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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 0 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? (b) A uniform, weak magnetic ±eld of amplitude B 0 is applied along the z -axis. What are the new energy eigenvalues and corresponding wavefunctions? (c) Instead of a weak magnetic ±eld along the z -axis, a uniform electric ±eld of magnitude E 0 is applied along the x -axis. Find an approximation for the low-lying energy levels that is valid in the limit qr 0 E 0 ≫ p 2 /Mr 2 0 . Hint: try expanding around the potential about a stable equilibrium point. 2. [10 pts] Write out the fully-normalized hydrogen wavefunctions for all of the 3p orbitals. Expand out
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