Reference no: EM132481901

Physics /Advanced Quantum Mechanics /Questions

Question # 1
Assume A and B are two Hilbert space operators, and that A has an inverse A^-1 such thatAA^-1= A^-1A = 1. Let λ be a small parameter, 0 < λ <<1. Find the explicit form of the inverse,(A - λ B)^-1, of the combined operator (A - λ B), up to first order in λ. Prove that the inverse youpropose indeed yields the expected result (up to first order in λ) if multiplied with the originalcombined operator. Note: Do not automatically assume that any 2 operators commute!

Question # 2
A particle is in the state |φ> = cos(θ/2), sin(θ/2) but the angle θ is unknown, and has a 1/3 chance of being in each of the following values θ = 0, Π/2, Π.
(a) Find the state operator ρ.
(b) Find the expectation value of the operators S_x and S_z for this state operator.
(c) Show that if the Hamiltonian is H = a S_x , the state operator will be time independent.

Question # 3

This Question is to be worked entirely in the Heisenberg formulation of quantum mechanics.

Consider a particle of mass m in the one-dimensional linear potential .

(a) What is the Hamiltonian? Find expressions for the derivatives of the position operator X and momentum operator P.

(b) Solve for the position and momentum operators at time t in terms of the operators at time 0.

(c) Show that there is a minimum uncertainty relation between the uncertainty of the initial position Δx (0)and the position at time t, Δx (t).

Question # 4
Obtain the Clebsch Gordan coefficients for the addition of a spin 1 and a spin 2 particles.