\documentstyle[12pt]{article}
\pagestyle{empty}
\textheight22.5cm
\textwidth15cm
\normalbaselineskip=12pt
\normalbaselines
\newcommand{\dps}{\displaystyle}
\title{The Geometric Phase in Quantum Physics}
\author{A.\ Bohm\\
Center for Particle Theory, The University of Texas\\
Austin, Texas 78712}
\date{}
\begin{document}
\maketitle
Berry's phase has
been fashionable in many areas of physics and in
chemistry and also among mathematicians and mathematical
physicists. The mathematical people are attracted to this
area because it is related to the beautiful mathematics
of fibre bundles which underlay gauge theories. In fact
this is the most accessible example of a gauge theory for
people who know just the elementary facts of
nonrelativistic quantum mechanics.
The chemists and physicists are interested because the
geometric or Berry phase has observable consequences,
which could not be explained before. It is this aspect
which distinguishes the Berry phase from the many other
fashions of mathematical physics.
Though the Berry phase
may turn out not to be as important as its present
popularity suggests,
it is a discovery which will remain forever.
The surprising thing about it is that its importance has
been realized 60 years too late. The reason for this
was that many people (including myself) thought that phases
are unimportant in quantum mechanics, because a quantum
mechanical state is not described by a vector $\psi$ but
by a ray or a projection operator $\mid\psi><\psi\mid$ and
that phase factors could always be removed by a suitable
phase or gauge transformation.
That this is not always possible and under which
conditions this is not possible was shown by Berry in his
famous 1984 paper.$^{1)}$ And the Berry phase fashion
started when Simon$^{2)}$ explained that Berry's phase is
the holonomy (element of the holonomy group) for a fibre
bundle with a particular connection, the adiabatic
connection.
But the physical effect of the Berry phase had been known
for quite some time. It was observed as some anomalies
in the spectra of mole\-cules$^{3)}$ and then, (1978),
explained in a series of remarkable papers by C.A. Mead
and Truhlar$^{4)}$ by the introduction of a gauge
potential, which is identical with the one derived by
Berry and which is now called Berry connection.$^{5)}$
This gauge potential emerges naturally from the
Born-Oppenheimer procedure in molecular physics$^{6)}$ if
one does not make the drastic Born-Oppenheimer
approximation.$^{7)}$
The Born-Oppenheimer method is concerned with the study
of complicated mole\-cules by dividing them into two
parts: the electronic motion described by a set of
``fast" variables, and the collective motion described by
a set of ``slow" variables. Berry connection and Berry
phase, therefore, arise in the dissection of
complicated quantum physical systems into simpler
subsystems. This reduction to the simpler is the basic
meaning of understanding in science.
In the old, drastic approximation the dissection results
in the trivial direct product of the
states for the
two subsystems. In
the less-drastic, adiabatic approximation or in the
exact theory, the motion of one subsystem
(the ``fast" subsystem) alters the dynamics of the
``slow" subsystem by inducing in it a Berry gauge
potential. Thus, the parts of a complicated quantum
physical system turn out to be different from what we
naively expected. Our discussions here will be mainly
concerned with these aspects of the Berry connection.
The Hamiltonian for a (diatomic) molecule is,
\begin{equation}
H={{\bf P}^2\over2\mu}+
{\hat{\bf p}^2\over2m}+V({\bf X},{\hat{\bf r}})
%\eqno(1)
\end{equation}
where ${\bf\hat p}$, ${\bf\hat r}$ stand for the
observables of the fast electrons and ${\bf P}$,
${\bf X}$ stand for the observables of the slow nuclei.
Since the light electrons instantaneously follow the
motion of the heavy nuclei, the slow variables can also
be understood as being the variables of the molecule as
a whole, i.e. the collective variables.
In particular, for the diatomic molecule ${\bf X}$ will
be the vector along the internuclear axis and ${\bf P}$
its conjugate momentum (in addition there are the center
of mass position and momenta which are, as always in a
non-relativistic theory, ignored).
The potential $V({\bf X},{\hat{\bf r}}$) is a complicated
function of the operators ${\bf X}$,${\hat{\bf r}}$ and
possibly some other operators like spin.
The Hamiltonian $H$ of $(1)$ is split into
two parts,
\begin{eqnarray}
%\eqalignno{
\displaystyle H&\displaystyle =&
\displaystyle {{\bf P}^2\over2\mu}+h({\bf X})\\
%&(2)\cr
\displaystyle h({\bf X})&\displaystyle =&\displaystyle
{\hat{\bf p}^2\over2m}+
V({\bf X},{\hat{\bf r}}) %&(3)
%\cr}
\end{eqnarray}
where $h({\bf X})$ denotes the ``fast" or
electronic Hamiltonian that depends upon
the
``slow" operator ${\bf X}$.
The eigenvalue problem,
\begin{equation}
H\mid\psi^E\rangle=E\mid\psi^E\rangle
%\eqno(4)
\end{equation}
is solved in the Born-Oppenheimer procedure
by first solving the eigenvalue problem for
the operator $h({\bf X})$.
In the drastic Born-Oppenheimer approximation $\bf X$ is
considered as a classical parameter $\bf x$ which is
fixed.
With ${\bf X}={\bf x}\ =\ fixed,\ h({\bf X})$ commutes
with ${\bf P}$ and thus $h({\bf X})$ and $H$ can be
diagonalized together (which amounts to ignoring the
effect of the kinetic energy of the slow variables in (1)):
\begin{eqnarray}
%\eqalignno{
\dps\mid N,n;{\bf x}\ \rangle&\dps=&\dps\mid
N\rangle\otimes\mid n({\bf x})\ \rangle\\ %&(5)\cr
\dps h({\bf x})\mid
n({\bf x})\rangle&\dps=&\dps\varepsilon_n(x)\mid n({\bf
x})\ \rangle\\ %&(6)\cr
\dps H\mid N,n;{\bf x}\rangle&\dps=&\dps
\Bigl(
{{\bf P}^2\over2\mu}
+\varepsilon_n(x)\Bigr) \mid N,n;
{\bf x}\rangle=E_{N,n}\mid N,n;{\bf x}\rangle %&(7)\cr}
\end{eqnarray}
One
first solves (6) for every value of the {\it fixed}
parameter $\bf x$, and obtains the electronic energy
values $\varepsilon_n(x_e)$ where $x_e$ is the minimum
(equilibrium) value of the
``potential curve"
$\varepsilon_n(x)$.
The eigenvectors
$\{\mid n({\bf x})
\rangle\mid n=1,2,\ldots\}$ for every value of ${\bf x}$
form
a complete system of basis vectors for the space of
physical states ${\cal H}^{fast}$ for the fast\vspace*{3pt}\\
\parbox{180pt}{
\setlength{\unitlength}{0.5pt}
\begin{picture}(300,250)(-50,15)
\linethickness{1.2pt}
\put(0,-40){\makebox(300,20){\scriptsize Fig.1 Schematics of
typical molecular spectra}}
\put(50,10){\line(1,0){200}}
\put(260,10){\makebox(0,0)[l]{$\varepsilon_1$}}
\put(260,240){\makebox(0,0)[l]{$\varepsilon_2$}}
\put(220,60){\makebox(0,0)[l]{\scriptsize E$_{1,0;1}$}}
\put(220,77){\makebox(0,0)[l]{\scriptsize E$_{1,3;1}$}}
\put(220,160){\makebox(0,0)[l]{\scriptsize E$_{3,0;1}$}}
\put(5,10){\makebox(0,0)[l]{\scriptsize $\nu=0$}}
\put(5,60){\makebox(0,0)[l]{\scriptsize $\nu=1$}}
\put(5,110){\makebox(0,0)[l]{\scriptsize $\nu=2$}}
\put(5,160){\makebox(0,0)[l]{\scriptsize $\nu=3$}}
\put(5,240){\makebox(0,0)[l]{\scriptsize $\nu=0$}}
\linethickness{0.8pt}
\put(70,13){\line(1,0){30}}
\put(70,20){\line(1,0){30}}
\put(70,27){\line(1,0){30}}
\put(70,40){\line(1,0){30}}
\put(110,27){\makebox(0,0)[l]{\scriptsize $j=3$}}
\put(110,40){\makebox(0,0)[l]{\scriptsize $j=4$}}
\put(85,47){\circle*{2}}
\put(85,50){\circle*{2}}
\put(85,53){\circle*{2}}
\put(50,60){\line(1,0){160}}
\put(110,63){\line(1,0){30}}
\put(110,70){\line(1,0){30}}
\put(110,77){\line(1,0){30}}
\put(110,90){\line(1,0){30}}
\put(150,77){\makebox(0,0)[l]{\scriptsize $j=3$}}
\put(150,90){\makebox(0,0)[l]{\scriptsize $j=4$}}
\put(125,97){\circle*{2}}
\put(125,100){\circle*{2}}
\put(125,103){\circle*{2}}
\put(50,110){\line(1,0){160}}
\put(50,160){\line(1,0){160}}
\put(130,170){\circle*{2}}
\put(130,173){\circle*{2}}
\put(130,176){\circle*{2}}
\put(130,179){\circle*{2}}
\put(130,182){\circle*{2}}
\put(50,212){\line(1,0){160}}
\put(220,212){\makebox(0,0)[l]{\scriptsize E$_{\nu,j;1}$}}
\linethickness{1.2pt}
\put(50,240){\line(1,0){200}}
\linethickness{0.8pt}
\put(70,243){\line(1,0){30}}
\put(70,250){\line(1,0){30}}
\put(70,257){\line(1,0){30}}
\put(70,270){\line(1,0){30}}
\put(85,277){\circle*{2}}
\put(85,280){\circle*{2}}
\put(85,283){\circle*{2}}
\put(50,290){\line(1,0){160}}
\put(130,300){\circle*{2}}
\put(130,303){\circle*{2}}
\put(130,306){\circle*{2}}
\put(130,309){\circle*{2}}
\put(130,312){\circle*{2}}
\put(220,290){\makebox(0,0)[l]{\scriptsize E$_{1,0;2}$}}
\end{picture}
}\hfill
%\vspace{25pt}\\
\parbox{225pt}{ subsystem.
After $\varepsilon_n(x)$ has been obtained from (6) for
every value of the fixed parameter {\bf x}, one inserts
it into the right-hand side of (7)
as an ``induced scalar
potential" and solves (7) for a given value of the
electronic quantum number $n$. As $\varepsilon_n(x)$ is
(often) approximately an oscillator potential,
$\varepsilon_n(x_e)$ splits into vibrational excitations
with quantum number $\nu$.
And as the diatomic molecule (dumbbell) also rotates about
its center of mass, each vibrational excitation splits
into rotational bands
with quantum number $j$.
The
collective quantum numbers $N$
are thus the vibrational
quan-}\vspace*{3pt}\\
tum number $\nu$ and the angular momentum $j:N=\nu,j$;
and one obtains the typical spectrum of
molecules, Fig. 1.
The time evolution of the fast system is described by the
Schr\"odinger equation
\begin{equation}
i\hbar{d\mid\psi(t)\rangle\over dt}=
h(X)\mid\psi(t)\rangle
%\eqno(8)
\end{equation}
and if initially the state vector is an electronic energy
eigenstate,
\begin{equation}
\psi(0)=\mid n({\bf x})\rangle,%\eqno(9)
\end{equation}
then the solution of (8) is
\begin{equation}\psi(t)
=e^{-{i\over\hbar}\varepsilon_n(x)t}\mid
n({\bf x})\rangle
=e^{-{i\over\hbar}\int^t_0dt^{\prime}
\varepsilon_n(x(t'))}\mid
n({\bf x})\rangle
%\eqno(10)
\end{equation}
iff ${\bf x}$=``fixed" parameter.
We will now consider the less drastic, adiabatic
approximation.$^{8)}$ If the internuclear distance and
direction ${\bf X}$ is considered a classical parameter
${\bf x}(t)$ which changes
slowly
in time (fast quantum system in
a slowly changing classical environment) then an initial
eigenstate of $h({\bf X}(t))$ ; $\psi(0)=\mid n({\bf
x}(0))\rangle$ may ``jump" into a state which also has
different electronic quantum numbers $n{^\prime}\not=
n$.
The adiabatic approximation is an evolution in which
${\bf x}(t)$ changes so slowly that an eigenstate of
$h({\bf X}(t))$
always remains in the same
eigenstate:$^{9)}$
\begin{equation}
\mid\psi(t)\rangle\langle\psi(t)\mid=\mid
n({\bf x}(t))\rangle\langle n({\bf x}(t))\mid
%\eqno(11)
\end{equation}
The solution of (8) for an initial eigenstate (9)
with time-dependent ${\bf x}(t)$
is then
given by
\begin{equation}
\psi(t)=e^{-{i\over\hbar}\int^t_0dt^{\prime}
\varepsilon_n(x(t'))}
e^{i\gamma_n(t)}
\mid n({\bf x}(t))\rangle
%\eqno(12)
\end{equation}
In addition to the dynamical phase factor of (10), there
appears another phase factor $e^{i\gamma_n(t)}$.
This phase factor had always been omitted in the old
adiabatic approximation because it was believed that it
can always be absorbed into the eigenvector
$\mid n({\bf x})>$ by a phase (gauge) transformation:
\begin{equation}
\mid n({\bf x})>\to\mid n({\bf x})>^{\prime}=
e^{i\zeta_n({\bf x})}\mid n({\bf x})>
%\eqno(13)
\end{equation}
where $\mid n({\bf x})>^{\prime}$ is again a normalized
eigenvector in (6).
For cyclic time evolution
\begin{equation}
{\cal C}:{\bf x}(0)\to{\bf x}(t)\to{\bf x}(T)=
{\bf x}(0)\ ,
%\eqno(14)
\end{equation}
when the internuclear axis returns to its original
position after a period $T$, the solution of the
Schr\"odinger equation (8) for the vector of the state
$\mid\psi(T)><(\psi(T)\mid=$\hfil\break$\mid n({\bf
x}(T))>< n({\bf x}(T))\mid$ is
\begin{eqnarray}
%\eqalignno{
\dps \psi(T)&\dps =&\dps e^{-{i\over\hbar}\int^T_0dt'
\varepsilon_n(
{\bf x}(t'))} e^{i\gamma_n(T)}\mid
n({\bf x}(0))\rangle\\
%&(15)\cr
\noalign{\hbox{where}}
\dps \gamma_n(T)&\dps =&\dps \int_{\cal C}d{\bf x}{\bf A}_n
({\bf x})=
\int_S\int d{\bf S}\cdot{\bf B}_n\\ %&(16)\cr
\noalign{\hbox{with}}
\dps {\bf A}_n&\dps \equiv&\dps i\langle n({\bf x})\mid\nabla_x\mid
n({\bf x})\rangle\quad ;\quad {\bf B}_n=
\nabla_x\wedge{\bf A}_n
%&(17)\cr}
\end{eqnarray}
and where
${\cal C}$ is the closed path in the parameter
space (14) and $S$ is a surface spanned by
${\cal C}$.
It is convenient and a standard convention to choose
eigenvectors which are single valued functions of the
parameter ${\bf x}$ in the region that
contains ${\cal C}$:
\begin{equation}
\mid n({\bf x}(T))\rangle =
\mid n({\bf x}(0)\rangle
%\eqno(18)
\end{equation}
Under this convention ${\bf A}_n({\bf x})$ is called Berry
connection or Berry gauge potential, ${\bf B}_n$ is
called Berry curvature and $\gamma_n(T)$ is called the
Berry phase angle.
It can be shown that $e^{i\gamma_n(T)}$ is
an invariant with respect to the transformation (13)
(gauge invariant) which may be different from
unity.$^{1)}$ It can therefore not be transformed
away by (13).
An example of a system where the Berry phase
$\gamma_n(T)$ and Berry connection are non-trivial
($i.e.$ not removable by a gauge transformation)
is
the quantum magnetic moment ${\bf m}=-{e\over2mc}g{\bf j}$
(of the electrons with spin ${\bf j}$) in a slowly
rotating magnetic field ${\bf B}^{mag}=B\hat{\bf X}(t)$
(along the internuclear axis of the molecule caused by
the rapidly orbiting electrons). The fast Hamiltonian
for this case is
\begin{equation}
h(t)=h({\bf X}(t))=-{\bf m
}\cdot{\bf B}^{mag}(t)=-{e\over2mc}gB{\hat{\bf X}}
(t)\cdot{\bf j}=b{\hat{\bf X}}(t)\cdot{\bf j}
%\eqno(19)
\end{equation}
The eigenvectors
$\mid n({\bf x})\rangle=\mid
k(x,\theta(t),\varphi(t))\rangle$ depend upon the polar
coordinates $(\theta(t)$, $\varphi(t))$
of the unit vector
$\hat{\bf X}(t)$, and $k$ denotes the component of
angular momentum of the fast system along the internuclear
axis
\begin{equation}{\bf x}(t)\cdot{\bf j}\mid k(\theta,\varphi)\rangle=
k\mid k(\theta,\varphi)\rangle\quad ;\quad
\varepsilon_k=\hbar bk %\eqno(20)
\end{equation}
By a straightforward calculation using
\begin{eqnarray}
%\eqalignno{
\dps \mid k(\theta,\varphi)\rangle&\dps =&\dps e^{-i\varphi
j_3}e^{-i\theta j_2}e^{i\varphi j_3}\mid
k(\theta=0,\varphi=0)\rangle\qquad{\rm
for}\qquad \theta<\pi\\ %&(21)\cr
\noalign{\hbox{and}}
\dps \mid
k(\theta,\varphi)\rangle^{\prime}&\dps =&\dps e^{-i\varphi
j_3}e^{-i\theta j_2}e^{-i\varphi j_3}\mid
k(\theta=0,\varphi=0)\rangle\qquad{\rm for}
\qquad \theta>0 %&(22)\cr}
\end{eqnarray}
one obtains for the Berry connection
\begin{equation}
{\bf A}^{k'k}=i\langle
k^{'}(\theta,\varphi)\mid\nabla\mid
k(\theta,\varphi)\rangle={\bf e}_x\hat
A_x^{k'k}+{\bf e}_\theta\hat
A_\theta^{k'k}+
{\bf e}_e\hat A_\varphi^{k'k}:
%\eqno(23)
\end{equation}
the following results
\begin{eqnarray}
%\eqalignno{
\hspace*{2.3cm}\dps {\bf A}_x=0\qquad A^{k'k}_\theta=0\ ,\
A^{k'k}_\varphi&\dps =&\dps -{k(1-\cos\theta)\over
x\sin\theta}\qquad\theta<\pi\hspace{1.7cm}(24)\nonumber \\ %&(24)\cr
\dps A'^{k'k}_\varphi&\dps =&\dps
\phantom{-}{k(1+\cos\theta)\over
x\sin\theta}\qquad\theta>0\hspace{1.7cm}(24^\prime)\nonumber %\cr}
\end{eqnarray}
\addtocounter{equation}{1}
The two vectors $\mid
k(\theta,\varphi)\rangle$ and $\mid
k(\theta,\varphi)\rangle^{\prime}$ (and the corresponding
two vector potentials ${\bf A}$ and ${\bf A}^{\prime}$)
had to be chosen
differently
for the domain $(\theta<\pi)$ and
the domain $(\theta>0)$ so that they are single-valued in
each domain.
The Berry curvature (17) calculated from (24) and
$(24^{\prime})$ is
\begin{equation}
{\bf B}^k=-{k\over x^2}\hat{\bf X}
%\eqno(25)
\end{equation}
And the Berry phase (16) calculated from (24)
(or (24$^{\prime}$))
is given by the standard result
\begin{equation}
\gamma_k({\cal C})=
-k ({\rm solid\ angle\ subtended\ by\ {\cal C}}) =
-k\Omega. %\eqno(26)
\end{equation}
The result (25) is identical with the field strength of
Dirac's magnetic monopoles$^{10)}$
$e{\bf B}={eg\over4\pi}{\hat{\bf X}\over X^2}$,
except that the
electromagnetic constant ${eg\over4\pi}$ is replaced by
the motion constant $k$, the component of angular
momentum along the internuclear axis, which in this
approximation is a fixed number.
>From this result we already suspect that something
like a magnetic monopole must be part of the
diatomic molecule (except if $k=0$). This
motional or mechanical ``monopole" will remain
uncovered if
one uses the drastic Born-Oppenheimer
approximation.$^{11)}$
If the eigenvalues of the fast Hamiltonian
$\varepsilon_n(x_e)$ are degenerate or close to each
other
(compared with the splitting between
$E_{N',n}$ and
$E_{N'',n}$) then the adiabatic approximation (11) is
apparently not good and one may consider in place of the
$U(1)$ gauge transformation (13) an $U({\cal N})$ gauge
transformation
(where ${\cal N}$ is the degeneracy of $\varepsilon_n$) and
obtain a non-abelian Berry connection$^{12)}$
${\bf A}^{mn}({\bf x})$ in place of the abelian
${\bf A}_n({\bf x})={\bf A}^{n}({\bf x})$ of (17).
Alternatively one can
treat the slow variables ${\bf P}$ and
${\bf X}$ as operators and
solve the problem quantum
mechanically by the Born-Oppenheimer method.$^{13)}$
Then these non-abelian Berry connections will emerge
naturally. This we will discuss now:
The space of physical states is, according to the basic
principles of quantum mechanics, the direct product of
the space for the fast motion ${\cal H}^{fast}$
and the space for the slow motion
${\cal H}^{slow}$:
${\cal H}={\cal H}^{slow}\ \otimes\ {\cal H}^{fast}$.
The slow variables ${\bf P}$ and ${\bf X}$ are the
operators ${\bf P}={\bf P}\otimes 1^{fast}$ and
${\bf X}={\bf X}\otimes1^{fast}$ and the operator $h({\bf
X})= h(\hat{\bf p},\hat{\bf r};{\bf X})$ acts in both
factors of ${\cal H}$. As basis of ${\cal H}^{fast}$ one
takes the $\mid{\bf n}({\bf x})\rangle$, as basis of
${\cal H}^{slow}$ one takes a basis of generalized
eigenvectors of ${\bf X}:\mid{\bf x}\ldots\rangle$ and the
basis system of ${\cal H}$ is something like the direct
product basis $\mid{\bf x}\ldots n\rangle=\mid
{\bf x}\ldots\rangle\tilde\otimes\ \mid n({\bf
x})\rangle$.
A straightforward$^{6,13)}$ but lengthy calculation
shows that the effective Hamiltonian
$H^{eff}$
for
the slow motion in
${\cal H}^{slow}$ is not given as in the drastic
approximation by (7) but by
\begin{equation}
H^{eff}=
{1\over2\mu}{\bf\Pi}^2+\varepsilon_n(X)
%\eqno(27)
\end{equation}
where
\begin{eqnarray}
%\eqalignno{
\dps {\bf\Pi}^{mn}({\bf x})&\dps =&\dps {\bf P}\delta^{mn}-{\bf A}^{mn}
({\bf X})\\ %&(28)\cr
\noalign{\hbox{and where}}
\dps {\bf A}^{mn}({\bf x})&\dps =&\dps i\langle m({\bf
x})\mid\nabla\mid n ({\bf x})\rangle=({\bf
A}^{mn})^{\dag} %&(29)\cr}
\end{eqnarray}
$H^{eff}$,
${\bf\Pi}$ and
${\bf A}$ are
${\cal N}\times{\cal N}$ matrices and
${\bf\Pi}^2$ in (27) means matrix multiplication
$\Sigma_m{\bf\Pi}^{n'm}\cdot{\bf\Pi}^{mn}$.
Thus the fast
motion induces in the dynamics of the slow motion not only
a scalar potential $\varepsilon_n(X)$ as in the drastic
approximation (7) but also a non-abelian vector potential
${\bf A}^{mn}({\bf X})$ and in place of the canonical
momentum ${\bf P}$ of (7) one has the gauge-covariant
momentum (28).
If one makes no approximation then $m,n$ in (29) range over
all values of the electronic quantum numbers
(${\cal N}=\infty$) and the Berry
connection ${\bf A}^{mn}$ is an infinite matrix.
But this ${\bf A}^{mn}$
is trivial (can be gauged away by a $U(\infty)$ gauge
transformation).
The eigenvalue equation of $H^{eff}$ is then an infinite
system of coupled differential equations, which is
useless for practical calculations. One obtains a workable
eigenvalue equation only if one can restrict oneself to a
small number ${\cal N}$ of eigenvalues $\varepsilon_n$ and
eigenvectors $\mid n({\bf x})\rangle\ n=1,2,\ldots{\cal
N}$ (Born-Huang approximation).$^{14)}$
${\bf A}^{mn}({\bf x})$ of (29) is then a connection of a
$U({\cal N})$ gauge theory, which is in general
non-trivial.
As a special case we consider the doubly degenerate
$\Lambda$-levels of a diatomic molecule for which ${\cal
N}=2$ and $m,n$ takes the two values $k=\pm\Lambda$ where
$k$ is again the component of angular momentum along the
internuclear axis and the $\mid{\bf n}({\bf x})\rangle$ in
(29) are given by (21) (or (22)).
The space of physical states of the slow system is
\begin{equation}{\cal H}^{slow}=
{\cal H}^{k=1}\oplus{\cal H}^{k=-1}
%\eqno(30)
\end{equation}
and (27), (28), (29) are $2\times2$ operator matrices:
\begin{eqnarray}
%\eqalignno{
\dps H^{k'k}&\dps =&\dps
{1\over2\mu}\sum^{+1}_{k''=-1}{\bf\Pi}^{k'k''}
\cdot{\bf\Pi}^{k''k}+
\varepsilon(x)\\ %&(31)\cr
\dps {\bf\Pi}^{k'k}&\dps =&\dps \delta^{k'k}{\bf P}-{\bf A}^{k'k}.
%\cr}
\end{eqnarray}
By a straightforward calculation one obtains as in (24)
for the spherical components of the Berry connection
${\bf A}^{k^\prime k}$:
\begin{equation}A_\theta^{k'k}={\Lambda\over x\sin\theta}
\left(\matrix{
-(1-\cos\theta) &0\cr
0&(1-\cos\theta)\cr}\right)\quad;\quad
A^{k'k}_\varphi=\left(\matrix{
0 &0\cr
0 &0\cr}\right)\ . %\eqno(33)
\end{equation}
The rapidly orbiting and spinning motion of the
electrons about the internuclear axis of a diatomic
molecule thus leads to an induced vector potential in the
dynamics of the slow (collective) motion, which is the
same as that of a pair of magnetic monopoles with the
monopole strength $g$ given by
${eg\over4\pi}=\pm\Lambda$. The components of the gauge
covariant momentum operator do no more commute but
fulfill the commutation relation
\begin{equation}
\Bigl[\Pi_i^{k'k''}\ ,\ \Pi_j^{k''k}\Bigr]=
-i\varepsilon_{ij_\ell}{X_\ell\over
X^3}\Lambda\left(\matrix{ 1 &0\cr
0 &-1\cr}
\right)
=iB^{k'k}_{ij}\ ,%\eqno(34)
\end{equation}
where
$B_{ij}^{k^{\prime}k}
=\varepsilon_{ij\ell}B_\ell^{k^{\prime}k},
$
is the Berry curvature (17) for the
${\bf A}^{k^{\prime}k}$ given by (33).
The commutation relations (34) are the well known $c.r.$
for a charge-monopole system and the first term in the
Hamiltonian (31) is the monopole
Hamiltonian with free radial motion,
$H_{monopole}={1\over2\mu}{\bf\Pi}^2$.
Due to the induced scalar potential, which is
approximately a radial oscillator potential
$\varepsilon(x)\approx{f\over2}(x-x_e)^2$, the radial
motion of (31) is not free. To dissect the system
described by (31) into a radial part and an angular part,
we use the radial momentum operator$^{15)}$
\begin{equation}P_{rad}={1\over2}\Bigl\{{X_i\over X}\ ,P_i\Bigr\}
%\eqno(35)
\end{equation}
and the angular momentum operator of a
monopole:$^{16)}$
\begin{equation}J_i=
\varepsilon_{ij_\ell}
X_j\Pi_\ell-X_i{1\over2}
\varepsilon_{mn_\ell}X_mB_{n_\ell}=
\varepsilon_{ij_\ell}X_j\Pi_\ell+k{X_i\over X}\ .
%\eqno(36)
\end{equation}
Then we obtain after a straightforward calculation for
$H$ of (31):
\begin{eqnarray}
%\eqalignno{
\dps H&\dps =&\dps {1\over2\mu}\vec\Pi^2+\varepsilon(X)=
\underbrace{
{1\over2\mu X^2}
({\bf J}^2-k^2)}
+
\underbrace{
{1\over2\mu}P_{rad}^2+
\varepsilon(X)}\nonumber\\ %\cr
\noalign{\hbox{or}}
\dps H&\dps =&\dps H_{monopole}+\varepsilon(X)=\phantom{22}
H_{rot}
\phantom{2221}+\phantom{22212}H_{radial \ oscillator}\ .
%&(37) \cr}
\end{eqnarray}
$H_{rot}$ is the Hamiltonian of a rotating dumbbell with
flywheel on its axis$^{17)}$ whose doubly degenerate
rotator
spectrum
${1\over2\mu x_e^2}j(j+1)$
starts at $j=k$.
Thus the spectrum is
again something like shown in Fig. 1 except
that the rotational levels do not start at
$j=0$ but, as observed for diatomic molecules,
at $j=k$ (because from (36)
follows that
${1\over X}X_iJ_i=k$). Therewith
we have obtained the standard
result in a way, which shows that it is
caused by the monopole dynamics induced by
the fast motion in the slow collective
motion.
Although we used the diatomic molecule as an
example, the same arguments should hold for
all kinds of quantum systems in which the fast
subsystem is a rapid rotation about a slowly
moving axis like a spinning quark about the
axis of a thin flux tube.
Fifty years after Dirac conceived the idea of
magnetic monopoles he wrote (in a letter to Abdus
Salam):
``I am inclined now to believe that monopoles do not
exist."
Though magnetic monopoles of the electromagnetic kind
may not exist, physical systems with the same dynamics
as that of monopoles do exist.
These monopoles are ``parts" of complicated physical
systems, like the ``part" which performs the collective
motion of molecules or the collective motion of a flux
tube.
%\vfil\eject
\begin{thebibliography}{[18]}
%\centerline{References}
%\vskip15pt
\bibitem[1]{}
M.V. Berry, {\it Proc. Roy. Soc. London Ser.} A{\bf
392}, 45 (1984).
%\vskip10pt
\bibitem[2]{}
B. Simon, Phys. Rev. Letters {\bf 51}, 2167 (1983).
%\vskip10pt
\bibitem[3]{}
G. Herzberg and H.C. Longuet-Higgins, {\it Discuss.
Faraday Soc.} {\bf 35}, 77 (1963); H.C.
Longuet-Higgins, {\it Proc. Roy. Soc. London Ser.}
A{\bf 344}, 147 (1975).
%\vskip10pt
\bibitem[4]{}
C. Mead and D. Truhlar, {\it J. Chem. Phys.} {\bf 70}, 2284
(1979); C.A. Mead, {\it Chem. Phys.} {\bf 49}, 23, 33
(1980).
%\vskip10pt
\bibitem[5]{}
For earlier ``anticipations" of the Berry phase, see
M. Berry, {\it Physics Today} December, 1990, p. 34.
%\vskip10pt
\bibitem[6]{}
J. Moody, A. Shapere, F. Wilczek, {\it Phys. Rev. Lett.}
{\bf 56}, 893 (1986); and in
``Geometric Phases in Physics,"
A. Shapere and F. Wilczek (editors) World Scientific
(1989);
R. Jackiw, {\it Int. J. Mod. Phys. A}{\bf 3}, 285 (1988);
A. Bohm, in {\it Symmetries in Science} {\bf 3}, p. 85, B.
Gruber and F. Iachello (editors) Plenum Press (1988).
%\vskip10pt
\bibitem[7]{}
M. Born and J. Oppenheimer,
{\it Ann. Phys.}
{\bf 84}, 457 (1927).
%\vskip10pt
\bibitem[8]{}
M. Born and V. Fock, {\it Z. Phys.} {\bf 51}, 165 (1928).
%\vskip10pt
\bibitem[9]{}
Note that (8) and (11) are two conditions which are in
general not compatible.
This means that there does not exist a solution of (8)
with the initial condition
$\psi(0)=\mid n({\bf x}(0))>$ such that
(11) is fulfilled. However there exists a solution of (8)
with the initial condition\hfill
$\psi(0)=\mid n(\tilde{\bf x}(0))\rangle=
\mid n(\tilde{\bf x}
(T))\rangle$ \hfill such that\\
$\mid\psi(t)><\psi(t)\mid=\mid n({\bf
x}(t)>$ is not an eigenvector of
$h(t)$ but of some other observables. With these
sections one can define a geometric phase and a connection
which describe a realistic time evolution and which go in
the adiabatic limit into the Berry phase and the adiabatic
connection. Y. Aharonov and J.
Anandan, {\it Phys. Rev. Lett.} {\bf 58}, 1593 (1987); J.
Anandan and Y. Aharonov, {\it Phys. Rev. D}{\bf 38}, 1863
(1988).
%\vskip10pt
\bibitem[10]{}
P. Goddard and D.I. Olive,
{\it Rep. Prog. Phys.}
{\bf 41}, 1357 (1978); S. Coleman, in (Erice lectures
1981), ``The Unity of the Fundamental Interactions,"
p. 21, H.
Zichichi (editor) Plenum Press (1981). P. Dirac, Proc.
Roy. Soc. London Ser. A{\bf 133}, 60 (1931).
%\vskip10pt
\bibitem[11]{}
That a bead, which has a constant angular momentum
component along an axis on which it can freely slide, has
the same dynamics as a magnetic monopole $g$ in the field
of an electrical charge $e$, has been discussed in J.M.
Leinaas, {\it Physica Scripta} {\bf 17}, 483 (1978).
\bibitem[12]{}
F. Wilczek and A. Zee,
{\it Phys. Rev. Lett.} {\bf 52}, 2111 (1984).
%\vskip10pt
\bibitem[13]{}
A. Bohm,
B. Kendrick and M.E. Loewe, Internat. Journ. Quantum
Chemistry {\bf 41}
(1992). B. Zygelman, Phys. Rev. Lett. {\bf 64}, 256
(1990), T. Pacher, C.A. Mead, L.S. Cederbaum, H.
K\"oppel,
J. Chem. Phys. {\bf 91}, 7057 (1989).
%\vskip10pt
\bibitem[14]{}
M. Born, K. Huang, ``Dynamical Theory of Crystal Lattices,"
Oxford University Press (1954).
%\vskip10pt
\bibitem[15]{}
A. Bohm, ``Quantum Mechanics" (Second Edition)
Eq. (VII.2.2) Springer (1986).
%\vskip10pt
\bibitem[16]{}
This is the conserved observable that fulfills all
conditions of an angular momentum operator for the system
described by $H$ of (31) with position operator $X_i$ and
momentum operator $\Pi_i$, cf ref. 10.
%\vskip10pt
\bibitem[17]{}
Section V.4 of
reference 15 or
G. Herzberg, ``Molecular Spectra and Molecular
Structure,"
Vol. I, D. Van Nostrand Publishers, NY (1955).
\end{thebibliography}
\end{document}