. ### Bipartite state purity

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?
19. ### Bipartite state purity

Yes I know. That is what I am searching for, I do not have it in my notes and there is no textbook :( Do you know this equation?
20. ### Bipartite state purity

Ah, that makes it work! Alright so the only part I am confused about then is part d) and how to compute the entanglement entropy mathamatically
21. ### Bipartite state purity

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...
22. ### Expectation value of observable in Bell State

I'm still a little confused about the abstract representation :(
23. ### Bipartite state purity

Could anybody help with this please :smile:
24. ### Bipartite state purity

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...
25. ### Superposition of Hilbert space of qutrit states

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?
26. ### Superposition of Hilbert space of qutrit states

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...
27. ### Superposition of Hilbert space of qutrit states

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?
28. ### Expectation value of observable in Bell State

No worries, I actually do have to do it with a matrix representation and an abstract representation anyway
29. ### Superposition of Hilbert space of qutrit states

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)
30. ### Expectation value of observable in Bell State

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