ch16 - CHAPTER 16 1 The perturbation caused by the magnetic...

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CHAPTER 16. 1 . The perturbation caused by the magnetic field changes the simple harmonic oscillator Hamiltonian H 0 to the new Hamiltonian H H = H 0 + 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 0 - 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 M v 2 = 1 2 ( M 2 v 2 R 2 ) M R 2 = 1 2 L 2 I
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where L = MvR is the angular momentum, and I the moment of inertia. Extending this to
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