hw8solutions(first4) - HOMEWORK ASSIGNMENT 8 PHYS852...

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Unformatted text preview: HOMEWORK ASSIGNMENT 8 PHYS852 Quantum Mechanics II, Spring 2008 New topics covered: Scattering amplitude, cross-section, partial wave expansion . 1. Spherical Bessel functions: The spherical Bessel function j ( ) is defined as j ( ) = ( 1) parenleftbigg 1 d d parenrightbigg sin . Show that this state satisfies the radial wave equation. Answer: Let j ( ) = ( 1) u ( ) / , where u ( ) = +1 parenleftbigg 1 d d parenrightbigg sin The radial wave equation for u ( ) is then bracketleftbigg d 2 d 2 ( + 1) 2 + 1 bracketrightbigg u ( ) = 0 Let H = bracketleftbigg d 2 d 2 ( + 1) 2 + 1 bracketrightbigg Strategy: We will show that if we assume H u = 0 then we must also have H +1 u +1 = 0. Then if we establish that H u = 0, then H u = 0 for all . First we derive the recursion relation: u +1 = +2 parenleftbigg 1 d d parenrightbigg +1 sin = +2 parenleftbigg 1 d d parenrightbigg 1 +1 parenleftbigg 1 d d parenrightbigg sin = +2 parenleftbigg 1 d d parenrightbigg 1 u Let A = +2 parenleftbigg 1 d d parenrightbigg 1 = bracketleftbigg d d ( + 1) bracketrightbigg Then we have u +1 = Au . Now we want to compute the commutator [ H ,A ] = H A = AH . First computing H A , we find bracketleftbigg d 2 d 2 ( + 1) 2 + 1 bracketrightbiggbracketleftbigg d d ( + 1) bracketrightbigg = d 3 d 3 + ( 2 + 2)( + 1) 3 + (2 )( + 1) 2 d d ( + 1) d 2 d 2 d d + ( + 1) 1 Similarly, for AH we find bracketleftbigg d d ( + 1) bracketrightbiggbracketleftbigg d 2 d 2 ( + 1) 2 + 1 bracketrightbigg = d 3 d 2 ( + 1) 2 d d + (2 + 1)( + 1) 3 + d d ( + 1) d 2 d 2 ( + 1) . Subtracting the two gives H A AH = 2( + 1) 2 d d 2( + 1) 2 2 = 2 2( + 1) 2 bracketleftbigg d d ( + 1) bracketrightbigg = 2( + 1) 2 A. Starting from H A = AH + 2( +1) 2 A , we operate on u and use H u = 0 to find H u +1 = 2( + 1) 2 u +1 which gives bracketleftbigg H 2( + 1) 2 bracketrightbigg u +2 = 0 , But we see that H 2( + 1) 2 = d 2 d 2 ( + 1) 2 + 1 2( + 1) 2 = d 2 d 2 ( + 1)( + 2) 2 + 1 = H +1 Thus we have proven that H +1 u +1 = 0 under the assumption H u = 0 ....
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hw8solutions(first4) - HOMEWORK ASSIGNMENT 8 PHYS852...

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