{Position Uncertainty Measures on the Sphere}
D.A. Trifonov
72 Tzarigradsko chaussee
1784 Sofia
Bulgaria
Position uncertainty (delocalization) measures for a particle on the sphere are proposed and illustrated on several examples of states. The new measures are constructed using suitably the standard multiplication angle operator variances. They are shown to depend solely on the state of the particle and to obey uncertainty relations of the Schr\"odinger--Robertson type. A set of Hermitian operators with continuous spectrum is pointed out the variances of which are complementary to the longitudinal angle uncertainty measure. \\[-2mm] PACS numbers: 03.65.-w,\, 02.30.Gp,\, 45.50.Dv
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\begin{document}

\title{{\bf Position Uncertainty Measures on the Sphere}}
\author{D.A. Trifonov\\
Institute for nuclear research,\\ 72 Tzarigradsko chaussee, 1784 Sofia,
Bulgaria}
\maketitle   \date{}
\vspace{-65mm} 

{\small \noi 
quant-ph/0404087\\
To appear in Proceedings of Int. Conf. on Geometry, \\
Integrability and Quantization, held in Varna 2003} 
\vspace{55mm}

\begin{abstract}
Position uncertainty (delocalization) measures for a particle on the
sphere are proposed and illustrated on several examples of states.  
The new measures are constructed using suitably the standard
multiplication angle operator variances. They are shown to depend solely
on the state of the particle and to obey uncertainty relations of the
Schr\"odinger--Robertson type. A set of Hermitian operators with
continuous spectrum is pointed out the variances of which are
complementary to the longitudinal angle uncertainty measure. \\[-2mm]

PACS numbers: 03.65.-w,\, 02.30.Gp,\, 45.50.Dv
\end{abstract}
%%\maketitle


\section*{{\large\bf 1. Introduction}}

Recently an interest is shown in the literature to the problem of a
quantum particle on the circle \ci{FA01,GP,GO98,Kast,KRP96,KR02,T03} and 
on the sphere \ci{GP,HM,KR00}.  In \ci{GO98,HM,KRP96,KR00} overcomplete
families of states (coherent states) for these systems are constructed.

\noi
 One of the difficulties for these systems is the position (and momentum)
uncertainty measures for the particle (or equivalently, the wave function
spread measure). This is a consequence of the issue related the choice of
the operator for the azimuthal angle $\varphi$. For a particle on the
sphere there is a second problem, related to the non-hermiticity of the
operator $-\ri\partial/\partial\vthe$, where $\vthe$ is the
longitudinal angle. The problem of correct definitions of uncertainty
measures is (and should be) closely related to the construction and
justification of the uncertainty relations (UR's), and of coherent and
squeezed states  as well.

\noi
From the Dirac correspondence rule between Poisson bracket $\{f,g\}$ of
two classical quantities $f$ and $g$ and the commutator of the
corresponding operators $\hat{f}$ and $\hat{g}$,

\beq\lb{PB}
\{f,g\} \longrightarrow {\ri} [\hat{f},\hat{g}]
\eeq
it follows that $[\hat{p}_\vphi,\hat{\vphi}] = -{\ri}$, where $\hat\vphi$ is
the azimuthal angle operator, and $\hat{p}_\vphi$ is the angular momentum
operator. Formally this commutation relation is satisfied by $\hat\vphi =
\vphi$, and $\hat{p}_\vphi = -{\ri}\pd/\pd\vphi$. Then one can write the
standard Heisenberg-Robertson UR
$(\Dlt p_\vphi)^2(\Dlt\vphi)^2\geq 1/4$.

\noi However on the eigenstates $\psi_m(\vphi)$ of $\hat{p}_\vphi$,

\beq\lb{psi_m}
\psi_m(\vphi) = \exp({\ri} m\vphi)/\sqrt{2\pi},\qquad m=0,\pm1,\dots
\eeq
the above UR breaks down.

\noi The reasons for this contradiction is the fact, that $\hat{p}_\vphi$ 
is not Hermitian on the $\vphi$-transformed state
$\psi^\prime(\vphi)=\vphi\,\psi(\vphi)$, since $\psi^\prime(\vphi)$ is no
more invariant under translations on $2\pi$ (no more $2\pi$ invariant).
Therefore  many authors try to adopt some $2\pi$-invariant position
operator \ci{CN68,FA01,GO98,KR02}, or even another definition of the
uncertainty  on the circle \ci{KR02}.  However almost all of the
associated uncertainty measures are not in good consistency with the
localization of the particle on the circle \ci{T03}.

\noi In the case of a particle on the sphere the situation is even worth
since in addition to the  problems with the  $\vphi$ and  $p_\vphi$, one
encounters the subtle  of non-hermiticity of the operator
$-i\pd/\pd\vthe$, related to the longitudinal angle coordinate $\vthe$,
$0\leq \vthe \leq \pi$.

\noi In this paper we provide an approach to these issues with minimal (in
our opinion) deviation from the standard commutation relation and standard
measure of uncertainty. In the case of a particle on the circle the main
idea has been sketched in the second paper in \ci{T03}, and developed in
greater detail (providing some proofs and examples) in the first paper in
\ci{T03}.

\noi In section 2 a brief review of the properties of main previous
position uncertainty measures on the circle (based on $2\pi$-periodic
operator $\hat\vphi$) is provided. In section 3 two different position
uncertainty measures for a particle on the sphere are constructed and
discussed. At $\vthe = \pi/2$ the corresponding states and measures on a
circle are recovered. The new measures are constructed using suitably the
standard expressions of the first and second moments of the angle
variable, calculated by integration over $2\pi$ intervals. They are of the
form of positive state functionals, the values of which depend solely on
the state considered. In section uncertainty relations (UR's) (of the
Robertson--Schr\"odinger type)  are established for the position
delocalization measures and the appropriate {\it complementary measures}.
The latters are of the form of standard variances of $\hat{p}_\vphi$ and
new Hermitian operators $\hat{p}_{n\vthe}$\,, $n=1,2,\ldots$. 

\section*{{\large\bf 2. Uncertainty measures on the circle}}

 For a particle on the real line the standard measure of the position
uncertainty is given by the second moment $(\Dlt x)^2:= \la (x -
\la x\ra)^2\ra$  of the position operator $\hat{x}=x$, or
equivalently by the standard deviation $\Dlt x$. Mathematically both $\Dlt
x$ and $\la x\ra$ are one-to-one functionals on state space.  The quantity
$(\Dlt x)^2$ is also called variance, or dispersion,  of $x$ and is also
denoted as $Dx$ or $M^{(2)}x$. The variance of $x$ is regarded as a
measure of spread, or delocalization, of the state wave function
$\psi(x)$.  More precisely this is a measure of spread of the probability
distribution $p(x):=|\psi(x)|^2$.  Here the means $\la  x\ra$ and $\la
x^2\ra$ are calculated by integration with respect to $x$:\, $ \la x\ra =
\int x|\psi(x)|^2 {\rd} x $.\,

\noi However in the case of angle operator $\hat{\vphi} = \vphi$ it was
not clear how to calculate and interpret the analogous quantity
$\Dlt\vphi$, since the operator $\hat{\vphi} = \vphi$ is not invariant
under translation $\vphi \rightarrow \vphi+2\pi$ (not $2\pi$-periodic),
while the wave functions $\psi(\vphi)$ are $2\pi$-periodic by definition.
This trouble seems to be the main reason why many authors look for
$2\pi$-invariant position operators in order to  construct relevant
uncertainty measures on the circle.

\noi The first such operators used probably were $\sin\vphi$ and $\cos\vphi$ 
\cite{CN68}. The variances of these operators satisfy correct inequalities 
\cite{CN68}

\beq\lb{URsin}
(\Dlt p_\vphi)^2(\Dlt\sin\vphi)^2 \geq |\la\cos\vphi\ra|^2/4,\qquad
(\Dlt p_\vphi)^2(\Dlt\sin\vphi)^2 \geq |\la\cos\vphi\ra|^2/4
\eeq
However one easily find states in which the variances $(\Dlt\sin\vphi)^2$
and $(\Dlt \cos\vphi)^2$  take values greater than the corresponding ones
in the uniform distribution $p_{\rm uni}(\vphi) = 1/2\pi =
|\psi_m(\vphi)|^2$: in $\psi_m(\vphi)$ one has $(\Dlt\sin\vphi)^2 =
(\Dlt\cos\vphi)^2 = 1/2$, while in $\psi_{\rm cos}(\vphi) =
(1/\sqrt{\pi})\cos\vphi$ these variances are $(\Dlt\cos\vphi)^2=3/4$,
$(\Dlt\sin\vphi)^2=1/4$. In $\psi_{\rm sin}(\vphi) =
(1/\sqrt{\pi})\sin\vphi$ they are interchanged -- $(\Dlt\cos\vphi)^2=1/4$,
$(\Dlt\sin\vphi)^2=3/4$.  The two states $\psi_{\rm cos}(\vphi)$ and
$\psi_{\rm sin} (\vphi)$ coincide under the shift $\vphi \rightarrow \vphi\pm
\pi/2$, therefore it is reasonable to have coinciding (or close) measures
of spread for them, which should be less than those in the eigenstates
$\psi_m(\vphi)$. These deficiencies are partially removed by the
"uncertainty measure" \ci{CN68} $(\tilde{\Dlt}\vphi)^2 =  (\Dlt\cos\vphi)^2
+ (\Dlt\sin\vphi)^2$, which can be written also in the forms

\beq\lb{tldDlt}
(\tilde{\Dlt}\vphi)^2 =  1 - \la\cos\vphi\ra^2 - \la\sin\vphi\ra^2
= 1 - |\la U(\vphi)\ra|^2, \qquad U(\vphi)=e^{i\vphi}
\eeq
The quantity $\tilde{\Dlt}\vphi$ has been considered also in \ci{GO98}
and \ci{FA01}. In \ci{FA01} it was noted that $\tilde{\Dlt}\vphi$ has the
meaning of radial distance of the centroid of the ring distribution
$p(\vphi)$ from the circle line (and $\la\cos\vphi\ra^2
+\la\sin\vphi\ra^2$ is the squared centroid's distance from the center of
the circle -- see figure 1 in \ci{FA01}).
From (\ref{tldDlt}) and (\ref{URsin}) it follows that \ci{CN68}

\beq\lb{tldURsin}
(\Dlt p_\vphi)^2({\tilde\Dlt}\vphi)^2 \geq \frac 14(\la\cos\vphi\ra^2 +
\la\sin\vphi\ra^2)
\eeq
This UR is approximately minimized in the canonical
coherent states (CS) $|\alpha,\beta\ra$ of the two dimensional oscillator with
large value of Re$^2\alpha+{\rm Re}^2\beta$ \ci{CN68}.

\noi However if one consider the quantity $(\tilde{\Dlt}\vphi)^2$, eq.
(\ref{tldDlt}), as a delocalization measure on the circle one encounters 
some unsatisfactory results. For example, it produces the same maximal
delocalization (i.e. $\tilde{\Dlt}\vphi = 1$) for the eigenstates
$\psi_m(\vphi)$ of $\hat{p}_\vphi$ and for all states $\psi(\vphi)$
with the property $|\psi(\vphi)|=|\psi(\vphi+\pi)|$. The centroid for
those $\pi$-periodic distributions
$|\psi(\vphi)|^2$ is in the center of the ring. On figure 1 graphics of
three $\pi$-periodic distributions are shown: uniform one $p_{\rm uni}(\vphi) =
1/2\pi = |\psi_m(\vphi)|^2$,
$p_{\rm sin}(\vphi)= \psi_{\rm sin}(\vphi)^2=\sin^2\vphi\,/\pi$ and
$p_{\rm sin2}(\vphi) = \sin(2\vphi)^2/\pi$. It is clear that
the localization of that distributions is quite different, and it is
desirable to have an uncertainty measure that distinguishes between them.


\noi A rather nonstandard expressions for position and angular momentum
uncertainties for a particle on the circle were introduced and discussed
in \ci{KR02}:

\beq\lb{hatDlt}
\Dlt^2(\hat{p}_\vphi) =
\frac 14 \ln(\la e^{-2\hat{p}_\vphi}\ra\la e^{2\hat{p}_\vphi}\ra),\quad
\Dlt^2(\hat{\vphi}) = -\frac 14 \ln|\la U(\vphi)^2\ra|^2
\eeq
For a large sets of states these quantities obey the inequality
$\Dlt^2(\hat{p}_\vphi) + \Dlt^2(\hat{\vphi}) \geq 1 $, the equality being
reached in the eigenstates $|\xi\ra$ of the operator
$Z=\exp(-\hat{p}_\vphi+1/2)U(\vphi)$. The family of $|\xi\ra$ is overcomplete
and the states $|\xi\ra$ are called CS on the circle
\ci{KRP96,GO98,KR02}.

\noi The functional $\Dlt^2(\hat{\vphi})$, based on the $2\pi$-invariant
operator $U(\vphi)$ ($U(\vphi+2\pi) = U(\vphi))$ was proposed as a position
uncertainty on the circle.  However this uncertainty measure was found
\ci{T03} to be not quite consistent with state localization: on CS $|\xi\ra$
it equals $1/2$, while on the visually worse localized states
$|\xi\ra-|\!-\!\xi\ra$ (Schr\"odinger cat states on the circle) it can
take rather less value of $0.33$ (see \ci{T03} and figure 2 therein).
On the above noted states $\psi_{\rm sin}(\vphi)$, $\psi_{\rm sin2}(\vphi)$ and
$\psi_m(\vphi)$ it takes values $0.346$, $\infty$, $\infty$. Thus it makes
distinction between  $\psi_{\rm sin}(\vphi)$ and $\psi_{\rm sin2}(\vphi)$ and
$\psi_m(\vphi)$, but identifies $\psi_{\rm sin2}(\vphi)$ with the uniform state
$\psi_m(\vphi)$ (see figure 1).
\vspace{15mm}

\bc
\hspace{20mm}\begin{tabular}{ll}
\hspace{15mm}\makebox(90,65)[rc]{\input{pumsfig1.pic}}&
\makebox(48,55)[lc]{\begin{minipage}{45mm}
{\small {\bf Figure 1.}\, $\pi$-periodic, $\pi/2$-periodic 
and uniform distributions on the circle:  
$p_{\rm sin}(\varphi)= (\sin\vphi)^2/\pi$ (dot line), 
$p_{\rm sin2}(\varphi) = (\sin(2\vphi))^2/\pi$ (solid line) and
$p_{\rm uni}(\varphi)  = 1/2\pi= |\psi_m(\vphi)|^2$. 
The functional  $\tilde{\Dlt}\vphi$, eq. (\ref{tldDlt}), on all these
distributions takes the same maximal value of $1$, while
$\Dlt^2(\hat{\vphi})$, eq. (\ref{hatDlt}), takes the values $0.346$,
$\infty$ and $\infty$ respectively.}
\end{minipage}} \end{tabular}
\vspace{15mm}  \ec

\noi  New position uncertainty measures, that are better consistent with
the localization on the circle are constructed in the next section as
particular cases ($\vthe=\pi/2$) of the measures on the sphere.


\section*{{\large\bf 3. Uncertainty measures on the sphere}}

A  point on the sphere ${\mathbb S}^2$ is determined by the spherical
angles $\vphi$ and $\vthe$. The Hilbert space of states for a particle on
the sphere is defined as the space of square integrable functions on
${\mathbb S}^2$ with respect to the normalized measure (on the unit sphere)
${\rd}\mu(\vphi,\vthe) = \sin\vthe~{\rd}\vthe~{\rd}\vphi/4\pi \equiv {\rd}
S/4\pi$. 
Wave functions $\psi(\vartheta,\vphi)$ have to be $2\pi$-periodic in $\vphi$
(periodicity in $\vthe$ is not required).

\noi  The measure of uncertainty of $\vthe$ in a state $\psi(\vthe,\vphi)$
may be adopted as the ordinary variance
$(\Dlt\vthe)^2=\la\psi|\vthe^2|\psi\ra - \la\psi|\vthe|\psi\ra^2$. 
The uncertainty measure for $\vphi$ can not be taken in a similar way.  In
view of the nonperiodicity of $\vphi$ the standard "means"
$\la\vphi^k\ra$, $k=1,\ldots$, are ill defined: their values depend on the
limit $\vphi_0$ of integration (on the unit sphere: ${\rd} S=\sin\vthe
{\rd}\vthe {\rd}\vphi$),

\beq\lb{mphik}
\la\psi|\vphi^k|\psi\ra =\int_0^\pi\sin\vthe {\rd}\vthe
\int_{\vphi_0-\pi}^{\vphi_0+\pi} \vphi^k|\psi(\vphi,\vthe)|^2 {\rd}\vphi
\equiv M^{(k)}\vphi(\vphi_0)
\eeq

\noi  Following the scheme of refs \ci{T03} for the case of a circle we
define the $\vphi$-uncertainty measure on the sphere as

\beq\lb{cDlt}
(_c\Dlt\vphi)^2 = \int_0^\pi\sin\vthe {\rd}\vthe
\int_{\vphi_c-\pi}^{\vphi_c+\pi} \vphi^2|\psi(\vphi,\vthe)|^2 {\rd}\vphi -
\left(\int_0^\pi\sin\vthe {\rd}\vthe
\int_{\vphi_c-\pi}^{\vphi_c+\pi} \vphi|\psi(\vphi,\vthe)|^2
{\rd}\vphi\right)^2 
\eeq
where $\vphi_c$ is the $\vphi$-coordinate of the {\it center of the
packet} $|\psi(\vphi,\vthe)|^2$. For packets $|\psi(\vphi,\vthe)|$ that
{\it are not} $2\pi/k$-periodic, $k=2,\ldots$, in $\vphi$ the angle $\vphi_c$
can be determined as the polar angle of a pont in the plane with cartesian
coordinates $x_c = \la\cos\vphi\ra$, $y_c = \la\sin\vphi\ra$. For packets
that are $2\pi/k$-periodic, $k=2,\ldots$, in $\vphi$, one obtains
$x_c=0=y_c$, i.e. the center of the packet $\vphi_c$ remains undefined in
this way.  To overcome this difficulty suffice it to observe that the packet
center $\vphi_c$, when determined from the above $x_c$ and $y_c$, satisfies the
conditions (as checked on several examples)

\beq\lb{phi_c}
\begin{tabular}{l}
$\displaystyle M\vphi(\vphi_c) = \vphi_c$,\\[3mm]
$\displaystyle
\int_0^\pi\sin\vthe|\psi(\vphi_c+\pi,\vthe)|^2{\rd}\vthe \leq \frac{1}{2\pi}$
\end{tabular}
\eeq
where $M\vphi(\vphi_0)$ is the limit-dependent "mean" of $\vphi$, defined
in eq. (\ref{mphik}) for $k=1$. Therefore it is reasonable to define
$\vphi$-coordinate $\vphi_c$ of the center of the wave packet on the
sphere as solution of the system (\ref{phi_c}). It is straightforward to
check that conditions (\ref{phi_c}) ensure {\it the minimum} of the
limit-dependent variance $\Dlt\vphi(\vphi_0)$ as a function of $\vphi_0$,

\beq\lb{Dlt_0}
(\Dlt\vphi)^2(\vphi_0) = \int_0^\pi\sin\vthe {\rd}\vthe
\int_{\vphi_0-\pi}^{\vphi_0+\pi} \vphi^2|\psi(\vphi,\vthe)|^2 {\rd}\vphi -
(M\vphi(\vphi_0))^2
\eeq
One can verify that the function $(\Dlt\vphi)^2(\vphi_0)$ is
$2\pi$-periodic [i.e. $(\Dlt\vphi)^2(\vphi_0+2\pi)=(\Dlt\vphi)^2(\vphi_0)$],
therefore the minimum always exists.

\noi  For $2\pi/k$-periodic, $k=2,\ldots$, in $\vphi$ wave packets the
conditions (\ref{phi_c}) may have more than one solutions (in fact $k$
solutions). We call these $2\pi/k$-periodic packets {\it multi-centered}.

\noi The $\vthe$-coordinate $\vthe_c$ of the packet center on the sphere
can be defined as the mean of $\theta$: $$\vthe_c =
\int\vthe|\psi(\vphi,\vthe)|^2{\rd} S. $$ Thus the coordinates of the wave
packet center on the sphere are $(\vphi_c, \vthe_c)$.
The uncertainty (or delocalization) measure of a state $|\psi\ra$ on the
sphere can be defined in two complementary ways: as a sum or as a product
of the corresponding $\vphi$- and $\vthe$- measures,

\beq\lb{UMs}
M_+(\psi) := (_c\Dlt_\psi\vphi)^2+(\Dlt_\psi\vthe)^2,\quad{\rm or}\quad
M_{\mbox{\,\circle*{3}}}(\psi) := (_c\Dlt_\psi\vphi)^2(\Dlt_\psi\vthe)^2
\eeq
\vspace{5mm}

\noi 
{\sl\large\bf Examples.} Let us illustrate the relevance of the above 
constructed position uncertainty measures on two families of states
$f_{uv\gamma k}(\vphi,\vthe)$ and $\psi_{uv\tau}(\vphi,\vthe)$,
\beq\lb{fuvk}
f_{uv\gamma k}(\vphi,\vthe) = N(u,v,\gamma,k) \,
\left[2 + \cos(k\vphi-u) + \cos(3(\vthe-v)/2)\right]^\gamma
\eeq
\beq\lb{psiCS}
\psi_{uv\tau}(\vphi,\vthe) = N(u,v,\tau)\,\sum_{l=0}^{\infty}e^{-\tau
l(l+1)/2}\,\sqrt{2l+1}\,P_{l}(\cos\theta)
\eeq
where $N(u,v,\gamma,k), N(u,v,\tau)$ are
normalization factors, $\gamma$ and $\tau$ are real positive parameters,
$k$ is a positive integer, $P_l(x)$ are Legendre polynomials, and $\theta$
is the angle between radii of the current point $(\vphi,\vthe)$ and a
fixed point $(u,v)$ on the unit sphere.

\noi The function $f_{uv\gam k}(\vphi,\vthe)$ is constructed as a $k$-peak
state on the unit sphere ($k=1,2,\ldots$), the width of the peaks being
decreasing with $\gam$. Thus $1/\gam$ plays a role of a delocalization
parameter, and $\gam$ -- a {\it squeezing parameter} of the states $f_{uv\gam
k}$ (see figures 2 and 3 for the cases of $k=2$, $u=\pi, v=\pi/2$, and
$\gam =1$ (figure 2) and $\gam = 5$ (figure 3) (or $1/\gam=1$ and $0.2$)).
In these states the above defined position uncertainty measures are
calculated as $(_c\Dlt\vphi)^2 = 2.94$, $(\Dlt\vthe)^2 = 0.329$  and
$(_c\Dlt\vphi)^2 = 2.57$, $(\Dlt\vthe)^2 = 0.146$ correspondingly. Both
$_c\Dlt\vphi$ and $\Dlt\vthe$ (and $M_+$ and $M_{\mbox{\,\circle*{3}}}$ as
well) are found decreasing when $1/\gam \rightarrow 0$, i.e. $\gam$ indeed 
appears as a {\it squeezing parameter} (the greater is $\gam$ the stronger is 
squeezing of the position uncertainty measures).
The packet centers $(\vphi_c,\vthe_c)$ do not depend on $\gam$ and for
$u=\pi, v=\pi/2, k=2$ the two centers are $(0,\pi/2)$ and $(\pi,\pi/2)$.

\bc
\begin{tabular}{cc}
%\begin{figure}
\includegraphics[width=65mm,height=45mm]{pumsfig2.eps}$\qquad$&
%\vspace{3mm}
\includegraphics[width=65mm,height=45mm]{pumsfig3.eps}\\[5mm]

\begin{minipage}{60mm}
{\small {\bf Figure 2.} Plot of the two peak distribution
$|f_{uv\gam k}(\vphi,\vthe)|^2$ with $u=\pi,  v=\pi/2$, $k=2$, $\gam=1$.
In this state the $\vphi$- and $\vthe$-uncertainties are
$(_c\Dlt\vphi)^2 = 2.94$, $(\Dlt\vthe)^2 = 0.329$.
The two packet centers are at
$(\vphi=0, \vthe=\pi/2)$ and $(\vphi=\pi, \vthe=\pi/2)$.}
\end{minipage} $\qquad$&
\begin{minipage}{60mm}
{\small{\bf Figure 3.}  Plot of the two peak distribution
$|f_{uv\gam k}(\vphi,\vthe)|^2$ with $u=\pi, v=\pi/2$,  $k=2$, $\gam=5$.
In this state the $\vphi$- and $\vthe$-uncertainties are
$(_c\Dlt\vphi)^2 = 2.57$, $(\Dlt\vthe)^2 = 0.146$.
The two packet centers are at
$(\vphi=0, \vthe=\pi/2)$ and $(\vphi=\pi, \vthe=\pi/2)$.}
\end{minipage}
\end{tabular}
\vspace{3mm}
\ec

\noi The function $\psi_{uv\tau}$ is taken from paper \ci{HM}, where the
set $\{\psi_{uv\tau}\}$ is shown  to form an overcomplete set of states
(for every $\tau$; and $u,v$ may be complex) on the sphere ${\mathbb
S}^2$, called {\it coherent states on the sphere} (CS on the sphere). In
\ci{HM} CS are constructed on $d$-dimensional sphere, and CS on ${\mathbb
S}^2$ were previously constructed in \ci{KR00}. The shapes of CS
$\psi_{uv\tau}$ are shown on figures 4 and 5 for the cases of $u=\pi,
v=\pi/2$, and $\tau =1$ (figure 4) and $\tau=0.2$ (figure 5).
Calculations show that the less $\tau$ is the less is the area of
${\mathbb S}^2$ in which CS are concentrated, confirming the suggestion of
\ci{HM}. In these CS the above defined position uncertainty measures take
the values $(_c\Dlt\vphi)^2 = 1.57$, $(\Dlt\vthe)^2 = 0.419$ (for
$\tau=1$) and $(_c\Dlt\vphi)^2 = 0.439$,  $(\Dlt\vthe)^2 = 0.185$ (for
$\tau=0.2$). Both $_c\Dlt\vphi$ and $\Dlt\vthe$ are decreasing (thereby
$M_+$ and $M_{\mbox{\,\circle*{3}}}$ also are decreasing) when $\tau
\rightarrow 0$, i.e. $1/\tau$ appears as a position squeezing parameter.
The packet center $(\vphi_c,\vthe_c)$ does not depend on $\tau$, and for
CS with $u=\pi, v=\pi/2$ it is $(\pi, \pi/2)$.

\noi  It is worth noting that the shapes of one-peak states $f_{uv\gam
1}(\vphi,\vthe)\equiv f_{uv\gam}(\vphi,\vthe)$ with $\gam=1/\tau$ and CS
$\psi_{uv\tau}(\vphi,\vthe)$ are quite similar: the packet centers of both
states are determined by $u$, $v$, and the position uncertainties
$_c\Dlt\vphi$ and $\Dlt\vthe$ vary with $\tau$ similarly.  In particular
in $f_{uv\gam}$ with $u=\pi, v=\pi/2$ and $1/\gam=\tau=1, 0.2$ one finds
$(_c\Dlt\vphi)^2 = 1.91, 0.418$, $(\Dlt\vthe)^2 = 0.329, 0.146$
respectively, which are to be compared with $(_c\Dlt\vphi)^2 = 1.57,
0.439$, $(\Dlt\vthe)^2 = 0.419, 0.185$ in the corresponding CS
$\psi_{uv\tau}$.

\noi Due to the factor $\sin\vthe$ in the surface element ${\rd} S$ the 
most delocalized states on the sphere is $\psi_\alf(\vphi,\vthe) =
\exp(i\alf(\vphi,\vthe))/(\pi\sqrt{2\sin\vthe})$ (and not the uniform one
$\psi_{\rm uni}=1/\sqrt{4\pi}$): In $\psi_\alf$ the $\vphi$- and
$\vthe$-uncertainty measures take the values $(_c\Dlt\vphi)^2=(\Dlt\vphi)^2=
\pi^2/3 \simeq 3.29$,  $(\Dlt\vthe)^2=\pi^2/12\simeq 0.82$, while in
$\psi_{\rm uni}$\,  $(_c\Dlt\vphi)^2=\pi^2/3$, $(\Dlt\vthe)^2 =
\pi^2/4-0.2 \simeq 0.47$.

\noi As we have already noted position uncertainty measures are positive
maps of states (in fact of the corresponding  probability distributions),
associated with coordinate variables. It is then clear that one can
construct such measures using other coordinates $x_i(\vphi,\vthe), i=1,2,$
on the sphere, such as the stereographic projections $q_i$ and the
"wrapping" coordinates $(\eta, \xi)$ \ci{GP}.
\bc
\begin{tabular}{cc}
%\begin{figure}
\includegraphics[width=65mm,height=45mm]{pumsfig4.eps}$\qquad$&
%\vspace{3mm}
\includegraphics[width=65mm,height=45mm]{pumsfig5.eps}\\[5mm]

\begin{minipage}{60mm}
{\small {\bf Figure 4.} Plot of CS distribution on the sphere
$|\psi_{uv\tau}(\vphi,\vthe)|^2$ with $u=\pi,  v=\pi/2$, $\tau=1$. In this
state the $\vphi$- and $\vthe$-uncertainties are $(_c\Dlt\vphi)^2 = 1.57$,
$(\Dlt\vthe)^2 = 0.419$. The packet center is at $(\vphi=\pi, \vthe=\pi/2)$.}
\end{minipage} $\qquad$&
\begin{minipage}{60mm}
{\small{\bf Figure 5.}  Plot of CS distribution on the sphere
$|\psi_{uv\tau}(\vphi,\vthe)|^2$ with $u=\pi, v=\pi/2$, $\tau=0.2$. In this
state the $\vphi$- and $\vthe$-uncertainties are $(_c\Dlt\vphi)^2 = 0.439$,
$(\Dlt\vthe)^2 = 0.185$. The packet center is at $(\vphi=\pi, \vthe=\pi/2)$.}
\end{minipage}
\end{tabular}
\vspace{3mm}
\ec
However when $x_i(\vphi,\vthe)$ depend on $\vphi$ through $\cos\vphi$
and/or $\sin\vphi$ (the case of $q_i$ and $(\eta,\xi)$), the standard
variance will exhibit deficiencies like those on the circle, discussed in
previous section.
The stereographic coordinates $q_i$,
\beq\lb{q_i}
 q_1= 2r\cot(\vthe/2)\cos\vphi,\quad q_2= 2r\cot(\vthe/2)\sin\vphi 
\eeq 
exhibit an extra deficiency (coming from the factor $\cot(\vthe/2)$) --
their variances are infinite in all states $\psi(\vphi,\vthe)$ which do not
vanish sufficiently fast when $\vthe \rightarrow 0$. Examples of such states
are $f_{uv\gam k}(\vphi,\vthe)$ and CS $\psi_{uv\tau}(\vphi,\vthe)$
discussed above. These states do not vanish at $\vthe=0$. Therefore
$\langle q_i^2 \rangle = \infty$. In the most delocalized states
$\psi_\alf(\vthe)$ the averages of $q_i^2$ are also divergent.  
\vspace{3mm}

\section*{\large\bf 4. Uncertainty inequalities on the sphere}
% \section{Uncertainty inequalities on the sphere}

The uncertainty relations (UR's) for the position delocalization measure on
the sphere encounter two problems. The first one is related to a ill property
of the multiplication position operator $\vphi$: the function
$\vphi\psi(\vphi,\vthe)$ is no more $\pi$-periodic. The second subtle
comes from the non-hermiticity of the operator $-{\ri}\pd/\pd\vthe$
($\hbar=1$). There is a third problem (compared to the case of a circle)
related to several position and momentums operators for a particle on the
sphere: one has to formulate uncertainty  relations for several
observables.

\noi In order to overcome these difficulties we have to apply the scheme
of the Gram-Robertson matrix, developed in \ci{T00}. For one state
$|\psi\rangle$ and several observables $X_i$, $i=1,\ldots,n$, the
Gram-Robertson matrix $G$ is defined as \ci{T00} $G=\{G_{ij}\}$,

\beq\lb{G}
G_{ij} = \langle (X_i-\langle X_i\rangle)\psi|(X_j-\langle
X_j\rangle)\psi\rangle \equiv G_{ij}(\psi;\vec{X})
\eeq
It was shown that the characteristic coefficients of the symmetric part
$S$ of $G$ ($S=(G+G^T)/2$) are greater or equal to that of the
antisymmetric part $A$ ($A=-{\ri}(G-G^T)/2$, $G^T$ being the transposed $G$).
These inequalities are called (generalized) {\it characteristic} UR's
for the $n$ observables $X_i$ in a state $|\psi\rangle$ \ci{T00}. In
particular, the senior characteristic UR reads
\beq\lb{detG}
\det{S} \geq \det{A}.
\eeq
The real and symmetric matrix $S$ is defined as a (generalized)
uncertainty matrix, and $A$ is regarded as a generalization of the matrix of
mean commutators $-{\ri}\langle[X_j,X_k]\rangle$. When the action of $X_i$ is
well defined on $X_j|\psi\rangle$ the matrix $S$ coincide with the
standard uncertainty (or covariance) matrix $\{\langle\psi|(X_i-\langle
X_i\rangle)(X_j-\langle X_j\rangle)\psi\rangle\} = \{{\rm Cov}(X_i,X_j)\}
\equiv \sigma_{ij})$, where ${\rm Cov}(X_i,X_j)$ is the standard covariance of
$X_i$ and $X_j$. The other notations for the covariance ${\rm Cov}(X_i,X_j)$
and variance ${\rm Cov}(X_i,X_i)$ are  $\Dlt X_iX_j$ and $(\Dlt X_i)^2$.
In such "smooth" cases the senior characteristic inequality (\ref{detG})
reads
\beq\lb{RUR}
\det\sigma\geq\det C,\quad{\rm where}\quad C=\{-{\ri}\langle[X_k,X_j]
\rangle/2\}
\eeq
and this latter inequality was first obtained by Robertson \ci{R34}. It is a
generalization of the Schr\"odinger (or Schr\"odinger-Robertson) UR for two
observable $X_1$, $X_2$ (first established in \ci{SR30}),
\beq\lb{SUR}
(\Dlt X_1)^2(\Dlt X_2)^2 \geq \frac 14|\langle[X_1,X_2]\rangle|^2
+ ({\rm Cov}(X_1,X_2))^2
\eeq

\noi  If for some reason the repeated action $X_iX_j|\psi\rangle$ is not
correctly defined one has to resort to generalized UR (\ref{detG}), which
for two operators reads

\beq\lb{gSUR}
( _g\Dlt X_1)^2( _g\Dlt X_2)^2 \geq \frac 14| _g\la[X_1,X_2]\ra|^2 +
( _g{\rm Cov}(X_1,X_2))^2
\eeq
where  $_g{\rm Cov}(X_1,X_2) =
{\rm Re}\la (X_1-\la X_1\ra)\psi|(X_2-\la X_2\ra)\psi\ra$,
$( _g\Dlt X)^2 = \,_g{\rm Cov}(X,X)$, and
$_g\la[X_1,X_2]\ra =
-2i{\rm Im}\la (X_1-\la X_1\ra)\psi|(X_2-\la X_2\ra)\psi\ra$ \ci{T00}.
The UR (\ref{gSUR}) quite similar to (\ref{SUR}). Thus we may define the
generalized covariance and the generalized mean commutator as $_g{\rm
Cov}(X_1,X_2)$ and $_g\la[X_1,X_2]\ra $ respectively \ci{T00,ChD}.

\noi Let us note that for two operators the inequality $\det S \geq \det A$,
eq. (\ref{detG}), is equivalent to $\det G\geq 0$, and UR (\ref{SUR}) is
equivalent to $\det (\sigma+{\ri}C) \geq 0$.

\noi  The two Hermitian operators on the sphere $X_1=\vphi$ and
$X_2=-{\ri}\pd/\pd\vphi\equiv \hat{p}_\vphi$ constitute an example in
which of $X_2X_1\psi(\vphi,\vthe)$ is not properly defined:
$\hat{p}_\vphi$ is not Hermitian on states $\vphi\psi(\vphi,\vthe)$, since
$\psi^\prime = \vphi\psi(\vphi,\vthe)$ is not $2\pi$-periodic. Therefore
the $\vphi$--$p_\vphi$ UR should resort to eq. (\ref{gSUR}). However even
the generalized covariance of $\vphi$, $\hat{p}_\vphi$ and the generalized
variance of $\vphi$ depend on the limits of integration when calculating
means like 
$$\la\vphi\ra = \int_{\vphi_0 - \pi}^{\vphi_0+\pi}\vphi |\psi|^2 {\rd} S.$$ 
Fortunately, the variance of $\vphi$ is $2\pi$-periodic function of
$\vphi_0$, therefore its global extrema exist and we may define
$\vphi$-uncertainty measure on the sphere $_c\Dlt\vphi$ as explained in
the previous section.

\noi 
Then the $\vphi$--$p_\vphi$ UR on the sphere could be adopted in the form
\beq\lb{phi-p UR}
(_c\Dlt\vphi)^2\,(\Dlt p_\vphi)^2 \geq
\left| _c\la (\vphi-\la \vphi\ra)\psi|(\hat{p}_\vphi-\la \hat{p}_\vphi\ra)
\psi\ra\right|^2
\eeq
where $_c\la\,X\,\ra$ means that the average  of $X$ is calculated by
integration from $\vphi_c-\pi$ to $\vphi_c+\pi$, $\vphi_c$ being the wave
packet center.   Note that the right hand side of (\ref{phi-p UR}) may
vanish (on the eigenstates of $p_\vphi$ for example), so that the less
precise version of the inequality (\ref{phi-p UR}) is
$(_c\Dlt\vphi)^2\,(\Dlt p_\vphi)^2 \geq 0 $.  The variance $\Dlt
p_\vphi$ should be called {\it complementary measure} to the position
delocalization measure $_c\Dlt\vphi$.

\noi A natural definition of a {\it complementary measure} is the following: 
a state measure $M(\psi)$ is a complementary one to a state measure $N(\psi)$ 
if $M(\psi)$ tends to its global maximum (minimum) when $N(\psi)$ tends to
its global minimum (maximum). The variance of $\hat{p}_\vphi =
-{\ri}\pd/\pd\vphi$ is a complementary one to the delocalization measure
$(_c\Dlt\vphi)^2$. For a particle on the real line the variances of the
coordinate $x$ and momentum $\hat{p} = -{\ri} {\rd}/{\rd}x$ are complementary 
measures.
It is clear that a given measure $N(\psi)$ may have many complementary
measures.  Additional criteria should be used to specify the most
convenient complementary measure is every special case. Let us also note,
that measures  map states on the positive part on the real line, i.e.
these are many-to-one maps. Therefore they may reach their extremal values
on a large subset of states.  

\noi  Our aim now is to construct measure complementary to the well defined
$\vthe$-uncertainty measure (the variance) $(\Dlt\vthe)^2$ on the sphere.
The problem with such complementary measure, and the $\vthe$--$p_\vthe$
UR as well, is in the ill property of the operator $-{\ri}\pd/\pd\vthe\equiv
\tilde{p}_\vthe$: this operator obey formally the relation
$[\vthe,\tilde{p}_\vthe]=\ri$, however it is not Hermitian. Therefore the
variance $\la \tilde{p}_\vthe^2\ra-\la \tilde{p}_\vthe\ra^2$ may be complex.

\noi  It is easy to point out Hermitian operator $\hat{p}_{0 \vthe}$ with the 
same commutator as for $\vthe$ and $\tilde{p}_\vthe$. Such is the operator
$\hat{p}_{0 \vthe} = -\ri\pd/\pd\vthe -(\ri/2)\cot(\vthe)$. One has
$[\vphi,\hat{p}_{0 \vthe}] =\ri$. Then we can write the standard
Schr\"odinger UR $(\Dlt \vthe)^2\,(\Dlt p_{0 \vthe})^2 \geq 1/4 +({\rm
Cov}(p_{0 \vthe},\vthe))^2$.

\noi However the variance $(\Dlt \hat{p}_{0 \vthe})^2$ could hardly serve
as a {\it complementary measure} to the position measure $(\Dlt\vthe)^2$
since it is diverged in all states that are not vanishing at $\vthe=0$,
and $\vthe=\pi$.  Examples of such states are $f_{uv\gam k}(\vphi,\vthe)$
and CS $\psi_{uv\tau} (\vphi,\vthe)$, treated in section 3. This ill
property of $\hat{p}_{0 \vthe}$ stems from the fact that the functions
$\hat{p}_{0 \vthe} \psi(\vphi,\vthe)$ are not normalizable (where
$\psi(\vphi,\vthe)$ represent normalized state).

\noi  Fortunately, there are simple Hermitian operators the variances of
which could be regarded as complementary to $(\Dlt\vthe)^2$ measures.
These are the first order differential operators $\hat{p}_{n \vthe}$ of the 
form
\beq\lb{p_n}
\hat{p}_{n \vthe} = -{\ri}\sin^{n}\vthe\frac{\pd}{\pd\vthe} - 
{\ri}\frac{n+1}{2} \cos\vthe \sin^{n-1}\vthe,\quad n=1,2,\ldots 
\eeq
We have $[\vthe,\hat{p}_{n \vthe}] = \ri\sin^n\vthe$, and the transformed 
states $\hat{p}_{n \vthe}\psi(\vphi,\vthe)$ are normalizable for all $n\geq 1$.
The variances and covariances of $\vthe$ and $\hat{p}_{n \vthe}$ satisfy the
Schr\"odinger UR (\ref{SUR}). In view of $\sin\vthe\geq 0$ in the
interval $(0,\pi)$ we have $\la \sin^n\vthe\ra > 0$, therefore the right
hand side of Schr\"odinger UR never vanishes, i.e. $(\Dlt\vthe)^2(\Dlt
p_{n \vthe})^2 > 0$. This is a proof that the spectrum of operators
$\hat{p}_{n \vthe}$ is not discrete. 

\noi 
Finally we have to point out which (from all $(\Dlt p_{n \vthe})^2$) is
the {\it best} complementary measure to the position uncertainty measure
$(\Dlt\vthe)^2$.  We have to apply some criterions. One such criterion
could be the demand that the complementary measure $(\Dlt p_{n \vthe})^2$ 
be minimal (with respect to $n$) in the most delocalized state
$\psi_{\alf=0}(\vphi,\vthe)=1/(\pi\sqrt{2\sin\vthe})\equiv \psi_0(\vthe)$. 
Numerical calculations show that this criterion selects 
$\Dlt p_{1 \vthe}$ and $\Dlt p_{2 \vthe}$:

In $\psi_0(\vthe)$ we find 
$(\Dlt p_{n>2,\vthe})^2 > (\Dlt p_{2 \vthe})^2 \simeq (\Dlt p_{1 \vthe})^2 
\simeq 0.125$. 
Note that the variances of $\hat{p}_{n\vthe}$ on $\psi_\alf(\vphi,\vthe)$
do depend on the phase $\alf$ when the latter is a function of angle $\vthe$.

\noi Another natural criterion is the lower limit ot the product $(\Dlt
p_{n\vthe})^2 (\Dlt \vthe)^2$, for which one has the standard UR 
\beq
(\Dlt p_{n\vthe})^2 (\Dlt \vthe)^2 \geq |\la \sin^n\vthe\ra|^2/4
\eeq
It is clear that in any state the inequality $|\la \sin^n\vthe\ra| > |\la
\sin\vthe\ra|$ holds. So the second criteria picks up from the set $\{(\Dlt
p_{n\vthe})^2: n=1,2,\ldots\}$ the variance $(\Dlt p_{1 \vthe})^2$ as
the {\it best complementary measure} to the position delocalization
measure $(\Dlt \vthe)^2$.  In analogy with $(\Dlt\vthe)^2$ we may denote
this $\vthe$-complementary measure as $(\Dlt p_\vthe)^2$, i.e. we put
$(\Dlt p_{1 \vthe})^2 \equiv (\Dlt p_\vthe)^2$.  In the two states
$f_{uv\gam k}(\vphi,\vthe)$ and two CS $\psi_{uv\tau}(\vphi,\vthe)$
represented in figures 2, 3 and 4, 5 the values of $(\Dlt p_{1 \vthe})^2$
are $0.57, 1.54$ and $0.419, 1.38$ respectively.  It is remarkable that in
CS $\psi_{\pi\pi/2\,\tau}$ with $\tau=1$ the $\vthe$-position uncertainty
is approximately equal to the complementary one:  $(\Dlt\vthe)^2 = 0.419=
(\Dlt p_{1 \vthe})^2$. This is to be compared with the case of CS on the
plane, where the two complementary  uncertainties (position and momentum
uncertainties) are equal (in any CS however).
   
\noi Thus the Hermitian operator $\hat{p}_{1 \vthe}$ could be examined as 
a momentum $\hat{p}_\vthe$ complementary to the variable $\vthe$. The four
measures on the sphere $(_c\Dlt\vphi)^2$, $(\Dlt\vthe)^2$, $(\Dlt
p_\vphi)^2$, and $(\Dlt p_\vthe)^2$ satisfy the generalized Robertson UR
(\ref{detG}), where the integration with respect to $\vthe$ in all matrix
elements involving the variable $\vphi$ should be from $\vphi_c-\pi$ to
$\vphi_c+\pi$.

\noi  Let us note that the means $\la \psi_0|\hat{p}_i|\psi_0\ra$, where
$\hat{p}_i$ are the two Hermitian operators \ci{GP}, "conjugated" to the
stereographic coordinates $q_i$ (\ref{q_i}), are divergent.  Thus these
operators move the most delocalized state $\psi_0(\vthe)$ (and many other
states as well) away from the appropriate Hilbert space, and their
variances are not convenient as complementary to the position uncertainty
measures on the sphere.

\section*{{\large\bf 5. Conclusion}}

We have constructed two position uncertainty measures $(_c\Dlt\vphi)$,
$(\Dlt\vthe)^2$ and two related complementary measures $(\Dlt p_\vphi)^2$,
$(\Dlt p_\vthe)^2$ for a particle on the sphere. The $\vthe$-complementary
measure $(\Dlt p_\vthe)^2$ is a variance of the new operator
$\hat{p}_{1 \vthe}$, eq.(\ref{p_n}). 

\noi The four measures obey the genralized Robertson UR (\ref{detG}), any
two of them satisfying the Schr\"odin\-ger-Robertson type UR (\ref{gSUR}).
The relevance of the constructed measures are illustrated on the example
of two sets of states: $f_{uv\gam k}(\vphi,\vthe)$, eq. (\ref{fuvk}), and
coherent states on the sphere \ci{HM} $\psi_{uv\tau}(\vphi,\vthe)$, eq.
(\ref{psiCS}).  The relevance of  $(_c\Dlt\vphi)^2$ as a position measure
should not be considered as a proof that the right position operator
$\hat{\vphi}$ for the azimuthal angle is the multiplication by $\vphi$.

\noi  The presented approach to uncertainty measures on ${\mathbb S}^2$
could be easily extended to higher dimensional spheres.
\vspace{5mm}


\begin{thebibliography}{99}
%\begin{enumerate}
\def\bi{\bibitem}

\bi{CN68} Carruthers P. and Nieto M., {\it Phase and Angle Variables in
    Quantum Mechanics}, Rev. Mod. Phys. {\bf 40} (1968) 411--440.

\bi{ChD}  Chisolm E., 
 {\it Generalizing the Heisenberg Uncertainty Relation},
 Am. J. Phys. {\bf69} (2001) 368--371\, [quant-ph/0011115]\,;
 Dumitru S., 
 {\it On a Subject of Diverse Improvisations: The Uncertainty Relations
 on a Circle},  Preprint quant-ph/0206009.

\bi{FA01} Forbes G. and  Alonso M., {\it Measures of Spread
     for Periodic Distributions and the Associated Uncertainty Relations},
     Am. J. Phys. {\bf 69} (2001) 340--347.

\bi{GP}  Golovnev A., Prokhorov L., {\it Uncertainty Relations in
Curved Spaces}, J. Phys. A 37 (2004) 2765--2775\, [quant-ph/0306080].

\bi{GO98} Gonz\'ales J. and del Olmo M., 
 {\it Coherent States on the Circle}, J. Phys. A {\bf31} (1998)
8841--8857 [quant-ph/9809020].

\bi{HM} Hall B.  and Mitchell J., {\it Coherent States on Spheres},
J. Math. Phys. {\bf43} (2002) 1211--1236 [quant-ph/0109086].

\bi{Kast} Kastrup H., {\it Quantization of the Optical Phase Space $S^2
= \{\phi\,\, {\rm mod}\,\, 2\pi, I > 0\}$ in Terms of the Group $SO(1,2)$},
Fortsch. Phys. {\bf 51} (2003) 975--1134 [quant-ph/0307069]. 

% \bi{H1} B. Hall, J. Funct. Anal. \textbf{122}, 103 (1994). {\fns
% Additional results may be found in B. Hall, Comm. Math. Phys.
% \textbf{184}, 233 (1997); J. Funct. Anal. \textbf{143}, 98 (1997; and
% in the survey paper  Bull. (N.S.) Amer. Math. Soc. \textbf{38}, 43 (2001)}.

\bi{KRP96} Kowalski K., Rembieli\'nski J. and Papaloucas L.,
 {\it Coherent States for a Quantum Particle on a Circle},
  J. Phys. A  {\bf 29} (1996) 4149--4167 [quant-ph/9801029].
	 
\bi{KR02} Kowalski K. and Rembieli\'nski J., 
{\it On the Uncertainty Relations and Squeezed States for the Quantum
Mechanics on a Circle},  J. Phys. A {\bf35} (2002) 1405--1414 
[quant-ph/0202070].

\bi{KR00} Kowalski K. and Rembieli\'{n}ski J., {\it Quantum Mechanics on 
a Sphere and Coherent States}, J. Phys. A \textbf{33} (2000) 6035--6048;\,
Kowalski K. and Rembieli\'{n}ski J., {\it The Bargmann
Representation for the Quantum Mechanics on a Sphere}, J. Math. Phys.
\textbf{42} (2001) 4138--4147 [quant-ph/0011070].

\bi{R34} Robertson H., {\it An Indeterminacy Relation  for Several
         Observables and its Classical Interpretation}, Phys. Rev. {\bf 46} 
         (1934) 794--801.

\bi{SR30} Schr\"odinger E.,  
 {\it Zum Heisenbergschen Unsch\"arfeprinzip},
  Sitz. Preus. Acad. Wiss. (Phys.-Math. Klasse) {\bf19} (1930) 296--303;\,
 Robertson H.,
 {\it A General Formulation of the Uncertainty Principle and its Classical 
 Interpretation},
  Phys. Rev. {\bf35} (1930) 667--667 (abstract only).

\bi{T03} Trifonov D., {\it On the Position Uncertainty Measure on
the Circle},  J. Phys. A {\bf 36} (2003) 11873--11879\, [quant-ph/0307137];\,
 Trifonov D., {\it Comment on ``On the Uncertainty
Relations and Squeezed States for the Quantum Mechanics on a Circle''},
J. Phys. A  {\bf 36} (2003) 2197-2202 (Erratum: ibid. p. 5359) 
[quant-ph/0209032].

\bi{T00} Trifonov D.,  {\it State Extended Uncertainty Relations},
 J. Phys. A  {\bf33} (2000) L299--L304\, [quant-ph/0005086];\, 
 Trifonov D.,
 {\it Remarks on the Extended Characteristic Uncertainty Relations},
  J. Phys. A  {\bf 34} (2001) L75--L78 [quant-ph/0012031].

%%\end{enumerate}
\end{thebibliography}

\end{document}

quant-ph0404087.extracted_bib
\begin{thebibliography}{99}
%\begin{enumerate}
\def\bi{\bibitem}

\bi{CN68} Carruthers P. and Nieto M., {\it Phase and Angle Variables in
    Quantum Mechanics}, Rev. Mod. Phys. {\bf 40} (1968) 411--440.

\bi{ChD}  Chisolm E., 
 {\it Generalizing the Heisenberg Uncertainty Relation},
 Am. J. Phys. {\bf69} (2001) 368--371\, [quant-ph/0011115]\,;
 Dumitru S., 
 {\it On a Subject of Diverse Improvisations: The Uncertainty Relations
 on a Circle},  Preprint quant-ph/0206009.

\bi{FA01} Forbes G. and  Alonso M., {\it Measures of Spread
     for Periodic Distributions and the Associated Uncertainty Relations},
     Am. J. Phys. {\bf 69} (2001) 340--347.

\bi{GP}  Golovnev A., Prokhorov L., {\it Uncertainty Relations in
Curved Spaces}, J. Phys. A 37 (2004) 2765--2775\, [quant-ph/0306080].

\bi{GO98} Gonz\'ales J. and del Olmo M., 
 {\it Coherent States on the Circle}, J. Phys. A {\bf31} (1998)
8841--8857 [quant-ph/9809020].

\bi{HM} Hall B.  and Mitchell J., {\it Coherent States on Spheres},
J. Math. Phys. {\bf43} (2002) 1211--1236 [quant-ph/0109086].

\bi{Kast} Kastrup H., {\it Quantization of the Optical Phase Space $S^2
= \{\phi\,\, {\rm mod}\,\, 2\pi, I > 0\}$ in Terms of the Group $SO(1,2)$},
Fortsch. Phys. {\bf 51} (2003) 975--1134 [quant-ph/0307069]. 

% \bi{H1} B. Hall, J. Funct. Anal. \textbf{122}, 103 (1994). {\fns
% Additional results may be found in B. Hall, Comm. Math. Phys.
% \textbf{184}, 233 (1997); J. Funct. Anal. \textbf{143}, 98 (1997; and
% in the survey paper  Bull. (N.S.) Amer. Math. Soc. \textbf{38}, 43 (2001)}.

\bi{KRP96} Kowalski K., Rembieli\'nski J. and Papaloucas L.,
 {\it Coherent States for a Quantum Particle on a Circle},
  J. Phys. A  {\bf 29} (1996) 4149--4167 [quant-ph/9801029].
	 
\bi{KR02} Kowalski K. and Rembieli\'nski J., 
{\it On the Uncertainty Relations and Squeezed States for the Quantum
Mechanics on a Circle},  J. Phys. A {\bf35} (2002) 1405--1414 
[quant-ph/0202070].

\bi{KR00} Kowalski K. and Rembieli\'{n}ski J., {\it Quantum Mechanics on 
a Sphere and Coherent States}, J. Phys. A \textbf{33} (2000) 6035--6048;\,
Kowalski K. and Rembieli\'{n}ski J., {\it The Bargmann
Representation for the Quantum Mechanics on a Sphere}, J. Math. Phys.
\textbf{42} (2001) 4138--4147 [quant-ph/0011070].

\bi{R34} Robertson H., {\it An Indeterminacy Relation  for Several
         Observables and its Classical Interpretation}, Phys. Rev. {\bf 46} 
         (1934) 794--801.

\bi{SR30} Schr\"odinger E.,  
 {\it Zum Heisenbergschen Unsch\"arfeprinzip},
  Sitz. Preus. Acad. Wiss. (Phys.-Math. Klasse) {\bf19} (1930) 296--303;\,
 Robertson H.,
 {\it A General Formulation of the Uncertainty Principle and its Classical 
 Interpretation},
  Phys. Rev. {\bf35} (1930) 667--667 (abstract only).

\bi{T03} Trifonov D., {\it On the Position Uncertainty Measure on
the Circle},  J. Phys. A {\bf 36} (2003) 11873--11879\, [quant-ph/0307137];\,
 Trifonov D., {\it Comment on ``On the Uncertainty
Relations and Squeezed States for the Quantum Mechanics on a Circle''},
J. Phys. A  {\bf 36} (2003) 2197-2202 (Erratum: ibid. p. 5359) 
[quant-ph/0209032].

\bi{T00} Trifonov D.,  {\it State Extended Uncertainty Relations},
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