4 Problems in Midterm Exam | Quantum Mechanics | PHY 5645, Exams of Quantum Mechanics

Download 4 Problems in Midterm Exam | Quantum Mechanics | PHY 5645 and more Exams Quantum Mechanics in PDF only on Docsity! Quantum Mechanics PHY5645 Midterm Exam - Closed Books Thursday October 18, 2007 Time: 11:00AM-12:15PM P1 (25pts) Galilean transformations: a) Construct the unitary operator û which transforms a time dependent wavefunction ψ(r, t) → ψ(r + vt, t), i.e. find û such that ûψ(r, t) = ψ(r + vt, t). b) Now, consider the free one particle time-dependent Schrodinger equation ih̄ ∂ ∂t ψ(r, t) = p̂2 2m ψ(r, t). Use the result of the previous part of the problem to find a unitary operator Û = eiφ(r,t,v)û which leaves the Schrodinger equation invariant i.e. it satisfies Û ( ih̄ ∂ ∂t − p̂ 2 2m ) Û † = ( ih̄ ∂ ∂t − p̂ 2 2m ) P2 (25pts) Heisenberg picture: The 3D quantum mechanical Hamiltonian for a particle of charge e moving in a time-independent magnetic field B(r) = ∇×A(r) is Ĥ = 1 2m ( p̂− e c A(r̂) )2 , (1) where c is the speed of light. Find the Heisenberg equations of motion for each component of the position vector operator r̂ and the velocity vector operator ˆ̇r. P3 (25pts) Bohr-Sommerfeld quantization conditions: Use WKB approximation to find the eigenenergies for a Schrodinger particle moving in a 1D potential V (x) = Ax for x > 0 = −Bx for x < 0 where A > 0 and B > 0. P4 (25pts) Coupled harmonic oscillators and normal modes: Consider a system of two identical, interacting harmonic oscillators in 1D, described by the Hamiltonian Ĥ = p 2 1 2m + p22 2m + 1 2 mω2x21 + 1 2 mω2x22 + λx1x2. Assume that |λ| < mω2. a) Find the eigenenergies of this Hamiltonian. (Hint: Let X± = 1√2(x1 ± x2)) b) Find the coordinate representation of the normalized ground state wavefunction. (Hint:∫∞ −∞ e −αx2dx = √ π/α)