Sub­sec­tions


A.30 Three-di­men­sional scat­ter­ing

This note in­tro­duces some of the gen­eral con­cepts of three-di­men­sion­al scat­ter­ing, in case you run into them. For more de­tails and ac­tual ex­am­ples, a quan­tum me­chan­ics text for physi­cists will need to be con­sulted; it is a big thing for them.

Fig­ure A.21: Scat­ter­ing of a beam off a tar­get.
\begin{figure}\centering
\setlength{\unitlength}{1pt}
\begin{picture}(405,21...
...\theta,\phi)\frac{e^{{\rm i}p_\infty r/\hbar}}{r}$}}
\end{picture}
\end{figure}

The ba­sic idea is as sketched in fig­ure A.21. A beam of par­ti­cles is sent in from the far left to­wards a three-di­men­sion­al tar­get. Part of the beam hits the tar­get and is scat­tered, to be picked up by sur­round­ing de­tec­tion equip­ment.

It will be as­sumed that the col­li­sion with the tar­get is elas­tic, and that the par­ti­cles in the beam are suf­fi­ciently light that they scat­ter off the tar­get with­out trans­fer­ring ki­netic en­ergy to it. In that case, the tar­get can be mod­eled as a steady po­ten­tial en­ergy field. And if the tar­get and/or in­com­ing par­ti­cles are elec­tri­cally neu­tral, it can also be as­sumed that the po­ten­tial en­ergy de­cays fairly quickly to zero away from the tar­get. (In fact, a lot of re­sults in this note turn out not ap­ply to a slowly de­cay­ing po­ten­tial like the Coulomb one.)

It is con­ve­nient to use a spher­i­cal co­or­di­nate sys­tem $(r,\theta,\phi)$ with its ori­gin at the scat­ter­ing ob­ject and with its axis aligned with the di­rec­tion of the in­com­ing beam. Since the axis of a spher­i­cal co­or­di­nate sys­tem is usu­ally called the $z$-​axis, the hor­i­zon­tal co­or­di­nate will now be in­di­cated as $z$, not $x$ like in the one-di­men­sion­al analy­sis done ear­lier.

In the en­ergy eigen­func­tions, the in­com­ing par­ti­cle beam can be rep­re­sented as a one-di­men­sion­al wave. How­ever, un­like for the one-di­men­sion­al scat­ter­ing of fig­ure 7.22, now the wave is not just scat­tered to the left and right, but in all di­rec­tions, in other words to all an­gles $\theta$ and $\phi$. The far-field be­hav­ior of the en­ergy eigen­func­tions is

\begin{displaymath}
\fbox{$\displaystyle
\psi_E \sim C^{\rm{l}}_{\rm{f}} e^{{\...
...{for}\quad r\to\infty
\qquad p_\infty \equiv \sqrt{2mE}
$} %
\end{displaymath} (A.216)

Here $E$ is the ki­netic en­ergy of the in­com­ing par­ti­cles and $m$ the mass. There­fore $p_\infty$ is what clas­si­cal physics would take to be the mo­men­tum of the par­ti­cles at in­fin­ity. The first term in the far field be­hav­ior al­lows the in­com­ing par­ti­cles to be de­scribed, as well as the same par­ti­cles go­ing out again un­per­turbed. If some joker re­moves the tar­get, that is all there is.

The sec­ond term de­scribes the out­go­ing scat­tered par­ti­cles. The con­stant $C_{\rm {f}}(\theta,\phi)$ is called the “scat­ter­ing am­pli­tude.” The sec­ond term also con­tains a fac­tor $e^{{{\rm i}}p_{\infty}r/\hbar}$ con­sis­tent with wave pack­ets that move ra­di­ally away from the tar­get in the far field.

Fi­nally, the sec­ond term con­tains a fac­tor 1/$r$. There­fore the mag­ni­tude of the sec­ond term de­creases with the dis­tance $r$ from the tar­get. This hap­pens be­cause the prob­a­bil­ity of find­ing a par­ti­cle in a given de­tec­tion area should de­crease with dis­tance. In­deed, the to­tal de­tec­tion area is $4{\pi}r^2$, where $r$ is the dis­tance at which the de­tec­tors are lo­cated. That in­creases pro­por­tional to $r^2$, and the to­tal num­ber of par­ti­cles to de­tect per unit time is the same re­gard­less of where the de­tec­tors are lo­cated. There­fore the prob­a­bil­ity of find­ing a par­ti­cle per unit area should de­crease pro­por­tional to 1/$r^2$. Since the prob­a­bil­ity of find­ing a par­ti­cle is pro­por­tional to the square of the wave func­tion, the wave func­tion it­self must be pro­por­tional to 1/$r$. The sec­ond term above makes it so.

Con­sider now the num­ber of par­ti­cles that is de­tected in a given small de­tec­tion area ${\rm d}{A}$. The scat­tered stream of par­ti­cles mov­ing to­wards the de­tec­tion area has a ve­loc­ity $v$ $\vphantom0\raisebox{1.5pt}{$=$}$ $p_\infty$$\raisebox{.5pt}{$/$}$$m$. There­fore in a time in­ter­val ${\rm d}{t}$, the de­tec­tion area sam­ples a vol­ume of the scat­tered par­ti­cle stream equal to ${\rm d}{A}$ $\times$ ${v}{\rm d}{t}$. The chances of find­ing par­ti­cles are pro­por­tional to the square mag­ni­tude of the wave func­tion times that vol­ume. Us­ing the as­ymp­totic wave func­tion above, the num­ber of par­ti­cles de­tected will be

\begin{displaymath}
{\rm d}I = \mbox{[constant] }
\vert C_{\rm {f}}(\theta,\phi)\vert^2 \frac{{\rm d}A}{r^2} {\rm d}t
\end{displaymath}

The con­stant in­cludes the fac­tor $p_\infty$$\raisebox{.5pt}{$/$}$$m$. The con­stant must also ac­count for the fact that the wave func­tion is not nor­mal­ized, and that there is a con­tin­u­ous stream of par­ti­cles to be found, rather than just one par­ti­cle.

Ac­cord­ing to the above ex­pres­sion, the num­ber of par­ti­cles de­tected in a given area ${\rm d}{A}$ is pro­por­tional to its three-di­men­sion­al an­gu­lar ex­tent

\begin{displaymath}
{\rm d}\Omega \equiv \frac{{\rm d}A}{r^2}
\end{displaymath}

This is the so-called “solid an­gle” oc­cu­pied by the de­tec­tion area el­e­ment. It is the three-di­men­sion­al gen­er­al­iza­tion of two-di­men­sion­al an­gles. In two di­men­sions, an el­e­ment of a cir­cle with arc length ${\rm d}{s}$ oc­cu­pies an an­gle ${\rm d}{s}$$\raisebox{.5pt}{$/$}$$r$ when ex­pressed in ra­di­ans. Sim­i­larly, in three di­men­sions, an el­e­ment of a sphere with area ${\rm d}{A}$ oc­cu­pies a solid an­gle ${\rm d}{A}$$\raisebox{.5pt}{$/$}$$r^2$ when ex­pressed in “stera­di­ans.”

In those terms, the num­ber ${\rm d}{I}$ of par­ti­cles de­tected in an in­fin­i­tes­i­mal solid an­gle ${\rm d}\Omega$ is

\begin{displaymath}
{\rm d}I = \mbox{[constant] } \vert C_{\rm {f}}(\theta,\phi)\vert^2 {\rm d}\Omega {\rm d}t
\end{displaymath}

As noted, the con­stant of pro­por­tion­al­ity de­pends on the rate at which par­ti­cles are sent at the tar­get. The more par­ti­cles are sent at the tar­get, the more will be de­flected. The num­ber of par­ti­cles in the in­com­ing beam per unit beam cross-sec­tional area and per unit time is called the “lu­mi­nos­ity” of the beam. It is re­lated to the square of the wave func­tion of the in­com­ing beam through the re­la­tion

\begin{displaymath}
{\rm d}I_{\rm {b}} = \mbox{[constant] }
\vert C^{\rm {l}}_{\rm {f}}\vert^2 {\rm d}A_{\rm {b}} {\rm d}t
\end{displaymath}

Here ${\rm d}{A}_{\rm {b}}$ is a cross sec­tional area el­e­ment of the in­com­ing par­ti­cle beam and ${\rm d}{I}_{\rm {b}}$ the num­ber of par­ti­cles pass­ing through that area.

Physi­cist like to re­late the scat­tered par­ti­cle flow in a given in­fin­i­tes­i­mal solid an­gle ${\rm d}{\Omega}$ to an equiv­a­lent in­com­ing beam area ${\rm d}{A_b}$ through which the same num­ber of par­ti­cles flow. There­fore they de­fine the so-called “dif­fer­en­tial cross-sec­tion” as

\begin{displaymath}
\fbox{$\displaystyle
\frac{{\rm d}\sigma}{{\rm d}\omega} \equiv \frac{{\rm d}A_{\rm{b,equiv}}}{{\rm d}\Omega}
$}
\end{displaymath} (A.217)

The quan­tity ${\rm d}{A}_{\rm {b,equiv}}$ can be thought of as the in­fin­i­tes­i­mal area of the in­com­ing beam that ends up in the in­fin­i­tes­i­mal solid an­gle ${\rm d}\Omega$. So the dif­fer­en­tial cross-sec­tion is a scat­tered par­ti­cle den­sity ex­pressed in suit­able terms.

Note how well cho­sen the term “dif­fer­en­tial cross-sec­tion” re­ally is. If physi­cists had called it some­thing like scat­tered cross-sec­tion den­sity, or even sim­ply scat­tered cross-sec­tion, non­ex­perts would prob­a­bly have a pretty good guess what physi­cists were talk­ing about. But cross sec­tion by it­self can mean any­thing. There is noth­ing in the term to in­di­cate that it is a mea­sure for how many par­ti­cles are scat­tered. And pre­ced­ing it by dif­fer­en­tial is a stroke of ge­nius be­cause it is not a dif­fer­en­tial, it is a dif­fer­en­tial quo­tient. This will con­fuse math­e­mat­i­cally lit­er­ate non­ex­perts even more.

The dif­fer­en­tial cross sec­tion does not de­pend on how many par­ti­cles are sent at the tar­get, nor on wave func­tion nor­mal­iza­tion. Fol­low­ing the ex­pres­sions for the par­ti­cle flows given above, the dif­fer­en­tial cross sec­tion is sim­ply

\begin{displaymath}
\fbox{$\displaystyle
\frac{{\rm d}\sigma}{{\rm d}\omega}
...
...}(\theta,\phi)\vert^2}{\vert C^{\rm{l}}_{\rm{f}}\vert^2}
$} %
\end{displaymath} (A.218)

More­over, the par­ti­cle flow in an in­com­ing beam area ${\rm d}{A}_{\rm {b}}$ may be mea­sured us­ing the same ex­per­i­men­tal tech­niques as are used to mea­sure the de­flected par­ti­cle flow. Var­i­ous sys­tem­atic er­rors in the ex­per­i­men­tal method will then can­cel in the ra­tio, giv­ing more ac­cu­rate val­ues.

The to­tal area of the in­com­ing beam that gets scat­tered is called the “to­tal cross-sec­tion” $\sigma$:

\begin{displaymath}
\fbox{$\displaystyle
\sigma \equiv A_{\rm b,total}
$}
\end{displaymath} (A.219)

Of course, the name is quite be­low the nor­mal stan­dards of physi­cists, since it re­ally is a to­tal cross-sec­tion. For­tu­nately, physi­cist are clever enough not to say what cross sec­tion it is, and cross-sec­tion can mean many things. Also, by us­ing the sym­bol $\sigma$ in­stead of some­thing log­i­cal like $A_{\rm {b}}$ for the dif­fer­en­tial cross-sec­tion, physi­cists do their best to re­duce the dam­age as well as pos­si­ble.

If you re­main dis­ap­pointed in physi­cists, take some com­fort in the fol­low­ing term for scat­ter­ing that can be de­scribed us­ing clas­si­cal me­chan­ics: the “im­pact pa­ra­me­ter.” If you guess that it de­scribes the lo­cal physics of the par­ti­cle im­pact process, it is re­ally hi­lar­i­ous to physi­cists. In­stead, think cen­ter­line off­set; it de­scribes the lo­ca­tion rel­a­tive to the cen­ter­line of the in­com­ing beam at which the par­ti­cles come in; it has no di­rect re­la­tion what­so­ever to what sort of im­pact (if any) these par­ti­cles end up ex­pe­ri­enc­ing.

The to­tal cross sec­tion can be found by in­te­grat­ing the dif­fer­en­tial cross sec­tion over all de­flec­tion an­gles:

\begin{displaymath}
\sigma = \int_{\rm all} \frac{{\rm d}A_{\rm {b,equiv}}}{{\rm d}\Omega} { \rm d}\Omega
\end{displaymath}

In spher­i­cal co­or­di­nates this can be writ­ten out ex­plic­itly as
\begin{displaymath}
\fbox{$\displaystyle
\sigma = \int\limits_{\theta=0^+}^{\p...
..._{\rm{f}}\vert^2}
\sin\theta { \rm d}\theta{\rm d}\phi
$} %
\end{displaymath} (A.220)


A.30.1 Par­tial wave analy­sis

Jim Napoli­tano from RPI and Cor­nell notes:

The term Par­tial Wave Analy­sis is poorly de­fined and overused.
Gee, what a sur­prise! For one, they are com­po­nent waves, not par­tial waves. But you al­ready com­po­nently as­sumed that they might be.

This dis­cus­sion will re­strict it­self to spher­i­cally sym­met­ric scat­ter­ing po­ten­tials. In that case, the analy­sis of the en­ergy eigen­func­tions can be done much like the analy­sis of the hy­dro­gen atom of chap­ter 4.3. How­ever, the bound­ary con­di­tions at in­fin­ity will be quite dif­fer­ent; the ob­jec­tive is not to de­scribe bound par­ti­cles, but par­ti­cles that come in from in­fin­ity with pos­i­tive ki­netic en­ergy and are scat­tered back to in­fin­ity. Also, the po­ten­tial will of course not nor­mally be a Coulomb one.

But just like for the hy­dro­gen atom, the en­ergy eigen­func­tions can be taken to be ra­dial func­tions times spher­i­cal har­mon­ics $Y_l^m$:

\begin{displaymath}
\psi_{Elm}(r,\theta,\phi) = R_{El}(r) Y_l^m(\theta,\phi)
\end{displaymath} (A.221)

These en­ergy eigen­func­tions have def­i­nite an­gu­lar mo­men­tum in the $z$-​di­rec­tion $m\hbar$, as well def­i­nite square an­gu­lar mo­men­tum $l(l+1)\hbar^2$. The ra­dial func­tions $R_{El}$ will not be the same as the hy­dro­gen $R_{nl}$ ones.

The in­com­ing plane wave $e^{{{\rm i}}p_{\infty}z/\hbar}$ has zero an­gu­lar mo­men­tum in the $z$-​di­rec­tion. Un­for­tu­nately, it does not have def­i­nite square an­gu­lar mo­men­tum. In­stead, it can be writ­ten as a lin­ear com­bi­na­tion of free-space en­ergy eigen­func­tions with dif­fer­ent val­ues of $l$, hence with dif­fer­ent square an­gu­lar mo­men­tum:

\begin{displaymath}
e^{{\rm i}p_\infty z/\hbar}
= \sum_{l=0}^\infty c_{{\rm {w...
...\theta)
\qquad
c_{{\rm {w}},l} = {\rm i}^l \sqrt{4\pi(2l+1)}
\end{displaymath} (A.222)

See {A.6} for a de­riva­tion and the pre­cise form of the spher­i­cal Bessel func­tions $j_l$.

Now find­ing the com­plete en­ergy eigen­func­tion cor­re­spond­ing to the in­com­ing wave di­rectly is typ­i­cally awk­ward, es­pe­cially an­a­lyt­i­cally. Of­ten it is eas­ier to solve the prob­lem for each term in the above sum sep­a­rately and then add these so­lu­tions all to­gether. That is where the name par­tial wave analy­sis comes from. Each term in the sum cor­re­sponds to a par­tial wave, if you use suf­fi­ciently lousy ter­mi­nol­ogy.

The par­tial wave analy­sis re­quires that for each term in the sum, an en­ergy eigen­func­tion is found of the form $\psi_{El}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $R_{El}Y_l^0$. The re­quired be­hav­ior of this eigen­func­tion in the far field is

\begin{displaymath}
\fbox{$\displaystyle
\psi_{El} \sim
\left[c_{{\rm{w}},l}...
...hbar)\right]
Y_l^0(\theta) \qquad\mbox{for } r\to\infty
$} %
\end{displaymath} (A.223)

Here the first term is the com­po­nent of the in­com­ing plane wave cor­re­spond­ing to spher­i­cal har­monic $Y_l^0$. The sec­ond term rep­re­sents the out­go­ing de­flected par­ti­cles. The value of the co­ef­fi­cient $c_{{\rm {f}},l}$ is de­ter­mined in the so­lu­tion process.

Note that the above far field be­hav­ior is quite sim­i­lar to that of the com­plete en­ergy eigen­func­tion as given ear­lier in (A.216). How­ever, here the co­ef­fi­cient $C^{\rm {l}}_{\rm {f}}$ was set to 1 for sim­plic­ity. Also, the ra­dial part of the re­flected wave func­tion was writ­ten us­ing a Han­kel func­tion of the first kind $h_l^{(1)}$. This Han­kel func­tion pro­duces the same $e^{{{\rm i}}p_{\infty}r/\hbar}$$\raisebox{.5pt}{$/$}$$r$ ra­dial be­hav­ior as the sec­ond term in (A.216), {A.6} (A.25). How­ever, the Han­kel func­tion has the ad­van­tage that it be­comes ex­act as soon as the scat­ter­ing po­ten­tial be­comes zero. It is not just valid at very large $r$ like the bare ex­po­nen­tial.

To be sure, for a slowly de­cay­ing po­ten­tial like the Coulomb one, the Han­kel func­tion is no bet­ter than the ex­po­nen­tial. How­ever, the Han­kel func­tion is very closely re­lated to the Bessel func­tion $j_l$, {A.6}, al­low­ing var­i­ous help­ful re­sults to be found in ta­ble books. If the po­ten­tial en­ergy is piece­wise con­stant, it is even pos­si­ble to solve the com­plete prob­lem us­ing Bessel and Han­kel func­tions. These func­tions can be tied to­gether at the jumps in po­ten­tial in a way sim­i­lar to ad­den­dum {A.27}.

In terms of the as­ymp­totic be­hav­ior above, the dif­fer­en­tial cross sec­tion is

\begin{displaymath}
\fbox{$\displaystyle
\frac{{\rm d}\sigma}{{\rm d}\omega} =...
...}},{\underline l}} Y_l^0(\theta)Y_{\underline l}^0(\theta)
$}
\end{displaymath} (A.224)

This can be ver­i­fied us­ing {A.6} (A.25), (A.216), and (A.218). The Bessel func­tions form the in­com­ing wave and do not con­tribute. For the to­tal cross-sec­tion, note that the spher­i­cal har­mon­ics are or­tho­nor­mal, so

\begin{displaymath}
\fbox{$\displaystyle
\sigma = \frac{\hbar^2}{p_\infty^2} \sum_{l=0}^\infty \vert c_{{\rm{f}},l}\vert^2
$}
\end{displaymath}

One spe­cial case is worth men­tion­ing. Con­sider par­ti­cles of such low mo­men­tum that their typ­i­cal quan­tum wave length, $2\pi\hbar$$\raisebox{.5pt}{$/$}$$p_\infty$, is gi­gan­tic com­pared to the ra­dial size of the scat­ter­ing po­ten­tial. Par­ti­cles of such large wave lengths do not no­tice the fine de­tails of the scat­ter­ing po­ten­tial at all. Con­versely, nor­mally the scat­ter­ing po­ten­tial only no­tices the in­com­ing par­tial wave with $l$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0. That is be­cause the Bessel func­tions are pro­por­tional to

\begin{displaymath}
(p_\infty r/\hbar)^l
\end{displaymath}

for small ar­gu­ments. If the wave length is large com­pared to the typ­i­cal ra­dial size $r$ of the scat­ter­ing po­ten­tial, this is neg­li­gi­ble un­less $l$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0. Now $l$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0 cor­re­sponds to a wave func­tion that is the same in all di­rec­tions; it is pro­por­tional to the con­stant spher­i­cal har­monic $Y_0^0$. If only the par­tial wave that is the same in all di­rec­tions gets scat­tered, then the par­ti­cles get scat­tered equally in all di­rec­tions (if they get scat­tered at all.)

Co­in­ci­dently, equal scat­ter­ing in all di­rec­tions also hap­pens in an­other case: scat­ter­ing of clas­si­cal point par­ti­cles from a hard elas­tic sphere. That is very much the op­po­site case, be­cause neg­li­gi­ble un­cer­tainty in po­si­tion re­quires high, not low, en­ergy of the par­ti­cles. In any case, the sim­i­lar­ity be­tween the two cases is is su­per­fi­cial. If a beam of clas­si­cal par­ti­cles is di­rected at a hard sphere, only an area of the beam equal to the frontal area of the sphere gets scat­tered. But if you work out the scat­ter­ing of low-en­ergy quan­tum par­ti­cles from a hard sphere, you get a to­tal scat­ter­ing cross sec­tion that is 4 times big­ger.


A.30.2 Par­tial wave am­pli­tude

This sub­sec­tion gives some fur­ther odds and ends on par­tial wave analy­sis, for the in­cur­ably cu­ri­ous.

Re­call that a par­tial wave has an as­ymp­totic be­hav­ior

\begin{displaymath}
\psi_{El} \sim
\left[c_{{\rm {w}},l}j_l(p_\infty r/\hbar)...
...ty r/\hbar)\right]
Y_l^0(\theta) \qquad\mbox{for } r\to\infty
\end{displaymath}

The first term cor­re­sponds to the wave func­tion of the in­com­ing par­ti­cles. The sec­ond term is the ef­fect of the scat­ter­ing po­ten­tial.

Physi­cists like to write the co­ef­fi­cient of the scat­tered wave as

\begin{displaymath}
\fbox{$\displaystyle
c_{{\rm{f}},l} = {\rm i}k c_{{\rm{w}},l} a_l
$}
\end{displaymath} (A.225)

They call the so-de­fined con­stant $a_l$ the “par­tial wave am­pli­tude” be­cause ob­vi­ously it is not a par­tial wave am­pli­tude. Con­fus­ing peo­ple is al­ways funny.

Now every par­tial wave by it­self is a so­lu­tion to the Hamil­ton­ian eigen­value prob­lem. That means that every par­tial wave must en­sure that par­ti­cles can­not just sim­ply dis­ap­pear. That re­stricts what the par­tial wave am­pli­tude can be. It turns out that it can be writ­ten in terms of a real num­ber $\delta_l$:

\begin{displaymath}
\fbox{$\displaystyle
a_l = \frac{1}{k} e^{{\rm i}\delta_l} \sin\delta_l
$}
\end{displaymath} (A.226)

The real num­ber $\delta_l$ is called the “phase shift.”

Some physi­cist must have got con­fused here, be­cause it re­ally is a phase shift. To see that, con­sider the de­riva­tion of the above re­sult. First the as­ymp­totic be­hav­ior of the par­tial wave is rewrit­ten in terms of ex­po­nen­tials us­ing {A.6} (A.24) and (A.25). That gives

\begin{displaymath}
\psi_{El} \sim \ldots
\left[e^{-{\rm i}p_\infty r/\hbar} +
(-1)^{l+1}e^{{\rm i}p_\infty r/\hbar}(1+2{\rm i}k a_l)
\right]
\end{displaymath}

The dots stand for com­mon fac­tors that are not im­por­tant for the dis­cus­sion. Phys­i­cally, the first term above de­scribes spher­i­cal wave pack­ets that move ra­di­ally in­wards to­ward the tar­get. The sec­ond term de­scribes wave pack­ets that move ra­di­ally out­wards away from the tar­get.

Now par­ti­cles can­not just dis­ap­pear. Wave pack­ets that go in to­wards the tar­get must come out again with the same am­pli­tude. And that means that the two terms in the as­ymp­totic be­hav­ior above must have the same mag­ni­tude. (This may also be shown math­e­mat­i­cally us­ing pro­ce­dures like in {A.32}.)

Ob­vi­ously the two terms do have the same mag­ni­tude in the ab­sence of scat­ter­ing, where $a_l$ is zero. But in the pres­ence of scat­ter­ing, the fi­nal par­en­thet­i­cal fac­tor will have to stay of mag­ni­tude one. And that means that it can be writ­ten in the form

\begin{displaymath}
1+2{\rm i}k a_l = e^{{\rm i}2 \delta_l}
\end{displaymath} (A.227)

for some real num­ber $\delta_l$. (The fac­tor 2 in the ex­po­nen­tial is put in be­cause physi­cists like to think of the wave be­ing phase shifted twice, once on the way in to the tar­get and once on the way out.) Clean­ing up the above ex­pres­sion us­ing the Euler for­mula (2.5) gives the stated re­sult.

If you add in the time de­pen­dent fac­tor $e^{{{\rm i}}Et/\hbar}$ of the com­plete un­steady wave func­tion, you can see that in­deed the waves are shifted by a phase an­gle $2\delta_l$ com­pared to the un­per­turbed wave func­tion. With­out any doubt, the name of the physi­cist re­spon­si­ble for call­ing the phase an­gle a phase an­gle has been os­tra­cized from physics. She will never be heard of again.


A.30.3 The Born ap­prox­i­ma­tion

The Born ap­prox­i­ma­tion as­sumes that the scat­ter­ing po­ten­tial is weak to de­rive ap­prox­i­mate ex­pres­sions for the scat­ter­ing.

Con­sider first the case that the scat­ter­ing po­ten­tial is zero. In that case, the wave func­tion is just that of the in­com­ing par­ti­cles:

\begin{displaymath}
\psi_E = e^{{\rm i}p_\infty z/\hbar} \qquad p_\infty=\sqrt{2mE}
\end{displaymath}

where $E$ is the en­ergy of the par­ti­cle and $m$ its mass.

Born con­sid­ered the case that the scat­ter­ing po­ten­tial $V$ is not zero, but small. Then the wave func­tion $\psi_E$ will still be close to the in­com­ing wave func­tion, but no longer ex­actly the same. In that case an ap­prox­i­ma­tion to the wave func­tion can be ob­tained from the so-called in­te­gral Schrö­din­ger equa­tion, {A.13} (A.42):

\begin{displaymath}
\psi_E({\skew0\vec r}) = e^{{\rm i}p_\infty z/\hbar}
- \fr...
...\skew0\vec r}^{ \prime}) { \rm d}^3{\skew0\vec r}^{ \prime}
\end{displaymath}

In par­tic­u­lar, in­side the in­te­gral the true wave func­tion $\psi_E$ can be re­placed by the in­com­ing wave func­tion:
\begin{displaymath}
\psi_E({\skew0\vec r}) \approx e^{{\rm i}p_\infty z/\hbar}
...
...rm i}p_\infty z'/\hbar} { \rm d}^3{\skew0\vec r}^{ \prime} %
\end{displaymath} (A.228)

It is not ex­act, but it is much bet­ter than just set­ting the in­te­gral to zero. The lat­ter would make the wave func­tion equal to the in­com­ing wave. With the ap­prox­i­mate in­te­gral, you get a valid ap­prox­i­ma­tion to the par­ti­cle de­flec­tions.

To get the dif­fer­en­tial cross sec­tion, ex­am­ine the be­hav­ior of (A.228) at given scat­ter­ing an­gles $\theta$ and $\phi$ for large $r$. That pro­duces, {D.47}:

\begin{displaymath}
\fbox{$\displaystyle
\frac{{\rm d}\sigma}{{\rm d}\omega}(\...
...}^{\,\prime}\;
\right\vert^2 \qquad \mbox{($V$\ small)}
$} %
\end{displaymath} (A.229)

Here
\begin{displaymath}
\vp_\infty^{\,\rm {l}} = p_\infty {\hat k}
\qquad
\vp_\in...
...fty {\hat\imath}_r
\qquad \mbox{(with $p_\infty=\sqrt{2mE}$)}
\end{displaymath} (A.230)

are the clas­si­cal mo­men­tum vec­tors of the in­com­ing and scat­tered par­ti­cles. Note that the di­rec­tion of $\vp_\infty$ de­pends on the con­sid­ered scat­ter­ing an­gles $\theta$ and $\phi$. And that ap­par­ently the mo­men­tum change of the par­ti­cles is a key fac­tor af­fect­ing the amount of scat­ter­ing.

One ad­di­tional ap­prox­i­ma­tion is worth men­tion­ing. Con­sider par­ti­cles of such low mo­men­tum that their quan­tum wave length, $2\pi\hbar$$\raisebox{.5pt}{$/$}$$p_\infty$, is gi­gan­tic com­pared to the ra­dial size of the scat­ter­ing po­ten­tial. Par­ti­cles of such wave lengths do not no­tice the fine de­tails of the scat­ter­ing po­ten­tial at all. Math­e­mat­i­cally, $p_\infty$ is so small that the ar­gu­ment of the ex­po­nen­tial in the dif­fer­en­tial cross sec­tion above can be as­sumed zero. Then:

\begin{displaymath}
\fbox{$\displaystyle
\frac{{\rm d}\sigma}{{\rm d}\omega} \...
...\right\vert^2 \qquad \mbox{($V$ and $p_\infty$ small)}
$} %
\end{displaymath} (A.231)

The dif­fer­en­tial cross sec­tion no longer de­pends on the an­gu­lar po­si­tion. If par­ti­cles get scat­tered at all, they get scat­tered equally in all di­rec­tions.

Note that the in­te­gral is in­fi­nite for a Coulomb po­ten­tial.