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# hw9 - the spacing between the resonances Notice that the...

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HOMEWORK # 9 Physics 6572 Friday, 11/14/08; due 11/21/08 In this homework you will study some of the properties of the spherical square well potential: V ( r ) = braceleftbigg - V 0 for 0 < r < a , 0 for r > a . (1) 1. Show that the s-wave ( l = 0) phase shift is δ 0 ( k ) = - ka + tan - 1 parenleftbigg k k prime tan( k prime a ) parenrightbigg , (2) where k prime = radicalbigg k 2 + 2 mV 0 ¯ h 2 . (3) 2. Show that for a very deep well there are regularly spaced, low energy ( E lessmuch V 0 ) s-wave resonances, with energies E R given by k prime R = (2 n + 1) π/ 2 a . Show that the resonances have
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Unformatted text preview: the spacing between the resonances. Notice that the resonances have a background phase shift δ bg =-k R a . 3. Show that as k → 0, k cot δ =-1 a + 1 2 r 2 k 2 , (4) where a and r are the s-wave scattering length and the e±ective range, respectively. ²ind a and r in terms of the parameters of the potential and the particle....
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