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Consider a particle of mass m elastically scattering off of a spherically symmetric potential V(r)

Physics

Consider a particle of mass m elastically scattering off of a spherically symmetric potential V(r). In S&N they define a function f(q), which to first order in the Born approximation reads

f((1)q) = -m/2ph2 V(q) = -2m/qh ò¥0dr r sin(qr/h) V(r),     q = 2p sin (q/2)                          (1)

Where q = |q| is the momentum transfer and p is the incident momentum. (Beware that S&N define q differently: the relation is qus = hqthem.) In terms of this function the differential cross-section is simply

ds/dW = |f(q)|2.                                                                                                                             (2)

The function f(q), being a function of angles, admits a decomposition into spherical harmonics. Since it does not depend on the longitude f (a consequence of the rotational symmetry of the potential), the relevant spherical harmonics are the Ye,m=0’s, which are proportional to Legendre polynomials,

Ye,m=0(q, f) µ Pe(cos q). So we have

f(q) = h/p S¥?=0(2?+1)eid?sind?P?(cosq)                             (3)

Where d? is a real function of k, the phase shift describing the scattering of an incident momentum p into outgoing waves of angular momentum ?.

In practice one can find the phase shifts by exploiting that the Legendre polynomials are orthogonal functions, with

ò1-1dxP?(x)p?(x) = 2/2?’+1 d??                                                                                                    (4)

1. Consider the scattering of an electron off of a potential localized near x = 0 which can be described in a multiple expansion from the point of view of long distances,

V(x) = V(0)d(3)(x) + Vi(1)id(3)(x) + ½ Vij(2)ijd(3)(x) + …,                                                               (5)

Where V(0), vi(1), etc. are the multiple moments of the potential.

(a) Find the first order correction to the propagator when the initial and final points are far away from the potential. Observe that the multipole expansion corresponds to a small p expansion.

(b) Find the leading approximation to the differential cross-section ds/dW as a function of the magnitude of the momentum p and scattering angles (q, f) to second order in p.

(c) Find the total cross-section as a function of the momentum p, again to second order in small p.

(d) Suppose that the potential is rotationally symmetric, i.e.

  Vi(1) = 0,                Vij(2)  = V(2)dij.                                                                                 (6)

With this simplification, find the phase shifts d? for ? = 0 and ? = 1. You may safely assume that the phase shifts are weak at small momentum, so that eid? sin d? » d?.

2. Suppose that the potential energy V(r) = —ef(r) is the result of a charge distribution p(x) with

                             Ñ2f = 4pp,                                                                                                      (7)

and the charge density is “short-range” with p(r) going to zero exponentially fast as r ® ¥.

(a) Show that           V(q) = -4ph2e/p2 òd3xe-id.x/hp(x).

(b) Now suppose that p(x) has a delta-function piece coming from a nucleon with charge Ze at r = 0, and pe(x) describes the rest of the charge density, coming from an electron cloud around the nucleus. Find the function f(q) to first order in the Born approximation in terms of Z and pe.

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