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 w ...continues

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 ...continues

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.

Clebsch-Gordan coefficients for two spin-1/2 Hilbert spaces.

Basically I understand the first part to be 4: |-->, |++>, |-+> and |+->. The second part to be something like; J.J = (S1+S2).(S1+S2) = S1^2 + S2^2 + 2*S1.S2 = (3/2)*(h^2) + 2*S1z*S2z + S1+*S2- + S1-*S2+. After this it gets a little hazy and then for the Clebsch-Gordan coefficients I have read and read and am still a littl ...continues

According to the Hawking process, black holes evaporate because negative energy virtual particles fall into it while the positive energy virtual particles escape. Why don't positive energy virtual particles fall into the black hole?

Stephen Hawking has stated that black holes disappear over time. My understanding is that because of the uncertainty principle there are vacuum fluctuations causing positive and negative energy particles to be produced at the event horizon. If the negatively energy particles fall into the black hole they reduce the mass of the o ...continues

Exchange Interaction

Exchange Interaction. See attached file for full problem description.

Eigenfunction expansion, time evolution, Hamiltonian

Eigenfunction expansion, time evolution, Hamiltonian. See attached file for full problem description.

Ferromagnet

Ferromagnet. See attached file for full problem description.