Quantum Description of Scattering This is a tutorial on the quantum mechanical description of the (unpolarized) scattering process: we are NOT deriving these results, just going over them The wave function (“amplitude”) consists of two parts: Incident plane wave corresponding to the beam Outgoing scattered wave with amplitude that decays as 1/r (so that the probability density falls as 1/r2 (as it should) The overall wave function is the sum of the two, but the outgoing Here the BOLD characters represent vectors and k and k’ represent (divided by ) the momentum of the projectile particle before and after the scattering.
Interpretation of the wave function The incident and scattered “parts” of the wave function are f(r) = and fscat(r) = The probability flux for a wave function y(r) is given by: So that the incident and scattered flux are: The differential cross-section is given by the ratio and so
Momentum Transfer and Scattering Amplitude The factor f(k, k’) can be thought of as the angular amplitude f(q) for the scattering, and [ds/dW](q)=|f(q)|2. As in the case of the classical scattering process, q2, the square of the magnitude of change in the momentum q = (p’- p), is a key variable, and is related to the scattering angle by: The scattering amplitude function f(k, k’) is given by: f(k, k’) = Quantum Mechanically this is the matrix element for the potential V(r) between the incident wave and the overall wave function.
Integral Equation So there is a bit of circular argument here…because in order to calculate the overall wave function y(r), one needs to know what it is already… What this means is that y(r) is the solution to the “integral equation”: Here f(r) is the incident wave This integral equation is not particularly easy to solve However, in this whole formulation, we have considered the scattering potential V(r) to be a relatively small “perturbation” to the incident wave, and should modify f(r) only slightly
Born Approximation One strategy for solving the integral equation is to do so by “iteration”, starting with the substitution where y(r) appears inside the integral This allows us to calculate the “1st order” approximation for the scattering amplitude f(k, k’) f1(k, k’) = The resulting wave function y1(r) can be plugged back into the integral to calculate the 2nd order amplitude f2(k, k’) This Iterative Process is called the Born Approximation (after Max Born: ***grandfather of Olivia Newton-John***)
1st order Born Approximation For a Spherically symmetrical potential V(r) = where q is the magnitude of momentum change, or “transfer”, and f(k, k’) is just the Fourier transform of V(r) in variable q Curiously,, the integral for diverges for the 1/r potential Has anybody noticed that the total cross-section integrated from 0 to 180 degrees in q is INFINITE? Could these two statements be related in some way What do we do now Boss? (whose nickname is this?)
Born Approx. on Yukawa Potentuial We use a trick: we apply the approximation on the Yukawa Potential (which can be thought to describe the potential for nucleon-nucleon strong interaction Reason: the integral for f1(k, k’) converges We can recover the Rutherford scattering case by taking the limit m0, and The result: Note the function is in q2: And we have:
Back to Rutherford Scattering And taking the limit m0, and we have: Noting that k = p and E=p2/2m we rewrite the result as: Rather amazingly, this first order Born Approximation result (ok so we had to go through the Yukawa potential) is identical to the one we got from the classical calculation!