Eigenvalues, eigenvectors, and time evolution.

Dear Mitra, I in fact wrote the wrong matrix but I am still confused after PART A. (PART A) The actual matrix is: H = 1 2 0 2 0 2 0 2 -1 Where the eigenvalues are E1 =0, E2=3hw, E3=-3hw and as you said the trace(H) =0 = sum of eigenvalues. I also found the eigenvectors using Hx = Ex and they were: for E1 =0 : |v1> = <-1, 1/2, 1> for E2 =3: |v2> = <2, 2, 1> for E3 =-3: |v3> = <1/2, -1, 1> After this, I am confused. Do I need to normalize those eigenvectors? if so then I get the normalized vectors |v1>, |v2>,|v3> to be: |v1> = <-2/3 , 1/3, 2/3> |v2> = <2/3 , 2/3, 1/3> |v3> = <1/3 , -2/3, 2/3> (PART B) Now I write ... click for more

Observable in Heisenberg Picture

Consider the observable q_u = u_1q_1+u_2q_2+u_3q_3 , where the q_i are the Pauli matrices and where u = (u_1,u_2,u_3) in R^3 and the Hamiltonian is H = q3. Compute the observable (q_u)_H (t) derived from q_u in the Heisenberg picture.

Tensor Products observables.

In the Hilbert space C^2 x C^2 (where x is used in all of my notation to mean the cross product) Consider the state vector: Psi = 1/sqrt(2) ( e_1 X e_2) + 1/sqrt(2) ( e_2 X e_1 ) (Part a) What is the probability that the measurement of q_3 X I gives the value -1 and how does the state vector change in this case? (NOTE: where q_3 is sigm_3, X is tensor product, I is identity matrix, and we are in C^2 X C^2) (Part b) The same question for the measurement of I x q_3 and the value +1 (Part c) Find the state vector psi(t) at time t if the Hamiltonian is: q_3 x I + I x q_2

Simultaneous Equations

X^2 + y^2 = 1 y=|x| - a find all "a" which prevent the system to have no solutions

Charged Particle on Ring

See attached file for full problem description.