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Unformatted text preview: CHAPTER 16. 1 . The perturbation caused by the magnetic field changes the simple harmonic oscillator Hamiltonian H to the new Hamiltonian H H = H + q 2 m B • L If we choose B to define the direction of the z axis, then the additional term involves B L z . When H acts on the eigenstates of the harmonic oscillator, labeled by | n r , l , m l >, we get H | n r , l , m l 〉 = h ϖ (2 n r + l + 3 2 + qB h 2 m m l | n r , l , m l 〉 Let us denote qB /2 m by ϖ B . Consider the three lowest energy states: n r = 0, l = 0, the energy is 3 h ϖ / 2 . n r = 0, l = 1 This three-fold degenerate level with unperturbed energy 5 h ϖ / 2 , splits into three nondegenerate energy levels with energies E = 5 h ϖ / 2 + h ϖ B 1- 1 The next energy level has quantum numbers n r = 2, l = 0 or n r = 0, l = 2. We thus have a four-fold degeneracy with energy 7 h ϖ / 2. The magnetic field splits these into the levels according to the m l value. The energies are E = 7 h ϖ / 2 + h ϖ B 2 1 0,0- 1- 2 n r = 1,0 2 , The system has only one degree of freedom, the angle of rotation θ . In the absence of torque, the angular velocity ϖ = d θ /dt is constant. The kinetic energy is E = 1 2 Mv 2 = 1 2 ( M 2 v 2 R 2 ) MR 2 = 1 2 L 2 I where L = MvR is the angular momentum, and...
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This note was uploaded on 03/09/2008 for the course PHYS 401 taught by Professor Mokhtari during the Spring '05 term at UCLA.
- Spring '05