practice2_soln

practice2_soln - PHY 4604 Fall 2008 Practice Questions for...

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PHY 4604 Fall 2008 – Practice Questions for Exam 2 (With Solutions) 1. A particle of mass m moves in one dimension under the potential V ( x )= ( for x< 0 , V 0 ( x - a ) for x> 0 , where V 0 and a are positive quantities. (a) Write down the general form of a stationary-state wave function of energy E> 0 in the regions 0, 0 <x<a ,and x>a . Solution : The general form of the stationary-state wave function is ψ ( x 0 for 0 , A 1 e ikx + B 1 e - ikx for 0 , A 2 e ikx + B 2 e - ikx for , where k = + 2 mE/ ~ . (b) By applying the appropriate boundary conditions, express the stationary-state wave function from (a) in terms of just one unknown amplitude. Solution : Wemustimposecontinuityof ψ at x =0 ,where V ( x ) jumps to inﬁnity: A 1 + B 1 A 1 = - B 1 ψ ( x S sin kx for 0 . Given this, it simpliﬁes matters slightly to rewrite ψ ( x C cos + D sin for . Then we must impose continuity of ψ at x = a : C cos ka + D sin = S sin D + C cot = S. (1) Finally, we must enforce Δ ψ 0 ( a )=(2 mV 0 a/ ~ 2 ) ψ ( a ): - kC sin + kD cos = kS cos + 2 mV 0 a ~ 2 S sin D - C tan = S (1 + α tan ) , (2) where α =2 mV 0 a/ ( ~ 2 k ). The solution to Eqs. (1) and (2) is C = -

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practice2_soln - PHY 4604 Fall 2008 Practice Questions for...

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