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 ) = b - 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 p k k p tan( k p a ) P , (2) where k p = r k 2 + 2 mV 0 ¯ h 2 . (3) 2. Show that for a very deep well there are regularly spaced, low energy ( E l V 0 ) s-wave resonances, with energies E R given by k p R = (2 n + 1) π/ 2 a . Show that the resonances have width Γ = ¯ hk R /ma and check that for a suFciently deep well this width is much less than
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Unformatted text preview: the spacing between the resonances. Notice that the resonances have a background phase shift bg =-k R a . 3. Show that as k 0, k cot =-1 a + 1 2 r 2 k 2 , (4) where a and r are the s-wave scattering length and the eective range, respectively. ind a and r in terms of the parameters of the potential and the particle....
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