A.31 The Born series

The Born approxi­mation is concerned with the problem of a particle of a given momentum that is slightly perturbed by a non­zero potential that it encounters. This note gives a description how this problem may be solved to high accuracy. The solution provides a model for the so-called “Feynman diagrams” of quantum electro­dynamics.

It is assumed that in the absence of the perturb­ation, the wave function of the particle would be

$$\psi_0 = e^{{\rm i}k z}$$

In this state, the particle has a momentum ${\hbar}k$ that is purely in the $z$-direction. Note that the above state is not normalized, and cannot be. That reflects the Heisenberg uncertainty principle: since the particle has precise momentum, it has infinite uncertainty in position. For real particles, wave functions of the form above must be combined into wave packets, chapter 7.10. That is not important for the current discussion.

The perturbed wave function $\psi$ can in principle be obtained from the so-called integral Schrödinger equation, {A.13} (A.42):

$$\psi({\skew0\vec r}) = \psi_0({\skew0\vec r}) - \frac{m}{2... ...skew0\vec r}^{\,\prime}) {\,\rm d}^3{\skew0\vec r}^{\,\prime}$$

Evaluating the right hand side in this equation would give $\psi$. Unfortunately, the right hand side cannot be evaluated because the integral contains the unknown wave function $\psi$ still to be found. However, Born noted that if the perturb­ation is small, then so is the difference between the true wave function $\psi$ and the unperturbed one $\psi_0$. So a valid approxi­mation to the integral can be obtained by replacing $\psi$ in it by the known $\psi_0$. That is certainly much better than just leaving the integral away completely, which would give $\psi$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\psi_0$.

And note that you can repeat the process. Since you now have an approxi­mation for $\psi$ that is better than $\psi_0$, you can put that approxi­mation into the integral instead. Evaluating the right hand side then produces a still better approxi­mation for $\psi$. Which can then be put into the integral. Etcetera.

Figure A.22: Graphical inter­pretation of the Born series.

Graphi­cally, the process is illustrated in figure A.22. The most inaccurate approxi­mation is to take the perturbed wave function as the unperturbed wave function at the same position ${\skew0\vec r}$:

$$\psi \approx \psi_0$$

An improvement is to add the integral evaluated using the unperturbed wave function:

$$\psi({\skew0\vec r}) = \psi_0({\skew0\vec r}) - \frac{m}{2... ...skew0\vec r}^{\,\prime}) {\,\rm d}^3{\skew0\vec r}^{\,\prime}$$

To represent this concisely, it is convenient to introduce some shorthand notations:

$$\psi_0' \equiv \psi_0({\skew0\vec r}^{\,\prime}) \qquad ... ...}\vert}}{\vert{\skew0\vec r}-{\skew0\vec r}^{\,\prime}\vert}$$

Using those notations the improved approxi­mation to the wave function is

$$\psi \approx \psi_0 + \int \psi_0' v' g_{\prime}^{\vphantom{\prime}}$$

Note what the second term does: it takes the unperturbed wave function at some different location ${\skew0\vec r}^{\,\prime}$, multi­plies it by a “vertex factor” $v'$, and then adds it to the wave function at ${\skew0\vec r}$ multi­plied by a “propagator” $g_{\prime}^{\vphantom{\prime}}$. This is then summed over all locations ${\skew0\vec r}^{\,\prime}$. The second term is illustrated in the second graph in the right hand side of figure A.22.

The next better approxi­mation is obtained by putting the two-term approxi­mation above in the integral:

$$\psi \approx \psi_0 + \int \left[\psi_0' + \int \psi_0''... ...me\prime}^{\prime}\right] v' g_{\prime}^{\vphantom{\prime}}$$

where

$$\psi_0'' \equiv \psi_0({\skew0\vec r}^{\,\prime\prime}) ... ...kew0\vec r}^{\,\prime}-{\skew0\vec r}^{\,\prime\prime}\vert}$$

Note that it was necessary to change the notation for one integr­ation variable to ${\skew0\vec r}^{\,\prime\prime}$ to avoid using the same symbol for two different things. Compared to the previous approxi­mation, there is now a third term:

$$\psi = \psi_0 + \int \psi_0' v' g_{\prime}^{\vphantom{\pri... ...\prime\prime}^{\prime} \; v' g_{\prime}^{\vphantom{\prime}}$$

This third term takes the unperturbed wave function at some position ${\skew0\vec r}^{\,\prime\prime}$, multi­plies it by the local vertex factor $v''$, propagates that to a location ${\skew0\vec r}^{\,\prime}$ using propagator $g_{\prime\prime}^{\prime}$, multi­plies it by the vertex factor $v'$, and propagates it to the location ${\skew0\vec r}$ using propagator $g_{\prime}^{\vphantom{\prime}}$. That is summed over all combin­ations of locations ${\skew0\vec r}^{\,\prime\prime}$ and ${\skew0\vec r}^{\,\prime}$. The idea is shown in the third graph in the right hand side of figure A.22.

Continuing this process produces the Born series:

$$\psi = \psi_0 + \int \psi_0' v' g_{\prime}^{\vphantom{\pri... ...ime}^{\prime} \; v' g_{\prime}^{\vphantom{\prime}} + \ldots$$

The Born series inspired Feynman to formulate relativistic quantum mechanics in terms of vertices connected together into “Feynman diagrams.” Since there is a non­technical, very readable discussion available from the master himself, [18], there does not seem much need to go into the details here.