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Unformatted text preview: HOMEWORK ASSIGNMENT 8 PHYS852 Quantum Mechanics II, Spring 2008 New topics covered: Scattering amplitude, crosssection, partial wave expansion . 1. Spherical Bessel functions: The spherical Bessel function j ℓ ( ρ ) is defined as j ℓ ( ρ ) = ( − 1) ℓ ρ ℓ parenleftbigg 1 ρ d dρ parenrightbigg ℓ sin ρ ρ . Show that this state satisfies the radial wave equation. Answer: Let j ℓ ( ρ ) = ( − 1) ℓ u ℓ ( ρ ) /ρ , where u ℓ ( ρ ) = ρ ℓ +1 parenleftbigg 1 ρ d dρ parenrightbigg ℓ sin ρ ρ The radial wave equation for u ℓ ( ρ ) is then bracketleftbigg d 2 dρ 2 − ℓ ( ℓ + 1) ρ 2 + 1 bracketrightbigg u ℓ ( ρ ) = 0 Let H ℓ = bracketleftbigg d 2 dρ 2 − ℓ ( ℓ + 1) ρ 2 + 1 bracketrightbigg Strategy: We will show that if we assume H ℓ u ℓ = 0 then we must also have H ℓ +1 u ℓ +1 = 0. Then if we establish that H u = 0, then H ℓ u ℓ = 0 for all ℓ . First we derive the recursion relation: u ℓ +1 = ρ ℓ +2 parenleftbigg 1 ρ d dρ parenrightbigg ℓ +1 sin ρ ρ = ρ ℓ +2 parenleftbigg 1 ρ d dρ parenrightbigg ρ − ℓ − 1 ρ ℓ +1 parenleftbigg 1 ρ d dρ parenrightbigg ℓ sin ρ ρ = ρ ℓ +2 parenleftbigg 1 ρ d dρ parenrightbigg ρ − ℓ − 1 u ℓ Let A = ρ ℓ +2 parenleftbigg 1 ρ d dρ parenrightbigg ρ − ℓ − 1 = bracketleftbigg d dρ − ( ℓ + 1) ρ bracketrightbigg Then we have u ℓ +1 = Au ℓ . Now we want to compute the commutator [ H ℓ ,A ] = H ℓ A = AH ℓ . First computing H ℓ A , we find bracketleftbigg d 2 dρ 2 − ℓ ( ℓ + 1) ρ 2 + 1 bracketrightbiggbracketleftbigg d dρ − ( ℓ + 1) ρ bracketrightbigg = d 3 dρ 3 + ( ℓ 2 + ℓ − 2)( ℓ + 1) ρ 3 + (2 − ℓ )( ℓ + 1) ρ 2 d dρ − ( ℓ + 1) ρ d 2 dρ 2 − d dρ + ( ℓ + 1) ρ 1 Similarly, for AH ℓ we find bracketleftbigg d dρ − ( ℓ + 1) ρ bracketrightbiggbracketleftbigg d 2 dρ 2 − ℓ ( ℓ + 1) ρ 2 + 1 bracketrightbigg = d 3 dρ 2 − ℓ ( ℓ + 1) ρ 2 d dρ + (2 ℓ + 1)( ℓ + 1) ρ 3 + d dρ − ( ℓ + 1) ρ d 2 dρ 2 − ( ℓ + 1) ρ . Subtracting the two gives H ℓ A − AH ℓ = 2( ℓ + 1) ρ 2 d dρ − 2( ℓ + 1) 2 ρ 2 = 2 2( ℓ + 1) ρ 2 bracketleftbigg d dρ − ( ℓ + 1) ρ bracketrightbigg = 2( ℓ + 1) ρ 2 A. Starting from H ℓ A = AH ℓ + 2( ℓ +1) ρ 2 A , we operate on u ℓ and use H ℓ u ℓ = 0 to find H ℓ u ℓ +1 = 2( ℓ + 1) ρ 2 u ℓ +1 which gives bracketleftbigg H ℓ − 2( ℓ + 1) ρ 2 bracketrightbigg u ℓ +2 = 0 , But we see that H ℓ − 2( ℓ + 1) ρ 2 = d 2 dρ 2 − ℓ ( ℓ + 1) ρ 2 + 1 − 2( ℓ + 1) ρ 2 = d 2 dρ 2 − ( ℓ + 1)( ℓ + 2) ρ 2 + 1 = H ℓ +1 Thus we have proven that H ℓ +1 u ℓ +1 = 0 under the assumption H ℓ u ℓ = 0 ....
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 Spring '10
 MichaelMoore
 mechanics, Work, Boundary value problem, Bessel function, Scattering theory, scattering amplitude, eﬀective range, dr ′ r

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