Assignment Task: Physics /Advanced Quantum Mechanics /Questions
Problem 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 you propose indeed yields the expected result (up to first order in λ) if multiplied with the original combined operator. Note: Do not automatically assume that any 2 operators commute!
Problem 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 Sx and Sz for this state operator.
(c) Show that if the Hamiltonian is H = a Sx , the state operator will be time independent.
Problem 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).
Problem 4: Obtain the Clebsch Gordan coefficients for the addition of a spin 1 and a spin 2 particles.
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