Quantum Mechanics Question

The Hamiltonian of a certain three level system is represented by the matrix. See attached file for full problem description.

Quantum Problem

Eigen-Function. See attached file for full problem description.

Quantum Problem

Bohr Radius. See attached file for full problem description.

X-Ray Problem

An X-ray photon of wavelength 6 pm (1 pm = 10^-12 m) makes a head-on collision with an electron, so that the scattered photon goes in a direction opposite to that of the incident photon. The electron is initially at rest. (a) How much longer is the wavelength of the scattered photon than that of the incident photon? (b) What is the kinetic energy of the recoiling electron? (Use the conservation of energy)

Eigenvalues, eigenvectors, and time evolution.

a. Given matrix A = 1 2 0 where i found the eigenvalues to be 3hw, 0, -3hw. 2 0 2 0 2 -2 then if at t=0 state-vector is 1 THEN compute the state vector at time t. 0 0 Please explain me how to get this, Im very confused. I know its some sort of matrix multiplication but why? b. If A = 1 0 0 is an observable. FIND its expectation at time t and compute the 0 2 0 0 0 0 probability that measuring A at time t gives the result 1.