hwsol5 - Physics 31 Spring, 2008 x 3.1915 A Solution to HW...

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Physics 31 Spring, 2008 Solution to HW #5 Problem A The wave function of a particle is given by ψ ( x )= N exp " µ x 3 . 1915 ˚ A 2 # (When you evaluate this function, take all distances x to be in ˚ A.) (a) What is the correct numerical value of the normalization constant N ? Hint: You must explicitly evaluate N .U se the general result R −∞ exp( ax 2 ) dx = p π/a. (b) Make a plot of ψ ( x )and ¯ ¯ ψ ( x ) ¯ ¯ 2 . Indicate numerical values. (c) Estimate the probability that a measurement of the par- ticle’s position gives a value between x =1 . 45 ˚ Aand x =1 . 55 ˚ A. (d) Use the sketch you made in part (b) to estimate the uncertainty ∆ x in the particle’s position. Hint: Do not do any integrals to answer this part of the question. (a) We evaluate N for an arbitrary value of the scaling con- stant b (not just 3.1915 ˚ A) from the normalization condition 1= Z −∞ ¯ ¯ ψ ( x ) ¯ ¯ 2 dx = N 2 Z −∞ exp · 2 ± x b ´ 2 dx ¸ , The value of the integral is p π/a , with a =2 /b 2 . Hence 1= N 2 b p π/ 2 N = µ 2 πb 2 1 /
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This note was uploaded on 02/29/2008 for the course PHYSICS 31 taught by Professor Hickman during the Spring '07 term at Lehigh University .

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