hw207a - Physics 512 Homework Set #7 Solutions Winter 2003...

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Physics 512 Winter 2003 Homework Set #7 – Solutions 1. Consider the scattering of a beam of spinless particles of momentum ¯ hk initially traveling along the +ˆ z direction by a potential of the form V ( ~r )= V 0 ± δ ( x ) δ ( y b ) δ ( z )+ δ ( x ) δ ( y + b ) δ ( z ) ² a ) Calculate the scattering amplitude and diFerential cross section in the Born ap- proximation. Using the Born approximation, we compute f Born ( ~q m 2 π ¯ h 2 Z V ( ~r 0 ) e i~q · ~r 0 d 3 ~r 0 = mV 0 2 π ¯ h 2 ± e iq y b + e iq y b ² = mV 0 π ¯ h 2 cos q y b where ~q = ~ k 0 ~ k . Since the beam initially travels in the z direction, we must have k y =0 . Hence we may write f Born ( ~q mV 0 π ¯ h 2 cos k 0 y b so that d = | f Born | 2 = ³ mV 0 π ¯ h 2 ´ 2 cos 2 k 0 y b Using spherical coordinates and conservation of energy ( | ~ k 0 | = | ~ k | ) this may be written as d = ³ mV 0 π ¯ h 2 ´ 2 cos 2 ( kb sin θ sin ϕ ) b ) This potential represents scatterers located at y = b and y =+ b . How does the quantum result diFer from what one would expect classically? The quantum scattering cross section has the form d cos 2 ( scattering direction ) so it results in an interference pattern (very similar to that of a double slit). Classically, particles would scatter from either the Frst scatterer or the second, but not both simultaneously (multiple scattering could occur, but only sequentially). Hence there would be no interference classically (which sounds pretty obvious when one thinks about it). 1
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2. Merzbacher, Exercise 13.11. If V = C/r n , obtain the functional dependence of the Born scattering amplitude on the scattering angle. Discuss the reasonableness of the result qualitatively. What values of n give a meaningful answer? Since the potential is spherically symmetric, we may use the expression f Born ( θ )= 2 m ¯ h 2 Z 0 V ( r ) sin qr r 2 dr = 2 mC q ¯ h 2 Z 0 sin r n 1 dr where q =2 k sin θ 2 . Before attempting to integrate this, we may obtain the functional dependence of this expression on q by simply performing a change of variables, x = . The result is f Born ( θ 2 mC ¯ h 2 I n q n 3 = 2 mC ¯ h 2 I n (2 k ) n 3 sin n 3 θ 2 where I n = Z 0 sin x x n 1 dx (1) Ignoring for the moment the convergence of I n , we Fnd the general behavior f Born ( θ ) sin n 3 θ 2 Note that we assume n is non-negative (but not necessarily to be an integer), otherwise the potential would grow indeFnitely for large r , in violation of the scattering assump- tions. ±or n< 3 , this blows up in the forward direction, corresponding to enhanced small angle scattering. However there is an opposite e²ect of suppression for n> 3 . This at least seems reasonable, since the larger n is, the steeper (and more highly localized) the potential is. ±or a very sharp potential, most of the time the incoming particles would miss it altogether. But in the instances when they hit the potential, they would most likely get scattered by large angles.
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hw207a - Physics 512 Homework Set #7 Solutions Winter 2003...

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