Download Quantum Theory I: Assignment 5 and more Assignments Quantum Mechanics in PDF only on Docsity! Massachusetts Institute of Technology Physics Department Physics 8.321 Fall 2006 Quantum Theory I October 2, 2006 Assignment 5 Due October 13, 2006 Announcements • The 8.321 midterm exam will take place in class on October 31 (Hallowe’en). It will be an hour and a half exam. Reading topics for this period • Classical mechanics and canonical quantization; Schrödinger and Heisenberg “pic- tures” of time evolution; two state systems; simple harmonic oscillator. Reading Recommendations 5 • 8.321 lecture notes on time evolution in quantum mechanics (posted on the website), classical Hamiltonian mechanics, and canonical quantization. • Sakurai, §2.1 and 2.2 discussed the basics of time evolution including “pictures”. • Review of classical mechanics (in addition to 8.321 posted lecture notes): Shankar, §, especially §2.5-2.7. • The basics of motion in a magnetic field are presented in Gottfried & Yan, §4.3, which has been scanned and put on the 8.321 website. • Two state systems are presented in Gottfried & Yan, §4.1, which has been scanned and put on the 8.321 website. • The harmonic oscillator is discussed in almost every textbook. Sakurai §2.7; Shankar §7; and Gottfried & Yan §4.2. Problem Set 5 Topics covered in the problems • Motion of a charged particle in a magnetic field, and the importance of the vector potential in quantum mechanics. 1 MIT 8.321 Quantum Theory I Fall 2006 2 • Motion in the Schrödinger and Heisenberg pictures. • Time dependence of the density matrix. 1. Canonical Quantization in the Presence of Static Magnetic and Electric Fields This is an important subject that we will return to from time to time in 8.321 and 8.322. It also illustrates the power as well as the shortcomings of the canonical quantization method. You may have studied the Hamiltonian formulation of this motion in classical mechanics. In that case the first few sections of the problem are review. The classical equation of motion for a particle in constant electric and magnetic fields is the Lorentz force law, m d2~x dt2 = ~F = e ~E + e c ~̇x× ~B (using Gaussian units). Remember that static fields can be described by φ and ~A, the electrostatic and magnetic vector potentials, ~E = −~∇φ and ~B = ~∇× ~A (a) Consider the Lagrangian L = 1 2 m~̇x 2 + e c ~̇x · ~A(~x) − eφ(~x) (1) What is the canonical momentum, ~p = ∂L/∂~̇x? Note that it is not m~̇x. Show that the Euler Lagrange equations, d~p/dt = ∂L/∂~x, give the Lorentz force law. You will have to remember that, although both ~A and φ have no explicit time dependence, they depend implicitly on time via the argument ~x(t). Thus d dt ~A = (~̇x · ~∇) ~A. [You’ll also need some vector calculus identities, or the help of a text like Jackson’s.] (b) Find the Hamiltonian, H = ~̇x · ~p − L. Combining the results of parts (a) and (b), it appears that the energy can be written as E = 1 2 m~̇x2 +eφ (an elementary result since the magnetic field does no work). What is the conceptual difference between H and E in classical mechanics? (c) Quantize this system canonically: [xj , pk] = i~δjk, etc.. Then write the Schrödinger equation in coordinate space. (d) Show that ~A = −1 2 ~x× ~B0 is a vector potential corresponding to a constant field ~B0. Substitute this into the Schrödinger equation (with φ = 0) to obtain ( − ~ 2 2m ~∇ 2 − e 2mc ~L · ~B0 + e2 8mc2 ρ2B2 0 ) ψ(~x) = Eψ(~x) (2) Here ~L = ~x×~p and ρ = B̂0×~x is the radial coordinate in the plane perpendicular to ~B0.
