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:

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

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:

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

where

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

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:

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

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:

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

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, [19], there does not seem much need to go into the de­tails here.