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4604_ProblemSet7_fa07

# 4604_ProblemSet7_fa07 - PHY4604 Fall 2007 Problem Set 7 PHY...

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PHY4604 Fall 2007 Problem Set 7 Department of Physics Page 1 of 3 PHY 4604 Problem Set #7 Due Friday November 30, 2007 (in class) (Total Points = 115, Late homework = 50%) Reading: Griffiths Chapter 5 (Sections 5.1 and 5.2). Problem 1 (30 points): The state of a two particle system is described by the wave function ) , , ( 2 1 t r r r r Ψ . The time evolution is given by Schrödinger’s equation t i H Ψ = Ψ h with ) , , ( 2 2 2 1 2 2 2 2 2 1 1 2 t r r V m m H r r h h + = where 2 2 2 2 2 2 2 k k k k z y x + + = for k = 1,2. For time-independent potentials, we obtain a complete set of solutions of the form h r r r r / 2 1 2 1 ) , ( ) , , ( iEt e r r t r r = Ψ ψ , where ) , ( 2 1 r r r r ψ satisfies the time- independent Schrödinger equation ψ ψ ψ ψ E r r V m m = + ) , ( 2 2 2 1 2 2 2 2 2 1 1 2 r r h h . Typically the interaction potential depends only on the vector 2 1 r r r r r r = ( i.e. the separation between the two particles). Suppose ) ( ) , ( 2 1 r V r r V r r r = and we change variables from 1 r r and 2 r r to 2 1 r r r r r r = and ) /( ) ( 2 1 2 2 1 1 m m r m r m R + + = r r r ( i.e. the center-of-mass vector). (a) (5 points) Show that r m R r r r r ) / ( 1 1 μ + = and r m R r r r r ) / ( 2 2 μ = , where ) /( 2 1 2 1 m m m m + = μ is the “reduced mass”. (b) (5 points) Show that r R m + = r r r ) / ( 2 1 μ and r R m = r r r ) / ( 1 2 μ , where ) /( 2 1 2 1 m m m m + = μ is the “reduced mass”. (c) (5 points) Show that the time-independent Schrödinger equation becomes ψ ψ ψ μ ψ E r V M r R = + ) ( 2 2 2 2 2 2 r h h , where 2 1 m m M + = and ) /( 2 1 2 1 m m m m + = μ is the “reduced mass”. (d) (5 points) Separate the variables by letting ) ( ) ( ) , ( r R r R r R r r r r ψ ψ ψ = and show that ) ( R R r ψ satisfies the one-particle Schrödinger equation with mass M = m 1 +m 2 , potential zero, and energy E R ( i.e. the center-of-mass moves like a free particle).

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