851final - PHYS851 Quantum Mechanics I Fall 2009 FINAL EXAM NAME 1 A particle of mass M and wave vector k> √ 2 MV planckover2pi1 is scattered

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Unformatted text preview: PHYS851 Quantum Mechanics I, Fall 2009 FINAL EXAM NAME: 1. A particle of mass M and wave vector k > √ 2 MV / planckover2pi1 is scattered by the potential V ( x ) = V 1 ( x ) + V 2 ( x ), where V 1 ( x ) = braceleftbigg 0; x < V > 0; x > (1) V 2 ( x ) = gδ ( x ) (2) (a) Compute the reflection and transmission amplitudes, r and t . (b) Compute the reflection and transmission probabilities, R and T . (c) Show that | r | 2 + | t | 2 negationslash = 1, and explain why probability conservation is not violated. 2. Consider a three-state quantum system, with energy eigenvalues 0, planckover2pi1 ω , and 4 planckover2pi1 ω . The first two normalized energy eigenstates are: | ω 1 ) = 1 √ 2 ( | 1 ) + | 2 ) ) (3) | ω 2 ) = 1 √ 3 ( | 1 ) − | 2 ) + | 3 ) ) (4) For the following questions, each answer must be given in Dirac notation. (a) What is the third normalized energy eigenstate? The initial state of the system is | ψ ( t =0) ) = 1 √ 2 ( | 1 ) − | 2 ) )....
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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.

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851final - PHYS851 Quantum Mechanics I Fall 2009 FINAL EXAM NAME 1 A particle of mass M and wave vector k> √ 2 MV planckover2pi1 is scattered

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