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# hw4_002 - PHY 4604 Fall 2009 Homework 4 Due by 5:00 p.m on...

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PHY 4604 Fall 2009 – Homework 4 Due by 5:00 p.m. on Monday, October 19. Please turn in your homework in class before the deadline, or else bring it to NPB 2162. (You may push it under the door if NPB 2162 is unoccupied.) No credit will be available for homework submitted after the start of class on Friday, October 24. Answer all three questions. Please write neatly and include your name on the front page of your answers. You must also clearly identify all your collaborators on this assignment. To gain maximum credit you should explain your reasoning and show all working. You may find useful the following: cos 2 x + sin 2 x = 1 cos 2 x - sin 2 x = cos 2 x 2 sin x cos x = sin 2 x sin( x ± y ) = sin x cos y ± cos x sin y 1. Consider a piecewise constant potential V ( x ). (a) Show that in any region where V ( x ) = V j = constant, the general stationary state ψ ( x ) = A j exp( ik j x ) + B j exp( - ik j x ) can be rewritten in the equivalent form ψ ( x ) = C j cos k j x + D j sin k j x. (1) Provide expressions for C j and D j in terms of A j and B j . (b) Writing C = | C | e and D = | D | e where γ and δ are real, find an expression for the probability density ρ ( x ) in the stationary state specified in Eq. (1). You should find that ρ ( x

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