13.1: Born Series, Green Functions - A Hint of Quantisation of the Field

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Solving the Schrödinger equation using Green Functions automatically gives a solution in a form appropriate for scattering. By making the substitution \(E = \hbar^2 k^2/2\mu\) and \(U(r) = (2\mu /\hbar^2 )V (r)\) we can write the TISE as:

\[[\nabla^2 + k^2 ] \Phi = U(r)\Phi \nonumber\]

For \(U(r) = 0\) this gives \(\phi_0(r) = Ae^{i{\bf k.r}}\), a travelling wave. We now introduce a ‘Green’s Function’ for the operator \([\nabla^2 + k^2 ]\), which is the solution to the equation:

\[[\nabla^2 + k^2 ]G(r) = \delta (r)G(r) \\ G(r) = − \text{ exp}(ikr)/4\pi r \nonumber\]

\(\delta (r)\) is the Dirac delta-function as is \(\delta (r)G(r)\), since G(r) diverges at the origin. G(r) has the property that any function \(\Phi\) which satisfies

\[\Phi(r) = \phi_0 (r) +\int G \left(r-r^{\prime} \right) U \left(r^{\prime} \right) \Phi (r') d^{3} r^{\prime} \nonumber\]

where \(\phi_0(r)\) is the free particle solution, will be a solution to the TISE. Since \(\phi_0(r)\) is the unscattered incoming wave, the second term must represent the scattered wave.

Thus the general solution to the TISE is given by:

\[\Phi(r) = A e^{i k \cdot r}+\int G \left(r-r^{\prime} \right) U \left(r^{\prime} \right) \Phi (r') d^{3} r^{\prime} \nonumber\]

In this expression, \(\Phi\) appears on both sides. We can substitute for \(\Phi\) using the same equation:

\[\Phi(r) = A e^{i k \cdot r}+\int G \left(r-r^{\prime} \right) U \left(r^{\prime} \right) A e^{i k \cdot r^{\prime}} d^{3} r^{\prime}+\iint G \left(r-r^{\prime} \right) U \left(r^{\prime} \right) G \left(r^{\prime}-r^{\prime \prime} \right) U \left(r^{\prime \prime} \right) \Phi \left(r^{\prime \prime} \right) d^{3} r^{\prime} d^{3} r^{\prime \prime} \nonumber\]

Repeated substitutions gives the Born series, terminated by a term involving \(\Phi (r)\) itself. If the potential is weak, the higher order terms can be ignored. The first order term is just the matrix element between the incoming plane wave and the Green function: the Born approximation again! If we think of the potential \(U\) as an operator, the first term represents the incoming wavefunction being operated on once. The second term represents the incoming wavefunction being operated on twice. And so forth. This suggests a way of quantising the effect of the field: The first order term corresponds to a single scattering event, the second order term to double scattering etc.

13.1.PNG — Figure \(\PageIndex{1}\): Born Series - scattering as series of terms