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hwsol6 - Physics 31 Spring 2008 Solution to HW#6 Problem A...

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Unformatted text preview: Physics 31 Spring, 2008 Solution to HW #6 Problem A The wave function of a particle is given by ψ ( x ) = N exp " βˆ’ Β΅ x 3 . 1915 ˚ A ΒΆ 2 # (When you evaluate this function, take all distances x in ˚ A.) (e) Use the exact formula to evaluate βˆ† x for this wave func- tion: βˆ† x = p x 2 βˆ’ x 2 . Compare your answer to your estimate in part (d). Hint: You need the integral R ∞ x 2 exp( βˆ’ ax 2 ) dx = 1 4 p Ο€/a 3 , but you do not need the integral R ∞ x exp( βˆ’ ax 2 ) dx = 1 / (2 a ). Why not? We can evaluate βˆ† x rigorously using the expression βˆ† x = p x 2 βˆ’ x 2 . Note that the average value x = 0. This result follows from the integral expression x = Z ∞ βˆ’βˆž x Β― Β― ψ ( x ) Β― Β― 2 dx = N 2 Z ∞ βˆ’βˆž x exp Β· βˆ’ 2 x b Β΄ 2 ΒΈ dx which is zero because the integrand is an odd function [that is, f ( x ) = βˆ’ f ( βˆ’ x ), so the contributions to the integral from x < 0 and x > 0 exactly cancel]....
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