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Unformatted text preview: Physics 4141 Test 2 April 13, 2007 Answer all questions on a separate sheet of paper, showing as many steps as possible. 1. Consider an infinite square well with a delta function at the center: V ( x ) = ( x ) for- a < x < + a for | x | a. Show that the EVEN bound state solutions have energies E = h 2 k 2 2 m that satisfy tan( ka ) =- h 2 k m Solution : Construct even states in the two regions: I = A cos( kx ) + B sin( kx ) x < II = A cos( kx )- B sin( kx ) x > Requiring II ( x = a ) = 0 gives A = B tan( ka ). Matching the discontinuity in the derivative at x = 0 to 2 m h 2 (0) gives- 2 Bk = 2 m h 2 A . Substituting for A or B from the continuity condition and canceling factors gives the equation we wished to prove. 2. A particle with energy E is incident from the left on the the step potential V ( x ) = 0 x < V ( x ) = V x Assume that 0 < E < V ....
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