NRQM10_4 - Physics 216 Problem Set 4 DUE Spring 2010 1(a...

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Physics 216 Problem Set 4 Spring 2010 DUE: MAY 25, 2010 1. (a) Consider the Born approximation as the first term of the Born series. Show that: (i) the Born approximation for the forward scattering amplitude [i.e. at θ = 0] is purely real, and therefore (ii) the Born approximation fails to satisfy the optical theorem. Do not assume that the potential is spherically symmetric. However, you may assume that the potential is hermitian. (b) Consider the Yukawa potential: V = ge μr r . In class, we computed the (first) Born approximation to the scattering amplitude. Consider now the second Born approximation; i.e., the second term in the Born series. Compute the scattering amplitude in the forward direction , θ = 0, in the second Born approximation. 1 Check to see whether the optical theorem is now satisfied. HINT: You will need to evaluate ( vector k | V ( E H 0 + ) 1 V | vector k ) , where H 0 = vector P 2 / (2 m ). In class, we inserted a complete set of position eigenstates in order to convert this matrix element as a multiple integral over d 3 r 1 d 3 r 2 . However, it is easier to evaluate the matrix element by inserting a complete set of momentum eigenstates, | k ) . You will then only have to evaluate an integral over d 3 k . (c) Compare the magnitudes of the first and second terms of the Born series for the forward scattering amplitude. What condition do you find if you require the second term in the Born series to be smaller than the first term? Compare this condition with the one you would get for the validity of the Born approximation based on the formula derived in class. (d) Using the first Born approximation for the scattering amplitude, compute the s and p wave phase shifts. Under what circumstances does the s -wave phase shift dominate? Is the Born approximation valid in this case? 2. Consider the scattering of particles by the square well potential in three dimensions: V ( r ) = braceleftBigg V 0 , for r<a, 0 , for r>a, (1) 1 Do not attempt to compute the scattering amplitude in the second Born approximation for θ negationslash = 0. It is extremely messy! 1
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where V 0 is positive. (a) Obtain the differential cross-section in the Born approximation. (b) Using the results of part (a), evaluate the total cross-section in the limits of low and high energy. Specifically, show that at low energies, the cross section can be approximated by: σ σ 0 (1 + Ak 2 ) , where the energy E = planckover2pi1 2 k 2 / (2 m ). You should evaluate the constants σ 0 and A . In the high energy limit, show that σ C k 2 , where the constant C should be determined. HINT: To determine the total cross-section in the high-energy limit, you should con- vert the integral to a manageable form before making any approximations. First inte- grate over the azimuthal angle. Then, change variables from cos θ to y = 2 ka sin( θ/ 2) = ka [2(1 cos θ )] 1 / 2 , and express the total cross-section as an integral over y . Now, you can evaluate the integral by taking the infinite energy limit. However, the result- ing integral is difficult (warning—Mathematica will have a very hard time with this, although Maple can do it quite easily!).
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