数理解析研究所講究録980 巻 1997 年 162-170

162
Single-valued Wave Funct ion of Aharonov-Bohm Problem

rosh Kawamo to(川 $*$)$t1\mathrm{i}$$\mathrm{i}$$\gamma_{\mathrm{A}}^{\dot{\backslash }\sim}$Mar$\mathrm{i}\mathrm{t}$ime Safe ty Academy, TranspOrtat on Mi ni stry of Japan,$\mathrm{i}$Hiroshima 737

Kure,(Abstract)

The wave funct on of a charged particle orbi$\mathrm{i}$$\mathrm{t}\mathrm{i}$ng around a conf ned magneti c$\mathrm{i}$flux is single-valued. On the basis of this, the Aharonov-Bohm effect is studied

In conclus ion.

is found that the flux-dependent di ffract ion originated from$\mathrm{i}\mathrm{t}$the nterference between the two spli ted waves

impOss ble al \dagger ,hough the$\mathrm{i}$$\mathrm{t}$ampliof each wave funct on$\mathrm{t}\mathfrak{U}\mathrm{d}\mathrm{e}$a Per$\mathrm{i}\mathrm{s}$$\mathrm{i}$$\mathrm{i}\mathrm{o}\mathrm{d}\mathrm{i}_{\mathrm{C}}$$\mathrm{i}\mathrm{s}$$\mathrm{i}$funct on of the applied flux.
$\mathrm{i}$we obtai a greatly depressed cross sect on for a

Futhermore,$\mathrm{i}$$\mathrm{n}$$\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{d}$wa 11 tube.

(Text)

There has been a controversy whether the wave function of a part cle orbi$\mathrm{i}$around a conf ned magnet$\mathrm{i}$I

PaPer,$\mathrm{n}\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{s}$the$\mathrm{s}$flux$\mathrm{i}\mathrm{c}$$\Phi \mathrm{i}\mathrm{s}$si ngle-valued

ingle-valued wave function$\downarrow$))or mult -valued.

$\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{d}$. On thesingle-valued wave funct on,

we exami ne the so-called Aharonov-Bohm

cons

(i)

$\mathrm{i}$$\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{i}$ng of threeri se to a$\Phi$

ssues;

a charged beam

-dePendent di fraction pattern$\mathrm{i}\mathrm{s}\mathrm{i}$$\mathrm{i}$$\mathrm{i}\mathrm{s}$nfluenced bya funct on of$\mathrm{i}$Our wave funct on

(I)

6)

7)

8)

$\Pi$

$\mathrm{i}$where the Hami ltonian

present analysi\dagger ,heof rad us$\mathrm{i}$def ned$\mathrm{i}$$\mathrm{i}\mathrm{n}$

$a$

)

$5$

$\mathrm{t}$$\mathrm{s}$,running along$\mathrm{i}$a region$\gamma\geqq a$(\"u)

a scat$\mathrm{g}\mathrm{i}$vesan energy eigenvalue$\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}$$\mathrm{i}$ng cross section$\mathrm{i}$.

$i\hslash$$\partial\Psi/$$\Phi \mathrm{i}\mathrm{s}$(1)conf ned wi thin a rigid tube$\mathrm{i}$. Therefore the wave funct ion. Let usfect 5)$H\mathrm{i}\mathrm{s}$the magnet ic flux
$z-\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{s}$$\mathrm{e}\mathrm{f}$of a$\mathrm{i}$)$(\ddot{\dot{\mathrm{m}}})$$\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{s}$ted nto two parts

sat sf es the Schrod nger equat on$H= \frac{1}{\mathrm{z}\mu}(-i\hslash\nabla-qA)arrow 2+q\phi$I$\not\in$

$()$

10).

9)

5)

$\Phi$$\Psi(r, t)$$\mathrm{i}$$\partial t=H\Psi$3)$\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}$$\mathrm{f}$of a part cleof the beam$\mathrm{i}$B)ng$\mathrm{i}$obta$\mathrm{i}\mathrm{s}$$2$$\mathrm{t}\mathrm{i}$$\mathrm{d}\mathrm{i}_{\mathrm{V}\mathrm{i}\mathrm{d}\mathrm{e}}arrow A(r’ t-,)\mathrm{i}$$1\mathrm{t}^{\mathrm{r}}(arrow r, t)\mathrm{i}\mathrm{s}$nto two parts,$Aarrow(arrow r, t)=$163
$arrowarrow A_{0}(r)+arrow A_{1}$ $( arrow r, t)$,respect vely$\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}arrow A_{0}\mathrm{a}\mathrm{n}\mathrm{d}A_{1}arrow$a $d.c$ part$\mathrm{i}$and an $a.c$ part of the$\mathrm{t}0\mathrm{t}\mathrm{a}\mathrm{l}arrow A$$\mathrm{r}\mathrm{e}\mathrm{P}^{\mathrm{r}}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{I}\mathrm{l}\mathrm{t}$. Under the condi ion$\nabla\cross A=arrow \mathit{0}$$\mathrm{t}$the wave,funct ion takes a factori zed form$\iota \mathrm{r}(r, tarrow)=exP\mathrm{f}iq]_{arrow}^{arrow}A\mathrm{r}r_{\mathrm{o}}arrow$

The pseudo wave funct on$\psi$$\mathrm{i}$.

$( arrow r , , t)drarrow,$ $/\hslash \mathit{1}\psi(r, tarrow)$$(r, tarrow)$sat$\mathrm{i}\mathrm{s}\mathrm{f}\mathrm{i}$es a pseudo’$\mathrm{S}\mathrm{c}\mathrm{h}\Gamma \mathrm{o}\mathrm{d}\mathrm{i}$(2)nger equation$arrow$$\mathrm{r}arrowarrow E_{1}(r’, t)dr$,1$\psiarrow(r_{J}tarrow),$$i \hslash\frac{\partial}{\mathrm{a}\mathrm{t}}\psi(r, tarrow)=f-\frac{\hslash^{l}}{\mathrm{z}\mu}\nabla^{2}-q]$$(\mathit{3})$$\wedge \mathrm{r}_{\mathrm{O}}$where$\phi$$(rarrow , t)arrow$equali ty$\equiv \mathit{0}$has been chosen. To obtai$\nabla 5_{\gamma_{t}}arrow\overline{A}dr’=\mathrm{r}|-arrow Aarrow(r, t-*)\mathrm{y}\mathrm{i}$Wi th the use of Faraday’obtai$\mathrm{s}$a mult -valuedness of$\mathrm{i}$$\mathrm{n}$law$a_{1}$$( t)$$A_{0}arrow$$( arrow r)$$\nabla\cross A=arrow \mathit{0}$has been used.

we

Eq. (3),$\psi(r, tarrow)$

is a scaled flux

$a_{1}$

1$\psi(r, \theta, t)$$( t)+i\lambda \mathit{1}$$( t)\equiv q\Phi_{1}$integral constant composed of two parts,

froma path-independent

Eq. ( 3 )$\Sarrow E_{1}dr=arrow-\partial\Phi_{1}$ $( t)/\partial t\mathrm{i}\mathrm{n}$$\psi(r, \theta+\mathit{2}\pi, t)=exP\mathrm{f}-\mathit{2}\pi i$ $a$

Hereelded from$\mathrm{n}$$( t)/h$ , and$\lambda=\lambda_{0}+\lambda’$. A part

.

(4)

is an

$\lambda$$\lambda_{0}$is originatedand plays an essent ial role of the problem, whi le ’is an$\lambda$

irrelevant part.
explici$\mathrm{t}$potential

form of$A_{0}arrow(rarrow)$$\mathfrak{l}\mathrm{n}$$\lambda_{0}$this paper, we set,$\mathrm{i}\mathrm{c}$) magnetic

exists. In th

flux. For the purpose of$\mathrm{f}$inding thelet us consider a case where only the static vector$\mathrm{i}\mathrm{s}$takes the form of free part icle’(stat$\lambda’=\mathit{0}$$\Phi_{0}$$\mathrm{s}$case, the pseudo

type. However. Thi can be shown$\mathrm{s}$$\mathrm{i}\mathrm{f}$$\mathrm{S}\mathrm{c}\mathrm{h}\mathrm{r}\mathrm{o}\mathrm{d}||\mathrm{i}$$\psi(r, tarrow)$nger equat ion (3)is governed by thethe system is rotated on theaxis of the flux tube. In the system rotat ng at an angular veloci ty 2 $\pi/T$ ,$\mathrm{i}$our$arrow$$r_{0}$

in the factorized phase also rotates wi th the same angu lar veloci

Thus we obtain the pseudo$\mathrm{S}\mathrm{c}\mathrm{h}\mathrm{r}0|\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{e}\mathrm{r}$$\mathrm{t}\mathrm{y}$.

equation$i \hslash\frac{\partial}{\mathrm{a}\mathrm{t}}\psi_{\mathrm{r}}(rarrow, t)=\zeta-\frac{\mathrm{t}^{l}}{\mathrm{z}\mu}\nabla^{2}-q\frac{\mathrm{a}}{\mathrm{a}\mathrm{t}}r_{0}’(-t)\cdot A_{0}(\gamma_{0}(arrowarrow t))]\psi_{\mathrm{r}}(r-,, t)$

(S)

164
The solution of Eq. (5) has a form
where

$r$

$\psi_{r}(rarrow, t )$$( i7 ( t ) )$$=exD$

represents the rotation of the system$( t )$(6)$7_{\mathrm{t}\frac{\delta}{\mathrm{a}\mathrm{t}}7}’(t)=qA_{0}( arrowarrow r_{0}(t))\cdot\frac{\partial}{3t}r_{0}(arrow t)$

wh lea solut ion of the free Schrodi nger equat ion$( arrow\gamma, t)\mathrm{i}\mathrm{s}$$\mathrm{i}$$\chi_{0}$’I ntegrating Eq. (6) along a path$\Delta_{7}\equiv_{7}(T)$direct ion of the$\thetaarrow\theta+\mathit{2}\pi$acqui–7 (0)$\mathrm{i}\mathrm{n}$$=-\mathit{2}\pi$wide sense. as Berry’$(t)\mathrm{i}\mathrm{s}$$)$$arrow\theta_{0}$chosenreplaced byreplacement$arrow A=\mathit{0}$phase1 }$\downarrow$

,

$0$

)

$\psi_{r}(-\rangle r, t)$$\mathrm{o}\mathrm{f}arrow r_{0}$$\mathrm{i}$. The reason why the OPPOSi tesuch a$\mathrm{i}\mathrm{n}$$\mathrm{d}\mathrm{i}$$\mathrm{d}\mathrm{i}$rection.

. I other words. the role of$\mathrm{n}$readi ly seen from the expli. Considering thi$\thetaarrow\theta+\mathit{2}\pi$$\mathrm{i}\mathrm{s}$$( \theta - \theta_{0} ( t))$a more general$\mathrm{i}$$\mathrm{i}$$2$$\mathrm{i}$$\uparrow \mathrm{t}^{r}(arrow r, t)$gi ven$\mathrm{i}\mathrm{n}$)$t$$\thetaarrow\theta$

Hence,

Eq. (2)$\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{s}$$\mathrm{i}\mathrm{n}P_{\lrcorner}q$$\mathrm{i}$$\mathrm{v}\mathrm{i}$obta ned from$\mathrm{i}$$\mathrm{i}$$+\mathit{2}\pi$,$\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}$$\mathrm{i}\mathrm{s}$poss$\mathrm{i}\mathrm{b}\mathrm{l}\mathrm{e}$fi. Thatvalence between thewh ch$\mathrm{i}$$\nabla\cross$$\lambda 0=-\mathit{2}\pi a_{0}\mathrm{i}$ewpo nt

merely an example of thethe theory of quantum mechan cs 1$($

can beform of the factor zed phase

equ valence we have. from Eq. (7),the present equi valence

the wave function$\mathrm{c}\mathrm{i}\mathrm{t}$$\mathrm{i}$$\mathrm{s}$$\theta_{0}$$\gamma_{0}$Phys ically equivalent to the coordinate$\mathrm{i}\mathrm{s}$$( \gamma’)r’ d\theta’=\Phi_{0}/\mathit{2}\pi$$-\}$$\mathrm{i}$. The validi ty of$\zeta_{\theta.l)}^{\epsilon_{1}}A_{0}$a$\mathrm{i}\mathrm{n}$rect on of$\theta_{0}(t)arrow\theta_{0}$ $( t+T)=\theta_{0}(t)-\mathit{2}\pi^{\backslash }$$\mathrm{i}\mathrm{s}$$( t)$ ;(7)$\psi_{r}(r, \theta, t)$that the rotation of the system$\thetaarrow\theta+\mathit{2}\pi$

rect ion$\mathrm{i}$rotat on of a system and a transformat on$\mathrm{i}\mathrm{n}$ater a revolut on$\mathrm{d}\mathrm{i}$in Eq. (7) could be nterpreted,$0$

Eq. (4). As to such an equivalence,$\mathrm{i}\mathrm{s}$gn of the equali ty reflects the$\mathrm{i}$$-\mathit{2}\pi i$$(t+T)=\theta_{0}$ $( t)-\mathit{2}\pi$ ,transformation. (2),$\mathrm{i}\mathrm{s}$(wethe flux tube,

OPPOS te wi th respect to the$\mathrm{i}\mathrm{s}$$\Delta 7=-\mathit{2}\pi$ $a$$\mathrm{s}$$\mathrm{s}\mathrm{i}$$0$$\psi_{\mathrm{r}}(r_{J}\theta, t+T)=e\mathrm{x}p$$\mathrm{t}$a . A$\Delta 7=-\mathit{2}_{T}a_{0}\mathrm{a}\mathrm{f}$$\mathrm{t}\mathrm{i}$The addi ional Phase$( t)$ goi ng
aroundEq. (4). Thus the PSeudo wave funct ion

an addi onal phase$\mathrm{r}\mathrm{e}\mathrm{s}$$-r_{0}$which$\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h}arrow r_{0}(t)$introduced$i-\kappa\partial\chi_{0}/\partial$

the rest frame.

$t=-\mathrm{X}z/\mathit{2}\mu\cdot\nabla^{2}\chi_{0}\mathrm{i}\mathrm{n}$have$\chi_{0}(r, tarrow )$$\mathrm{i}\mathrm{s}\mathrm{w}\mathrm{i}$dely accepted

we are led to the conclus ion that$\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{i}$ngle-valued. That$\mathrm{i}\mathrm{s}$

,

an

165
addi ional phase $\mathit{2}\pi a(t)$ of the factori zed phase factor for the revolution$\mathrm{t}$(or equivalently$\theta_{0}(t)arrow\theta_{0}(t)-\mathit{2}\pi$cancelled out by the acquired phase-$\mathit{2}\pi$$a$1$(t)+\lambda_{0}\neg$is exactly),$\thetaarrow\theta\downarrow \mathit{2}\pi$$( \lambda_{0}=-\mathit{2}\pi a_{0})$.

the pseudo wave function$\psi(r,tarrow\cdot\}$

The single-valuedness of$1\Psi(r_{J}tarrow)$(for a static

is found by)$A_{0}(r)arrowarrow$another method. Let us consider a particle orbi ting around the fluxof radius$100\mathrm{P}$$\xi_{0}$. Here the flux$\Phi_{1}(t)$to a $d.c$foris obtained ;$t<\mathit{0}$$\psi_{\cap}$

at$\Phi_{0}$$t=\mathit{0}.$$( \theta, t)=Nex\beta[-i\mathrm{I}_{\mathrm{t}_{\mathrm{o}}}^{\mathrm{t}}E\hslash$

where

(

$\beta_{\cap}$$\theta$

,

. I

$)$

$t<\mathit{0}$$\mathrm{i}\mathrm{s}$$\mathrm{s}$,of$\lambda_{0}=\mathit{0}$$( t , )$$dt’/\hslash \mathit{1}\beta$$\lambda_{0}=\mathit{0}$.

$(\theta.

$\Phi_{0}$,$E_{\mathrm{n}}$$\psi$(8)t)$$( t)=\hslash 2(n-$. satisfies the. Since there is no $d.c$ Part$u$)$oe$) $e$

ingle-valued. The solution (8) develOps into,

approaches toan extrapOlated form$\psi_{\mathrm{n}0}$fwnct ionas$\mathrm{g}\mathrm{i}$$\mathrm{o}\mathrm{f}Aarrow(r, tarrow)$

ven by Eq. (2$\Phi_{\iota}(t)$smoothly$(\theta, t)$ $=NexPf-iE_{\mathfrak{n}}(\mathit{0})$is consistent wi th the wave function really existing in the region

The eigenvalue equation in$t\geqq \mathit{0}$$t\geqq \mathit{0}$Equat ion (9) has, forma$\pm \mathit{1}$X,$a_{0}$,$\pm \mathit{2}$$\pm \mathit{3},$$\cdot$...

$2/\mathit{2}/\mathit{1}\xi_{0^{2}}$reali st . To$\mathrm{i}\mathrm{c}$momentum,,

many solut ionsare canonical angular momenta$l_{\mathrm{n}j}$)$11\mathrm{y}$$\mathrm{w}\mathrm{h}$$\mathrm{f}\mathrm{i}$ich$\Theta$$l_{\mathrm{n}\mathrm{j}}=$.

$( \theta )$$\mathrm{i}\mathrm{s}$

sat

$exp$

$=$

$\mathrm{j}$( $n-j$ a

The energy eigenvalues of. Among these solutions, therend th$\mathrm{i}\mathrm{s}$

).

.

,

is

(9)

$- \frac{l^{\mathrm{z}}}{\mathrm{z}\mu;_{0}^{\chi}}(\frac{\mathrm{a}}{3\mathit{9}}-i\alpha_{\mathrm{O}})\Theta_{\mathrm{n}\mathrm{j}}2(\theta)=E_{n\mathrm{j}}\Theta_{\mathrm{n}\mathrm{j}}(\theta)$

where$t/\mathrm{X}$

J. This extrapOlated form (of the Pseudo wave function$JexP\mathrm{f}i(n-a_{0})\theta$$)$

wi thin a

Fortunately. the exact solution of Eq. (3)is reasonable. Thus the$\psi_{\cap}$$\Phi$

is assumed to swi tch over from an $a.c$

is important to note that$\mathrm{t}$circular condi tion (4) wi thin$\Phi$t) $=exP\zeta i(n-a_{1}(t))\theta \mathit{1}$ and$a_{1}(t))2/\mathit{2}\mu\xi_{0}2$of$\mathrm{i}\mathrm{s}$$0$$\Theta_{\mathfrak{n}\mathfrak{j}}$)$( it_{\mathfrak{n}j}\theta/\mathrm{X})$$\mathrm{X}(n$

are

and

$i$

are

$\mathit{0}$,$\mathcal{B}_{nj}=(l_{\mathrm{n}\mathrm{j}}-$one solut ion physical ly. let us consider the conservat ion law of energy-angular$\mathrm{i}\mathrm{s}\mathrm{f}\mathrm{i}$ed at the$1\mathrm{i}\mathrm{n}\mathrm{l}\mathrm{c}\mathrm{i}$ng of two$u$)$oe$) $efimcti\alpha\iota s$

respect vely$\mathrm{i}$166$\mathrm{g}\mathrm{i}$venthe regions$\mathrm{i}\mathrm{n}$and$\mathrm{t}<\mathit{0}$$t\geq \mathit{0}$valued. Therefore we are led to

Let us examine (i)$C2)$

An incident wave)’$\mathrm{i}$$\langle$$( P)$$4)$$|2cos^{2}(\pi\alpha_{0})s)$appearing in the mixed terms
Contrary to this. such an

is

$5)$

$\mathrm{i}$$\Psi_{\mathrm{c}1}$$(P)$ and$|Q(P)|\mathit{1}\mathrm{i}$ngle-valued,$0\dot{0}\mathrm{t}\mathrm{a}\mathrm{i}0\mathrm{e}\mathrm{d}$$( P)\Psi$nterference$\psi^{\wedge}\mathrm{C}1(P)$$\mathrm{s}\mathrm{i}$$\mathrm{i}\mathrm{s}$$( P)$

funct ionscancelled$0^{1}\mathrm{J}\mathrm{t}$4–seen. a period

funct on$\mathrm{i}\mathrm{c}$$|Q(P)|\tilde{2}\mathrm{i}\mathrm{s}$.

$0^{\mathrm{f}}$$a$wi th a per od$\mathrm{i}$$0$important that the special care should be used,
13)

no to confuse the,$\mathrm{t}$aunreal )$(\ddot{\mathrm{n}})$I$\mathrm{o}^{-\mathrm{d}\mathrm{e}}\mathrm{e}\beta \mathrm{n}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{t}$$\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{s}\mathrm{i}$Here$\mathrm{g}\mathrm{i}$ven$\mathrm{i}\mathrm{n}$,$(\nu\equiv n-a_{0}).(\mathit{1}\mathit{0})$quite$\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{i}$lar$\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{s}$tly conta ns$-q\partial E(\Phi)/\partial\Phi$

6)

7)

ve a

the

$0\mathrm{n}\mathrm{e}$

of

$|$

as readily,. Therefore,$|Q(\mathrm{P})$‘$cos^{\sim}$$\mathrm{i}$$\mathrm{g}\mathrm{i}$$\mathrm{t}0$$\mathrm{i}\mathrm{t}$seemseven in the excellent experiment-dependence ofEq. (9 ). Here, let us)$\theta_{\mathrm{p}}$$\mathrm{e}\mathrm{o}.\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}0\mathrm{D}$$\mathrm{b}\mathrm{r}\mathrm{i}$$a_{(\uparrow}$2$\mathrm{w}\mathrm{i}$tfi the$($$a_{()}$,

because$\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{s}$. An example has been

ef comne.

A probabili ty current de$|$$(\pi a_{0})$$\mathrm{S}$$\mathrm{c}\mathrm{i}$$($Bessel$\mathrm{t}\mathrm{h}_{\vee}^{\circ}$

found that the energy eigenvalue ’1 a funct on ofch expli$\psi_{\mathrm{C}\mathrm{i}}$(in$Q(P,)’=^{\nabla}\eta_{\overline{\vee}}-\infty\infty\sim eXp$$\alpha_{0}=I$

interference expressed by$\mathrm{w}\mathrm{h}\mathrm{i}$ng from$|\uparrow\Psi_{\mathrm{C}\mathrm{l}}*\mathrm{t},2\backslash (P)|2=$$\mathrm{i}$$\mathrm{t}\mathrm{i}\mathrm{s}$determi ned by Eq. (10)already$\mathrm{o}\mathrm{b}\mathrm{s}\mathrm{e}\Gamma \mathrm{v}\mathrm{e}\mathrm{d}a_{0}$.

the present theory,$\mathrm{i}\mathrm{n}$because$|Q\backslash (P)|2cos^{2}$ $( \pi a_{0})$$\mathrm{i}$

,

$(P)$

c2

,

$\mathrm{i}$$\Psi \mathrm{c}\downarrow*\mathrm{C}2(P)|2$${}^{\mathrm{t}}\mathrm{i}^{\ulcorner^{*}}\mathrm{c}\iota(P)\Psi$Hence we have$\gamma_{\mathrm{p}})\perp b_{\mathrm{n}}(|\nu|)J_{|\mathrm{U}\mathrm{I}}(r_{\mathrm{p}})\mathrm{J}$-dependence of the present$(P)|2=4$c2$’/\hslash=\pm \mathit{2}\pi a_{0}$$\mathrm{i}$$- \frac{\mathrm{f}^{l}}{z\mu}[\frac{\mathrm{a}}{\tau \mathrm{a}r}(\mathrm{r}\frac{\mathrm{a}}{\delta Y})\sim-\frac{|\mathrm{L}J|\mathrm{z}}{\tau^{f}}1\mathrm{R}\mathrm{t}\mathrm{r})=\mathrm{E}\mathrm{R}(r)$$a_{0}$$($

sting theory where‘

resPect vely

sat sfy the radial Part of the$I_{\mathrm{f}}\downarrow\mu|(\gamma)$-hand

on a screen.

$P$$|\Psi_{\mathrm{C}^{1}+}$. $( P)$ and

’
$($

int

$\mathrm{l}\mathrm{e}\mathrm{f}\mathrm{t}$. The factor $cos^{2}(r‘ a_{0})$. The present $Q(P)$ is expressed by$\mathrm{f}a_{\mathrm{n}}(|\nu|)J-|\vee|$ngle-$\mathrm{s}\mathrm{i}$$\mathrm{c},\cdot$$\mathrm{T}\mathit{2}\pi a_{0}$$\psi_{\mathrm{C}2}$$\mathrm{p}0$$\mathrm{e}\mathrm{x}\mathrm{i}$$\pm q\oint \mathrm{C}\downarrow- \mathrm{c}\mathrm{z}A_{0}arrow d\cdot\nuarrow$by the counter-interference phases$\psi_{\mathrm{c}2}$

alsc

sPli tted into two$\mathrm{i}\mathrm{s}$. According to theassumed to be$\mathrm{i}\mathrm{s}$comes from the interference phasesThe$( \theta)$$0$of. the flux. These Spli tted waves get together at a$|Q(P)$. the wave function

Uc2 travelling along the right-hand $(C1)$ and theand$\Psi_{\mathrm{C}1}$.

$\Theta$$\mathrm{i}_{\mathrm{t}}’\mathrm{j}\mathrm{n}$$\mathrm{i}$$\sim(\hat{\mathrm{m}})$the pseudO wave function$4$

restrictth.e si- ngle-valued wave funct on.

$\iota_{1^{\Gamma}\mathrm{c}1\mathrm{c}z}\succ(P)=(\mathrm{r}_{\mathrm{C}1}(P)+\Psi_{\mathrm{C}2}$$P)$$\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{s}$. wh ich is single-valued. links up only to$\mathrm{i}\mathrm{n}t<\mathit{0}$waves. Under$\prime \mathrm{t}\mathrm{t}$on the equal ty $j–$$.\eta \mathrm{s}\mathrm{i}\mathrm{t}\mathrm{y}jarrow(r, tarrow)$$\mathrm{i}$

167
$-$

$arrow$sa$\mathrm{t}\mathrm{i}$sfy ng$\Psi\nabla^{1}1r\cdot]-(.

$( arrow\gamma, t)=\frac{\mathrm{f}}{xf^{\lambda}i}$a part cle conf ned wi$\mathrm{f}$$\mathrm{i}$$\mathrm{i}$$\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{n}$aEq. (11), $j=\mathcal{T}\iota(n-a 0)$ /2$|\psi|2=(\mathit{2}\pi\xi_{0})-1$

we reach$\mathrm{d}\theta=\ddagger$$\rho=$$\psi^{*}(arrow r, t)\nabla\emptyset(r, tarrow)$

[

and

,
(A /2

$j=$

$\mu i$)$\mathrm{f}\Psi^{*}\nabla^{1}1^{\Gamma}$$\psi(r, tarrow)\nabla\psi^{*}(arrow r, t)\mathrm{J}$-of radius$100\mathrm{p}$$\pi\mu\xi_{0^{2}}$-. (11)

cons dered, we have, from$\xi_{0}\mathrm{i}\mathrm{s}$$\mathrm{i}$for a normali zed pseudo wave funct ion. Combining thi wi th $E_{\mathrm{n}}=\hslash 2(n-a_{0})2/\mathit{2}\mu\xi_{0}2$$\mathrm{s}$An angular momentum$j=-q\partial E(\Phi)/\partial\Phi-,\cdot$$(|n-a_{0})\mathrm{y}\mathrm{i}$2$|\iota_{1’}|$isq/\mu)|\Psi|2A)arrow$

$j-,$

I

(

$j=\mathit{0}$$\partial\rho/\partial t-\vdash\nabla$$\mathrm{i}$elded from$j(-|r, t)\mathrm{i}\mathrm{s}$to thi , the canonical angular momentum isa$\mathrm{k}\mathrm{i}$$l= \int_{\mathrm{o}}^{\iota\pi}(\xi_{0}\cross\mu j )$nema cal one

Contrary

14)$\mathrm{t}\mathrm{i}$$l\mathrm{c}*\mathfrak{n}=\mathrm{X}n2$$=$

as seen from)$\mathrm{s}$$\xi_{0}$$l\mathrm{k}\mathrm{i}\mathrm{n}=$$\hslash(n-a_{0})$$(\mathrm{i}_{\tilde{\mathbb{I}}})$

Supposi ng that$\mathrm{i}\mathrm{t}$seems cons derably$\mathrm{i}$$\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{i}$cult to measure the crosssection in the actual condi ion, let us study, providing for the future, the$\mathrm{t}$di fferential cross sect ion$\sigma(\theta)$. For the$=sin2$ $( \pi a_{0})/\pi ksin2$ $( \theta/\mathit{2})5)$

present case where the rad us of flux tube$\mathrm{i}$the addi ional$\mathrm{t}$Si$\mathrm{n}\mathrm{c}e$

the

$k$

$\mathrm{P}^{\mathrm{r}0\mathrm{b}1}\mathrm{e}\mathrm{m}$–dependence, wh le the$\mathrm{e}\mathrm{i}$has already been gi,$\mathrm{i}\mathrm{s}\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}$cally the same.

$\mathrm{w}\mathrm{i}11$

be used. An inc ident particle

rect on $n1$ $(r, \theta)=exP(ikrcos\theta)\mathrm{i}\mathrm{s}$ represented by$\mathrm{i}$$1L \mathrm{i}(\gamma_{J}\theta)=\sum_{\mathrm{r}^{\underline{-}}\sim}^{\circ}exp\mathit{6}\infty(in\pi/\mathit{2})J_{n}(kr)ex\beta(in\theta)$

ich carries a current dens ty$\mathrm{i}$$\mathrm{w}\mathrm{h}$

hand,

$u$

.

$j$

$=\mathcal{T}\iota k//l$(see Eq. (11 )).

(12)

,

On the otherthe scattered wave is represented by$(r, \theta)=\sum_{-\mathrm{r}\iota-}\infty-\infty A\mathfrak{n}(\nu)ex\beta(in\pi/\mathit{2})H\mathrm{t}\mathrm{I})(1\cdot \mathrm{I}k\gamma)ex\beta(i_{\mathcal{V}}\theta),$

where. In thedepressed by$\sigma(\theta)\mathrm{i}\mathrm{s}$-dependence$\mathrm{v}e\mathrm{n}$$\sigma(\theta)$$\mathrm{i}$$\mathrm{i}$$\chi-\mathrm{d}\mathrm{i}$lament $(a=\mathit{0} )$ ,ther by the wave funct on or by the pseudo

wave function, the pseudo wave funct onmoving in a

10)$\mathrm{i}\mathrm{s}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e}$$a_{0}$$\mathrm{i}$can be studi ed

9)$\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{n}\mathrm{f}\mathrm{i}$$H_{\mathrm{I}^{\mu}\mathrm{I}}(1)(kr)$describes the$\mathrm{o}\mathrm{u}\mathrm{t}\mathrm{g}0\mathrm{i}$ng wave and $exp(i\nu\theta )$$\mathrm{r}$$(l\mathit{3})$eflects the168
multi-valuedness of the Pseudo wave function.
of the flux tube. $ui(a, \theta)+u$

.

$A$

’

,

A

$(v)exp(-in\pi/\mathit{2})$$rom the$\mathrm{F}$$\mathrm{i}$nPene rat ng condi ion$\mathrm{t}$$\mathrm{t}$. we ob tai$(a, \theta)=\mathit{0}$$\mathrm{i}$$\mathrm{n}A$.

$( ,t )$-:.

In Eq. (14) we have used $sin(\pi|\nu|)\cdot H_{\mathrm{I}U1}(\downarrow)(z)---i\mathrm{f}J_{-\mathrm{I}^{\mu}l}(z)$Let us note$exP(-i\pi|\nu|)J_{\mathrm{I}^{\mu}\mathrm{I}}(z)]$Considering the asymptot ic behavior of

’

.
at

$H_{1\nu|}(\downarrow)(z)$$\mathrm{z}arrow\infty$. as

def ned by$\mathrm{i}$$(i\nu\theta )$$\sigma(\theta)\equiv|f(\theta)|z$$\sigma(\theta)=\frac{z}{\pi\hslash}$$\mathrm{t}$.

$k\gamma)$

is adop$\sigma(\theta)$$\iota \mathrm{e}\mathrm{d}$I).

$\mathrm{f}$is a$\mathrm{P}\mathrm{e}\mathrm{r}\mathrm{i}0\mathrm{d}\mathrm{i}\mathrm{C}$in place offferential cross section$H_{\mathfrak{l}\nu 1}(\iota)(k\gamma)$the$e\mathrm{x}\mathrm{p}$($a_{0}$$a_{0}$$\mathrm{P}0\mathrm{s}\mathrm{i}$tive order)

ress ion for $a=05$.

$a\ll \mathit{1}$where $J0$ (. In this case, an expression of$sin$ $( \pi v )$$\sigma(\theta)=\mathit{2}/\pi k$

$k$

$k$

a)

$\sigma(\theta)$$( \int\zeta)$

wi 11 be restricted$\negarrow \mathit{1}$$($$J_{|\mu|}$

),

and

$10$ ))$9$from Eq. (15).

$\negarrow \mathit{0}(n\neq \mathit{0})$$v$funct ion ofsome linear combinat ion of ($\mathrm{i}$where$(v)exD$ (- $i|v|$

isLet us study the interest ng casea)$\mathrm{d}\mathrm{i}$$(\theta)/\gamma_{\gamma}(\gammaarrow\infty$.

$\mathrm{n}\mathit{0}<a_{0}<\mathit{1}$$\mathrm{i}\mathrm{s}\mathrm{r}\mathrm{e}\mathrm{p}\mathrm{r}0\mathrm{d}\mathrm{u}\mathrm{c}e\mathrm{d}$$k$.

. Thus an expression of the$H_{1\nu 1}(\downarrow)(z)$$| \sum_{n_{\wedge}-}A\infty-\infty \mathrm{n}’(|v|)exP(-i\pi|v|/\mathit{2})exP(i\nu\theta)|2$

is found thatwi thi$( \gamma_{J}\theta)arrow ex\beta(ik\gamma)f$$f(\theta)=(\mathit{2}/\pi k)1/2e\mathrm{r}D(-i\pi/4)\cdot\Sigma A/$$\pi/\mathit{2}\mathit{1}\cdot ex\beta$I$\iota\iota s$.

we have a scattering$arrow(\mathit{2}/\pi z)1/2exP[i(z-(\mathit{2}|v|+\mathit{1})_{\pi}/4)\mathrm{J}$ ,ampli tude $f(\theta)$-. $( v )-A$ . $(|v|)$
$l$

$A$

resulting from $J\mathrm{w}(z)--(-\mathit{1})\cdot J\mathrm{r}(z)$$)$$(l+)$$( \nu)=\frac{-i\sin(\pi\mu)\mathrm{s}\mathrm{i}\mathrm{n}(\pi 1\mu 1)}{\pi \mathrm{r}\mathrm{J}_{-|y1}\mathrm{c}\mathrm{k}\alpha)-_{\mathrm{e}^{arrow}J_{1}(\mathrm{k}}\dot{\iota}1\Gamma|y|)|\alpha \mathrm{J}}.\sum_{--\infty}^{\infty}\frac{(-i)\mathrm{J}\mathrm{p}\mathrm{t}^{(}\mathrm{k}a)n\iota}{t\mathrm{n}-}\mathrm{n}\iota-\cdot$$\mathrm{n}$

$J$

.

$($

because of $|v|$$=sin(\pi|\nu|)$ , becomes$| \sum_{l\iota_{-}-}sin\infty-\infty(\pi|\nu|)ex\rho(i\nu\theta)/D(k ; |v|)$$|2(l\mathit{6})$

169
$D$

$(k ; |\nu|)=(ka/\mathit{2})-|1\text{ノ}\mathfrak{l}_{e}\chi\beta(i\pi|\nu|/\mathit{2})\Gamma(\mathit{1}+|v|)$$(I7)$$-(ka/\mathit{2})^{1^{\mu 1}}e\chi p(-i\pi|v|/\mathit{2})\Gamma(\mathit{1}-|v|)$

Here the equali ties
and$\ldots.)$$v\Gamma$$( \nu )$ $\Gamma(\mathit{1} - v)$ $-arrow\pi/sin(\pi\nu)$$\Gamma$$( \nu)$$=\Gamma(\mathit{1}+\nu)$$k$-dependence, besides thethrough the denominator

the flux$( k a\ll \mathit{1} )$$\mathrm{f}\mathrm{a}\mathrm{c}\iota_{0\mathrm{r}}\mathit{2}/\pi k$lament wi th $a=0$. Unfortunately,$\mathrm{f}\mathrm{i}$. appears in$\mathrm{i}\mathrm{t}$-comp0nen$\iota$ofmagnetic region$\sigma_{0}=\frac{7\ulcorner}{\mathrm{z}h}$where$\Gamma$.

$\mathrm{i}$$\mathrm{i}$$a_{0}\ll ka\ll \mathit{1}$obtained for$\sigma(\theta)$

seems formi dably di ff icult to

0.57721....

magnetregion

).
$k$

$7Z+$

-

$\mathrm{p}\mathrm{r}i$nc ipal ina$0$$\cdot$. . has been used$( k a\ll \mathit{1})$this expression wi th

...

$J$

$\mathrm{i}\mathrm{t}$$\sigma_{0}$is found that the present$<\mathit{1})$

. The

(note
$\mathrm{d}\mathrm{e}\mathrm{P}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{S}\mathrm{i}\mathrm{o}\Pi$$\Gamma$a non-( 7

is Euler’$\mathrm{i}\mathrm{s}$$\mathrm{s}$(18)number$7=$

found in the$(l\mathit{9})$$\Gamma(\mathit{1}+a_{0})\mathrm{J}2$$\mathrm{f}\mathit{1}\neq cos\theta+(cos\theta)2+$

is modi ied by the depressing factor$\mathrm{f}$$\sigma_{0}$$\mathrm{z}\alpha_{a}$$(ka/\mathit{2})$$\mathfrak{l}\mathrm{n}$obtained from $\sigma(\theta)=sin2$ $( \pi a_{0})/\pi$$(\pi a_{0})/\pi k$$ksin2(\theta/\mathit{2})=\mathit{2}sin2$

.

the

;

$\sigma_{0}=\frac{\mathrm{Z}}{\pi \mathrm{k}}\cdot sin2(\pi\alpha_{0})r(ka/\mathit{2})^{\alpha_{\circ}}/$$\mathrm{c}_{0}\mathrm{m}\mathrm{P}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}$$a\ll \mathit{1}$$|^{-2}[\mathit{1}+(\pi a_{0}f_{\omega}\nu)2J-1$

On the other hand, an interesting result$a\ll$$k$$\mathrm{t}0$,. we have$|7-i\pi/\mathit{2}+ln(ka/\mathit{2})$$( \mathit{1} +z)=\mathit{1}$$\mathrm{i}\mathrm{c}$whose con tr but on is$\sigma(\theta)$$J_{\mathrm{k}1\cup 1}$$\sigma(\theta)$

carry out the sumnat ion in Eq. (16). Therefore we restr ict our analysis$s$,$\pm \mathit{2}$$\pm \mathit{1},$. Let, us note that an$(k ; |\nu|)$ . This di ffers from$D$$\nu\neq \mathit{0},$have been used to rewri te the expansion$(ka)-\neg(ka/\mathit{2})^{\mathrm{f}l\iota|}/’\Gamma(l\pm|\nu| )$addi tional$($$( l)=\Gamma(\mathit{2})=\mathit{1}$andby $(ka/\mathit{2})^{2\alpha_{a}}$ or by$\Gamma(l+_{a_{0}})\neg\sim \mathit{1}$$|ln(ka/\mathit{2})|-2$

for$\mathit{0}<a_{0}$seemsreasonable. The incident wave, as widely known to the phys icists, becomes more

insensi$\mathrm{t}\mathrm{i}$ve to the tube as ts wave length becomes longer. The depression in the$\mathrm{i}$magnet ic region, explici tly depend ing on the value of$a_{0}$,$\mathrm{i}\mathrm{s}$much larger than

170
the one arising$\mathrm{i}\mathrm{n}$the case

extraordinari ly small
dependent scatter$\Phi-$$\sigma_{0}$$\mathrm{i}\mathrm{n}\mathrm{g}$.

$a_{0}=\mathit{0}$(see Eq. (18)).

Hence such an

can be cons idered as an exPerimental evidence of theobserved$\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{s}$$\mathrm{i}\mathrm{n}$the magnetic regi on.

The author would like to thank Professor S. Takagi for$\mathrm{i}\mathrm{s}$also grateful to colleagues of Kyoto Un vers ty for$\mathrm{i}$$\mathrm{i}$$\mathrm{s}\mathrm{t}$$\mathrm{h}\mathrm{e}\mathrm{l}_{\mathrm{P}^{\mathrm{f}\mathfrak{U}}1}\mathrm{d}\mathrm{i}$imulat ng$\mathrm{i}$scussi$\mathrm{d}\mathrm{i}$$\mathrm{o}\mathrm{n}\mathrm{s}$. Hescuss ons$\mathrm{i}$and the critical check of the calculations.

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