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Unformatted text preview: QUANTUM MECHANICS SUBJECT EXAM Spring 2009 ID NUMBER: IMPORTANT: for you own sake, please try not to leave any question blank. If you can’t remember a crucial element, write in words what you would do if you had it. Or use another approach, but try to show us what you do know on the topic in question. 1. The DC Stark effect: A hydrogen atom is placed in a uniform electric field of strength E , whose direction defines the z-axis. Assume that the Stark shift will be large compared to the fine-structure shifts, so that you can perturb the bare hydrogen energy levels, and ignore spin. This problem will focus on the Stark splitting of the n = 2 energy level. a.) What is the bare energy of the n = 2 level? b.) Give the degeneracy of the n = 2 level, and list the corresponding bare | n,ℓ,m ) states. c.) Recall that the dipole moment operator is vector D = e vector R , and write the operator for the in- teraction energy between the atom and the field. d.) In the n = 2 subspace, this operator becomes a 4 × 4 matrix. Determine which matrix elements are non-zero by using [ L z ,Z ] = 0. e.) Compute the energy shifts to first-order in E . You may leave your answer in terms of a single matrix element, which you do not need to calculate. 2. Two identical fermions in a box: Consider two particles (with coordinates x 1 and x 2 ) in a one-dimensional infinite square-well potential of length...
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This note was uploaded on 12/21/2011 for the course PHYS 852 taught by Professor Moore during the Spring '11 term at Michigan State University.
- Spring '11