348-problem44 - metry. Next, notice that ˆ ˆ [ , ] z L z...

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Chem. 540 Instructor: Nancy Makri PROBLEM 44 The Stark effect is the shift of an atom's energy levels caused by a uniform electric field. In this problem you will use 1 st order perturbation theory to explore the Stark effect on the hydro- gen atom. Suppose a hydrogen atom is subject to the following electric field along the z direction: ˆ ˆ cos V e z e r θ = = E E . (a) Show that the 1 st order correction to the ground state energy ( 1 n = ) is zero. (b) Now consider the shift to the lowest excited states ( 2 n = ). Notice there is a fourfold de- generacy. First, notice that all diagonal matrix elements of the perturbation must be zero because of sym-
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Unformatted text preview: metry. Next, notice that ˆ ˆ [ , ] z L z = . Use this fact to show that ˆ ( ) lm z l m m m ′ ′ ′-= . Thus, the matrix element of the perturbation is nonzero only for states of the s a me m quantum number. These observations lead to the conclusion that the perturbation has a single nonzero matrix ele-ment. Calculate this matrix element and apply 1 st order degenerate perturbation theory to evaluate the energy level shifts. Does this electric field remove the degeneracy completely?...
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This note was uploaded on 02/08/2012 for the course CHEM 540 taught by Professor Mccall during the Fall '08 term at University of Illinois, Urbana Champaign.

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