Problem Set 4 Solution

Using t he s ix c oefficients y ou d etermined c

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Unformatted text preview: ≤ Ⱥ Ⱥ Ⱥ , Ⱥ Ⱥ x < 0, x > L Ⱥ € € € where n is the quantum number and L is the width of the box. € Using t he s ix c oefficients y ou d etermined, c onstruct a r epresentation o f t he s tep p otential for the case in which L =2 and C =1. Note that since the expansion was truncated after only six terms, it is an approximation rather than an exact result. 2 1 . continued The coefficients c n to be calculated are ∞ cn = € ∫ * ψ n ( x ) f ( x ) dx . −∞ Substituting the form of the function f ( x ) and the particle in a box wavefunctions yields € cn = C € 2 L L ∫ 0 ȹ nπx ȹ sinȹ ȹ dx . ȹ L Ⱥ Here, since the particle in a box wavefunctions are 0 outside the range 0 ≤ x ≤ L , the limits of integration are reduced to 0 to L. Also, on that range, the function f ( x ) equals a constant C, which can be pulled out of the € integral. The integral to be evaluated is € L L ȹ € πx ȹ ȹ nπx ȹ n L sinȹ dx = − cosȹ ȹ ȹ ȹ L Ⱥ ȹ L Ⱥ 0 nπ ∫ 0 =...
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This document was uploaded on 12/05/2013.

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