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Unformatted text preview: PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 12 Topics Covered: Motion in a central potential, spherical harmonic oscillator, hydrogen atom, orbital electric and magnetic dipole moments 1. [20 pts] A particle of mass M and charge q is constrained to move in a circle of radius r in the x y plane. (a) If no forces other than the forces of constraint act on the particle, what are the energy levels and corresponding wavefunctions? If the particle is forced to remain in the xy plane, then it can only have angular momen tum along the zaxis, so that ~ L = L z ~e z and L 2 = L 2 z . The kinetic energy can be found two ways: Method 1: Using our knowledge of angular momentum. We start by choosing φ as our co ordinate H = L 2 2 I = L 2 z 2 Mr 2 (1) so that the eigenstates are eigenstates of L z →  i ~ partial φ , from which we see know that the energy levels are then E m = ~ 2 m 2 2 Mr 2 , where m = , ± 1 , ± 2 , ± 3 . . . ... , and the wavefunctions are h φ  m i = 1 √ 2 π e imφ . Method 2: Solution from first principles. We start by choosing s as our coordinate, where s is the distance measured along the circle. The classical Lagrangian is then L = M ˙ s 2 2 (2) the canonical momentum is p s = ∂ s L = M ˙ s . The Hamiltonian is then H = p ˙ s L = p 2 s 2 M (3) promoting s and p s to operators, we must have [ S, P s ] = i ~ , so that in coordinate representation, we can take S → s , and P s →  i ~ ∂ s , which gives H = ~ 2 2 M ∂ 2 s (4) the energy eigenvalue equation is then ~ 2 2 M ∂ 2 s ψ ( s ) = Eψ ( s ) (5) or equivalently ∂ 2 s ψ ( s ) = 2 ME ~ 2 ψ ( s ) (6) 1 This has solutions of the form: ψ ( s ) ∝ e ± i √ 2 ME ~ s (7) singlevaluedness requires ψ ( s + 2 πr ) = ψ ( s ) (8) which means √ 2 ME ~ 2 πr = 2 πm (9) where m is any integer. This gives E = ~ 2 m 2 2 Mr 2 (10) so that ψ m ( s ) = e ims/r √ 2 πr (11) Both methods agree because s = r φ . (b) A uniform, weak magnetic field of amplitude B is applied along the zaxis. What are the new energy eigenvalues and corresponding wavefunctions? Using the angular momentum method, we now need to add the term qB 2 M L z to the Hamil tonian to account for the orbital magnetic dipole moment, which gives H = L 2 z 2 Mr 2 qB 2 M L z (12) so that the eigenstates are still L z eigenstates, ψ m ( φ ) = e imφ √...
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This note was uploaded on 12/21/2011 for the course PHY 851 taught by Professor Moore,m during the Fall '08 term at Michigan State University.
 Fall '08
 Moore,M
 mechanics, Charge, Mass, Work

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