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

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.

Figure $\PageIndex{1}$: Born Series - scattering as series of terms