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Unformatted text preview: Harvard University Physics 143b: Quantum Mechanics II Problem Set 8 due Friday November 19 1. Consider non-relativisitic electrons constrained to move on a thin ring of radius R . Assuming we can ignore motion except along the angular direction of the ring, which is labelled by an angle , the Schrodinger equation for the wavefunction ( ) is- h 2 2 mR 2 d 2 d 2 = E. (1) The eigenfunctions are = e i / 2 , and the eigenenergies are h 2 2 / (2 mR 2 ), where is any integer. Now magnetic flux is inserted in the center of the ring. (a) The vector potential can be chosen to have only an angular component. Show that A = / (2 R ), and so the Schr odinger equation is modified to 1 2 mR 2 h i d d- q 2 ! 2 = E (2) where q is the charge of the electron. (b) List the eigenvalues of (2). (c) Plot the total ground state energy of 5 electrons as a function of (ignore electron spin, and you can use Mathematica if you wish)....
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