Download 2 Problems on the Quantum Mechanics - Homework 1 | PHYS 580 and more Assignments Quantum Mechanics in PDF only on Docsity! PHYSICS 580 – FALL 2007 PROBLEM SET 6 – DUE: THURSDAY, NOV. 29 1. (LQM 4-13) The energy eigenfunctions of the harmonic oscillator are a complete set, i.e., form a basis. What is the expansion of a delta function, δ(x − a), in terms of the Hermite orthogonal functions? 2. (LQM 4-14) a) Calculate the following matrix elements for the harmonic oscillator 〈n|x|m〉, 〈n|px|m〉, 〈n|p|m〉, 〈n|xp|m〉 〈n|x2|m〉, 〈n|a†|m〉, 〈n|H|m〉. b) Show that the expectation value of the potential energy in an energy eigenstate of the harmonic oscillator equals the expectation value of the kinetic energy in that state. 3. (LQM 4-15) Estimate the zero point energy for a particle of mass m in the following potentials: a) V (x) = vx4 in one dimension, where v is a positive constant. b) V (r) = −e2/r in three dimensions; this is the potential felt by an electron in a hydrogen atom. Express the estimate in eV where e and m are the charge and mass of the electron. c) V (r) = −q2e−kr/r in three dimensions. Take k = 1.3× 106 cm−1 and q2 = e2/14. This problem requires a graphical solution. A potential of this form, called a “screened Coulomb potential,” is the type felt by an electron due to an impurity of charge e in a semiconductor such as germanium. 4. (LQM 5-1) a) Derive and solve the equations of motion for the Heisenberg operators a(t) and a†(t) for the harmonic oscillator. b) Calculate [a(t), a†(t′)]. 5. (LQM 5-2) a) Show that [~r, f(~p)] = ih̄~∇pf(~p) where f(~p) is an arbitrary function of the momentum operator. b) Using this result show that eip·λ/h̄~re−ip·λ/h̄ = ~r + ~λ where ~λ is a numerical vector. c) Show that the wave function of the state |Φ〉 ≡ e−ip·λ/h̄|Ψ〉 is the same as the wave function of the state |Ψ〉, only shifted a distance ~λ. Write out 〈x|Φ〉 explicitly if |Ψ〉 is the ground state of the harmonic oscillator in one dimension. d) Show that |Φ〉 develops in the Schrödinger representation by |Φ(t)〉 = e−ip (−t)·λ/h̄|Ψ(t)〉, where p (−t) is the momentum operator in the Heisenberg representation at time −t. 6. (LQM 5-3) Let |Ψ〉 be an energy eigenstate of the harmonic oscillator in one dimension. Let |Φ〉 = eipλ/h̄|Ψ〉. Show that |Φ(t)〉 in the Schrödinger representation is a wave packet whose center oscillates from λ to −λ with frequency ω, the wave packet never spreading out in time. Hint: use the relation eA+B = eAeBe−[A,B]/2 where [A,B] is a c-number. This oscillation is precisely what one expects from a classical oscillator displaced a distance λ from the origin and then released. 7. (LQM 5-4) a) Calculate the correlation function 〈0|x(t)x(t′)|0〉 where |0〉 is the ground state of a one-dimensional harmonic oscillator and x(t) is the position operator in the Heisenberg represen- tation.
