4604_ProblemSet2_fa07

4604_ProblemSet2_fa07 - PHY4604 Fall 2007 Problem Set 2 PHY...

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PHY4604 Fall 2007 Problem Set 2 Department of Physics Page 1 of 3 PHY 4604 Problem Set #2 Due Wednesday September 19, 2007 (in class) (Total Points = 110, Late homework = 50%) Reading: Griffiths Chapter 2 (sections 2.1, 2.2, and 2.3). Useful Math: 4 2 cos 2 sin 8 1 4 6 cos 8 2 cos 4 2 sin 4 cos 4 2 sin 2 cos sin cos cos 4 2 cos 2 sin 8 1 4 6 sin 8 2 cos 4 2 sin 4 sin 4 2 sin 2 sin cos sin sin 2 sin cos sin 2 3 2 2 2 2 2 2 3 2 2 2 2 2 2 x x x x x xdx x x x x x xdx x x x xdx x x x xdx x x x x x x xdx x x x x x xdx x x x xdx x x x xdx x x xdx x + + = + + = + = + = = = = = = Problem 1 (10 points): The time-dependent Schrödinger’s equation is ) , ( ) ( ) , ( 2 ) , ( 2 2 2 t x x V x t x m t t x i Ψ + Ψ = Ψ h h , where we take the potential energy, V, to be only a function of x (not of time). We look for solutions of the form ) ( ) ( ) , ( t x t x Φ = Ψ ψ . The time-dependent equation separates into two equations (separation constant E). ) ( ) ( ) ( ) ( 2 2 2 2 x E x x V x d x d m = + h and ) ( ) ( t E dt t d i Φ = Φ h . Thus, h / ) ( iEt e t = Φ . (a) (3 points) For normalizable solutions, show that the separation constant E must be real. ( Hint : see Griffiths problem 2.1) (b) (3 points) Show that E corresponds to the total energy and that ) ( ) ( ) , ( t x t x Φ = Ψ corresponds to a state with definite energy ( i.e. Δ E = 0). (c) (4 points) Show that the time-independent wave function, ) ( x , can always be taken to be a real function. ( Hint : take V to be real and see Griffith’s problem 2.1) mn mn p p p nxdx mx nxdx mx p δ π ξ 2 cos cos 2 sin sin 8 7 1 5 1 3 1 1
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4604_ProblemSet2_fa07 - PHY4604 Fall 2007 Problem Set 2 PHY...

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