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test3b - Physics 463 Final Exam This exam is to be turned...

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Physics 463 - Final Exam : This exam is to be turned in 24 hours after it is received. In solving the following problems you may use any notes, homework sets, and/or solutions accumulated during the semester, one quantum mechanics textbook (including all volumes of a multi-volume set such as Cohen-Tannoudji, et al.), and one mathematical reference. Your score will be based upon your 5 highest scores to the following 6 problems. 1. Use the variational method to estimate the ground-state energy of an electron con f ned to the region x [0 ,L ] by an in f nite square-well potential to which an electric f eld E = E 0 ˆ x has been applied. Choose a family of trial states that satisfy appropriate boundary conditions at the edges of the well. 2. The wave function of a particle subjected to a spherically symmetric potential V ( r ) is given by ψ ( ~ r )=( x + y +3 z ) f ( r ) . (a) Is ψ an eigenfunction of L 2 ? If so, what is the eigenvalue? If not, what values can be obtained if L 2 is measured? (b) If L z is measured, what values can be obtained and with what probabilities? (c) Suppose it is somehow known that ψ ( ~ r ) is an eigenstate of H . Indicate how one may determine V ( r ) . 3.
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