On Quantum States for Angular Position and Angular Momentum of Light

Bo-Sture K. Skagerstam111Corresponding author,222Email address: bo-
<EMAIL_ADDRESS>
Department of Physics
Norwegian University of Science and Technology - NTNU
N-7491 Trondheim, Norway

and

Per K. Rekdal333Email address<EMAIL_ADDRESS>
Molde University College
PO Box 2110, N-6402 Molde, Norway

###### Abstract

In the present paper we construct a properly defined quantum state expressed
in terms of elliptic Jacobi theta functions for the self-adjoint observables
angular position $\theta$ and the corresponding angular momentum operator
$L=-id/d\theta$ in units of $\hbar=1$. The quantum uncertainties
$\Delta\theta$ and $\Delta L$ for the state are well-defined and are, e.g.,
shown to give a lower value of the uncertainty product $\Delta\theta\Delta L$
than the minimal uncertainty states of Ref.[1]. The mean value $\langle
L\rangle$ of the state is not required to be an integer. In the case of any
half-integer mean value $\langle L\rangle$ the state constructed exhibits a
remarkable critical behaviour with upper and lower bounds
$\Delta\theta<\sqrt{\pi^{2}/3-2}$ and $\Delta L>1/2$.

Motivated by the ingenious experimental procedures to generate restricted
values of angular position and angular momentum degrees of freedom (see, e.g.,
Refs.[1, 2, 3, 4]), we consider a description of such limitations in terms of
a well-defined pure quantum state. Restrictions of the angular phase of degree
of freedom must be accompanied by restrictions on angular momentum degrees of
freedom of, e.g., light pulses (see Ref.[5] for a general presentation and
Refs.[6] for a limited selected set of original papers on the notion of
angular momentum for light pulses). Experimental and theoretical explorations
have been extended to the observations of fractional angular momentum as well
as to the use of single photon sources (see, e.g., Refs.[7, 8, 9, 10, 11, 12,
13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]). Here we, in particular, notice
that the presence half-integer angular momentum appears to play a very special
role as emphasised in, e.g., Ref.[15]. The quantum state as discussed in the
present paper turns also out to lead to unique signatures for half-integer
angular momentum mean values.

In order to describe photon sources and the detection of angular momentum
degrees of freedom it is of importance to carry out a detailed second-
quantization of the electromagnetic field making use of appropriate normal
mode functions. This has been investigated in great detail in the literature
in various contexts requiring considerable efforts (see, e.g., Refs.[24, 25]
and references cited therein). At the single-photon level one predicts
fractional angular momentum in a reduced physical dimension (see, e.g.,
Refs.[15, 17] and references cited therein). In the present paper we limit
ourselves to a transverse orbital angular momentum degree of freedom for mode
functions in a reduced physical propagation dimension and a construction of a
possible well-defined quantum state according to the basic rules of quantum
mechanics. Elsewhere related charged quantum states have previously been
discussed [26] in the context of quantum states for charged q-bits. The
results presented here can be applied to such systems as well.

We consider the following periodic extension of the minimal dispersion
intelligent pure states as discussed in Ref.[1], later revisited in Ref.[2],
in terms of a parametric representation with a real-valued parameter
$\lambda>0$, i.e.,

$\displaystyle\psi(\theta)=N\sum_{n=-\infty}^{\infty}f(\theta-\bar{\theta}+2\pi
n)~{},$ (1)

and where

$\displaystyle f(x)=e^{\displaystyle{ix\bar{l}}}e^{\displaystyle{-\lambda
x^{2}/2}}~{}.$ (2)

The range of $\theta$ is restricted in terms of an in principle arbitrary off-
set value $\theta_{0}$ for the phase $\theta$ such that
$\theta_{0}\leq\theta\leq\theta_{0}+2\pi$. Apart from the
$\bar{\theta}$-dependence, the $n=0$ contribution in Eq.(1) corresponds to the
minimal dispersion state as first discussed in Ref.[1]. As in this reference
we find it convenient to make use of the choice $\theta_{0}=-\pi$ since this
naturally leads to $\langle\theta\rangle=0$ when $\bar{\theta}=0$. Here $N$ is
a real-valued normalization constant for the state $\psi(\theta)$ and
$\overline{l}$ an additional real-valued parameter all of which will be
discussed in more detail later on.

In passing we notice that a periodic extension of the form as in Eq.(1) has a
resemblance to the quantum states considered by M. Born [27] in a discussion
of the notion of probabilities in classical and quantum physics and the role
of real numbers in physics. We will comment on these fundamental issues below
where we, in particular, exhibit an exponential sensitivity to the difference
between rational and real numbers for the observables under consideration.

The state $\psi(\theta)$ can now be expressed in terms of a well-known
elliptic $\vartheta_{3}$-function (see, e.g., Refs.[28]), i.e.,

$\displaystyle\psi(\theta)=Ne^{\displaystyle{i\overline{l}(\theta-\bar{\theta})}}e^{-\displaystyle{\lambda(\theta-\bar{\theta})^{2}/2}}\vartheta_{3}[\pi\left(\bar{l}+i\lambda(\theta-\bar{\theta})\right),e^{\displaystyle{-2\lambda\pi^{2}}}]~{},$
(3)

using the definition

$\vartheta_{3}[z,q]=\sum_{n=-\infty}^{\infty}q^{\displaystyle{n^{2}}}e^{\displaystyle{2niz}}~{},$
(4)

within the unit disk $|q|<1$. The state $\psi(\theta)$ has well-defined
properties, i.e., it is continuous and differentiable for all values
$\theta\in[-\pi,\pi]$, including the boundary $\theta=\pm\pi$ in contrast to
the state considered in Ref.[1]. One can actually be more precise in a
mathematical sense and show that $\psi(\theta)$ belongs to states for which
$\theta$ and $L=-id/d\theta$ are self-adjoint operators (see, e.g., Example 1
in Ref.[29] and related discussions in Refs.[30, 31, 32, 33]) as required by
the fundamental rules of quantum mechanics. The representation of
$\psi(\theta)$ in the form of Eq.(3) is suitable for considerations of large
values of the parameter $\lambda$.

Figure 1: Uncertainty products $\Delta\theta\Delta L$ as a function of the
phase uncertainty $\Delta\theta$ all with the parameters $\bar{\theta}=0$ such
that $\langle\theta\rangle=0$. The solid curve is the properly renormalized
$n=0$ contribution in Eq.(1) with $\bar{l}=0$ corresponding to the state of
Ref.[1]. The dotted curve corresponds to all terms in Eq.(1) with $\bar{l}=0$.
The lower thin line corresponds to equality in the $\theta$ and $L$ general
uncertainty relation Eq.(18) with $\bar{l}=0$. As for a comparison the upper
thin line corresponds to Eq.(1) with $\bar{l}=l+0.45$ for any integer $l\geq
0$.

An alternative and explicit expression for $\psi(\theta)$ according to Eq.(3)
which is more adapted for a small $\lambda$ expansion can now be obtained by
making use of general properties of $\vartheta_{3}$-functions under modular
transformations (see, e.g., Refs.[28]). For our purposes, and for the
convenience of the reader, we make this explicit in terms of a Poisson
summation technique (see, e.g., Refs.[34, 35]) by noticing that the state
$\psi(\theta)$ according to Eq.(1) is a periodic function. It can then be
expanded in a Fourier series

$\displaystyle\psi(\theta)=\sum_{n=-\infty}^{\infty}c(n)e^{\displaystyle{in\theta}}~{},$
(5)

such that

$\displaystyle c(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi}d\theta
e^{-\displaystyle{in\theta}}\psi(\theta)=\frac{N}{2\pi}\int_{-\infty}^{\infty}d\theta
e^{\displaystyle{-in\theta}}f(\theta-\bar{\theta})=\frac{N}{\sqrt{2\pi\lambda}}e^{-\displaystyle{in\bar{\theta}}}e^{-\displaystyle{(n-\bar{l})^{2}/2\lambda}}~{}.$
(6)

Figure 2: As in Fig.1, the mean values $\langle L\rangle$ and the uncertainty
$\Delta L$ as a function of the uncertainty $\Delta\theta$ with
$\bar{\theta}=0$ for various values of the parameter $\bar{l}$. The horizontal
solid curve corresponds to a half-integer value $\langle L\rangle=3/2$ which
ends asymptotically at the exact critical value
$\Delta\theta<\sqrt{\pi^{2}/3-2}$ and with the exact lower bound $\Delta
L>1/2$. The uncertainties $\Delta\theta$ and $\Delta L$ are actually is
independent of the parameters $\bar{\theta}$ and $\bar{l}=l+1/2$ for any
positive integer $l$. The thin lines for $\langle L\rangle$, are here, and for
reasons of simplicity, illustrated with $l=1$, corresponding in general to the
values of $\bar{l}=l+\epsilon$ with $1/2<\epsilon<1$ or $0<\epsilon<1/2$,
respectively.

It then follows that

$\displaystyle\psi(\theta)=\frac{N}{\sqrt{2\pi\lambda}}\sum_{n=-\infty}^{\infty}e^{\displaystyle{in(\theta-\bar{\theta})}}e^{-\displaystyle{(n-\bar{l})^{2}/2\lambda}}\,\,,$
(7)

which also can be expressed in terms of an elliptic $\vartheta_{3}$-function,
i.e.,

$\displaystyle\psi(\theta)=\frac{N}{\sqrt{2\pi\lambda}}e^{\displaystyle{-\overline{l}^{2}/2\lambda}}\vartheta_{3}[(\theta-\bar{\theta})/2-i\overline{l}/2\lambda,e^{\displaystyle{-1/2\lambda}}]~{},$
(8)

which is rapidly converging for small values of the parameter $\lambda>0$. The
explicit representation of $\psi(\theta)$ in the form as in Eq.(7) leads to
the general normalization condition

$\displaystyle(\psi,\psi)\equiv\int_{-\pi}^{\pi}d\theta|\psi(\theta)|^{2}=\frac{N^{2}}{\lambda}\sum_{n=-\infty}^{\infty}e^{-\displaystyle{(n-\bar{l})^{2}/\lambda}}=N^{2}\sqrt{\frac{\pi}{\lambda}}\vartheta_{3}[-\bar{l}\pi,e^{\displaystyle{-\lambda\pi^{2}}}]=1\,\,.$
(9)

The normalization factor $N$ is therefore in general only a function of the
parameters $\bar{l}$ and $\lambda$. With the state $\psi(\theta)$ in the
representation as in Eq.(7), we can make use of Borns rule to find the
properly normalized probability distribution $p(l)$ to find the integer
angular momentum $l$ component in terms of the angular momentum eigenfunction
$\psi_{l}(\theta)=\exp(il\theta)/\sqrt{2\pi}$, i.e.,

$\displaystyle
p(l)=|(\psi_{l},\psi)|^{2}=|\int_{-\pi}^{\pi}d\theta\psi_{l}^{*}(\theta)\psi(\theta)\,|^{2}=\frac{N^{2}}{\lambda}e^{\displaystyle{-(l-\overline{l})^{2}/\lambda}}\,\,,$
(10)

which, of course, can be used to compute various expectation values of the
observable $L$. The discrete nature of this Gaussian-like distribution is now
such that $\lambda$ is not directly related to the uncertainties
$\Delta\theta$ or $\Delta L$. Below we, e.g., show that $\lambda={\cal
O}(1/(\Delta\theta)^{2})$ for all values of $\bar{l}$ in the large $\lambda$
limit and $\lambda={\cal O}(1/\log(1/\Delta L))$ in the small $\lambda$ limit
at least for integer values of $\langle L\rangle$. We notice that Eq.(7), or
making use of the probability distribution $p(l)$ above, implies the general
and useful relations

$\displaystyle\langle
L\rangle=\frac{N^{2}}{\lambda}\sum_{n=-\infty}^{\infty}ne^{\displaystyle{-(n-\overline{l})^{2}/\lambda}}\,\,\,\,,\,\,\,\,\langle
L^{2}\rangle=\frac{N^{2}}{\lambda}\sum_{n=-\infty}^{\infty}n^{2}e^{\displaystyle{-(n-\overline{l})^{2}/\lambda}}\,\,,$
(11)

independent of $\bar{\theta}$, and which can be used to relate the parameter
$\lambda$ to the uncertainty $\Delta L$ at least numerically.

Furthermore, and with $\psi(\theta)$ in the form as given in Eq.(7), and for
any value of $\theta_{0}$ , we obtain the following general expression

$\displaystyle\langle\theta\rangle=\int_{\theta_{0}}^{\theta_{0}+2\pi}d\theta\theta|\psi(\theta)|^{2}$
$\displaystyle=\frac{N^{2}}{\lambda}\sum_{m\neq
n}\frac{\sin[(m-n)(\theta_{0}-\bar{\theta})]}{m-n}e^{\displaystyle{-\left((m-\overline{l})^{2}+(n-\overline{l})^{2}\right)/2\lambda}}+(\theta_{0}+\pi)~{}.$
(12)

For any value of $\bar{l}$ we observe that due to Eq.(12) the choice
$\theta_{0}=-\pi$ leads to $\langle\theta\rangle=0$ only if the parameter
$\bar{\theta}=0,\pm\pi$. A similar expression can be obtained for
$\langle\theta^{2}\rangle$ in a straightforward manner with the general result

$\displaystyle\langle\theta^{2}\rangle=2\frac{N^{2}}{\lambda}\sum_{m\neq
n}\frac{\cos[(m-n)(\bar{\theta}-\theta_{0})]}{(m-n)^{2}}e^{\displaystyle{-\left((m-\overline{l})^{2}+(n-\overline{l})^{2}\right)/2\lambda}}$
$\displaystyle+\,2\frac{N^{2}}{\lambda}(\theta_{0}+\pi)\sum_{m\neq
n}\frac{\sin[(m-n)(\theta_{0}-\bar{\theta})]}{(m-n)}e^{\displaystyle{-\left((m-\overline{l})^{2}+(n-\overline{l})^{2}\right)/2\lambda}}+\frac{(\theta_{0}+2\pi)^{3}-\theta_{0}^{3}}{6\pi}\,\,.$
(13)

With $\theta_{0}=-\pi$ we, in particular, have that

$\displaystyle\langle\theta^{2}\rangle=2\frac{N^{2}}{\lambda}\sum_{m\neq
n}\frac{\cos[(m-n)(\bar{\theta}-\pi)]}{(m-n)^{2}}e^{\displaystyle{-\left((m-\overline{l})^{2}+(n-\overline{l})^{2}\right)/2\lambda}}+\frac{\pi^{2}}{3}~{}.$
(14)

In the course of the present work we have found that less general expressions
than Eqs.(9) and (14) have been discussed in the literature in the context of
a minimum entrophic angular position and angular momentum quantum state [36].
There it was also argued that a proposed quantum state could be of use in
theoretical and/or experimental explorations of the notion of a phase
observable in quantum physics which we also presume can be the case for the
more general quantum state Eq.(3).

It now follows form Eqs.(12) and (13) that the implicit dependence of the
parameter $\lambda$ can in general be eliminated in terms of the dispersion
$\Delta\theta$ which, however, requires numerical but straightforward
considerations. If, finally, the parameter $\bar{l}$ is any integer Eq.(7)
simplifies to

$\displaystyle\psi(\theta)=\frac{N}{\sqrt{2\pi\lambda}}e^{\displaystyle{i\bar{l}(\theta-\bar{\theta})}}\vartheta_{3}[(\theta-\bar{\theta})/2,e^{\displaystyle{-1/2\lambda}}]~{},$
(15)

and the normalization factor $N$ is determined by the relation

$\displaystyle\frac{N^{2}}{\lambda}\vartheta_{3}[0,e^{\displaystyle{-1/\lambda}}]=1\,\,.$
(16)

For integer values of $\bar{l}$ it also follows from Eq.(11) that

$\displaystyle\langle L\rangle=\bar{l}\,\,\,\,,\,\,\,\,(\Delta
L)^{2}=\frac{N^{2}}{\lambda}\sum_{n=-\infty}^{\infty}n^{2}e^{\displaystyle{-n^{2}/\lambda}}\,\,.$
(17)

In the analysis of the physical properties of the state $\psi(\theta)$ we are
going to make use of the general form of the uncertainty relation for the
observables $\theta$ and $L$, i.e., the Cauchy-Schwarz inequality, which now
takes the following exact form

$\displaystyle(\Delta{\theta})^{2}(\Delta{L}^{2})\geq|((\theta-\langle\theta\rangle)\psi,(L-\langle
L\rangle)\psi)|^{2}$
$\displaystyle=\frac{1}{4}|({\theta}\psi,{L}\psi)-({L}\psi,{\theta}\psi)|^{2}+\frac{1}{4}|({\theta}\psi,{L}\psi)+({L}\psi,{\theta}\psi)-2\langle\theta\rangle\langle
L\rangle|^{2}~{}~{}~{},$ (18)

where $(\Delta{\cal O})^{2}=(({\cal O}-\langle{\cal O}\rangle)\psi,({\cal
O}-\langle{\cal O}\rangle)\psi)$, and where the first term leads to the Kraus
lower inequality bound [31] for the uncertainty product, i.e.,
$\Delta{\theta}\Delta{L}\geq|1-2\pi|\psi(\pi)|^{2}|/2$ after a partial
integration. If we, as used in Fig.1, consider the special case with
$\bar{\theta}=0$, i.e., $\langle\theta\rangle=0$, we observe that
$({\theta}\psi,{L}\psi)$ is purely imaginary for the state Eq.(3). This is so
since, as can be verified using Eq.(7), that indeed

$\displaystyle({\theta}\psi,{L}\psi)=-i\frac{N^{2}}{2\lambda}\sum_{m\neq
n}\cos[(m-n)\pi])e^{\displaystyle{-\left((m-\overline{l})^{2}+(n-\overline{l})^{2}\right)/2\lambda}}$
$\displaystyle=\frac{i}{2}(1-2\pi|\psi(\pi)|^{2})~{}~{}~{},$ (19)

for all values of $\bar{l}$ in accordance with the general uncertainty
relation Eq.(18). Therefore a strict minimal uncertainty state must satisfy
the condition $\Delta{\theta}\Delta{L}=|1-2\pi|\psi(\pi)|^{2}|/2.$ With the
parameters as used in Fig.1 it then follows that the state in Eq.(3) is not a
minimal uncertainty state. Nevertheless, as is also illustrated in Fig.1, we
find for the same state lower values of the uncertainty product
$\Delta\theta\Delta L$ as compared to the result of Ref.[1].

For large values of the parameter $\lambda$ the state $\psi(\theta)$ according
to Eq.(1) is such that the mean value of the phase $\theta$ is close to
$\bar{\theta}$ due to the $n=0$ contribution. The state $\psi(\theta)$ can
then be approximated by the properly normalized expression

$\displaystyle\psi(\theta)=\left(\frac{1}{2\pi(\Delta\theta)^{2}}\right)^{1/4}\exp\left(i\bar{l}(\theta-\bar{\theta})\right)\exp\left(-(\theta-\bar{\theta})^{2}/4(\Delta\theta)^{2}\right)\,\,,$
(20)

where the range of $\theta$ can be extended to all real numbers with an
exponentially small error and using the identification
$\lambda=1/2(\Delta\theta)^{2}$. It then follows from Eq.(20) that $\langle
L\rangle=\bar{l}$, $\langle\theta\rangle=\bar{\theta}$, and
$\Delta\theta\Delta L=1/2$ as well as
$((\theta-\langle\theta\rangle)\psi,(L-\langle L\rangle)\psi)=i/2$ is purely
imaginary. The asymptotic expression Eq.(20) therefore corresponds to a strict
minimal dispersion state with equality in the uncertainty relation Eq.(18)
using the approximation $\psi(\pm\pi)\rightarrow\psi(\pm\infty)=0$. As
illustrated in Fig.1, it turns out that the approximation Eq.(20) also well
describes precise numerical considerations for a wide range of finite values
of $\lambda$.

For sufficiently small values of $\lambda$ we proceed without loss of
generality as follows. In terms of $\bar{l}=l+\epsilon$ with $1/2<\epsilon<1$
or $0<\epsilon<1/2$, respectively, we can make use of the small $\lambda$
expansions of the exact expressions Eqs.(9) and (11) in order to obtain the
corresponding expressions for $\langle L\rangle$ and $\Delta L$, i.e.,

$\displaystyle\langle
L\rangle=l+\frac{1}{1+e^{\displaystyle{(1-2\epsilon)/\lambda}}}\,\,\,\,,\,\,\,\,(\Delta
L)^{2}=\frac{e^{\displaystyle{(1-2\epsilon)/\lambda}}}{(1+e^{\displaystyle{(1-2\epsilon)/\lambda}})^{\displaystyle
2}}\,\,,$ (21)

independent of the parameter $\bar{\theta}$. Therefore the mean value $\langle
L\rangle$ in the small $\lambda$ limit approach the integer values $l+1$ or
$l$ depending on $1/2<\epsilon<1$ or $0<\epsilon<1/2$, respectively, with
limiting value $\Delta L=0$ as it should. This branching feature of the state
$\psi(\theta)$ is illustrated in Fig.2 for the case $l=1$. For integer values
of $\langle L\rangle$, corresponding to the limiting values $\epsilon=0$ or
$\epsilon=1$, we obtain from Eq.(21) that $\lambda=1/\log(1/(\Delta L)^{2})$
in the small $\lambda$ limit.

By inspection of Eq.(7) it, furthermore, follows that in the small $\lambda$
limit the quantum state $\psi(\theta)$ takes the following form

$\displaystyle\psi(\theta)=\frac{1}{\sqrt{2\pi}}\frac{1+e^{\displaystyle{i(\theta-\bar{\theta})}}e^{\displaystyle{-(1-2\epsilon)/2\lambda}}}{(1+e^{\displaystyle{-(1-2\epsilon)/\lambda}})^{\displaystyle
1/2}}e^{\displaystyle{il(\theta-\bar{\theta})}}\,\,.$ (22)

In this limit we therefore obtain the expected $l+1$ or $l$ angular momentum
eigenstates

$\displaystyle\psi(\theta)=\frac{1}{\sqrt{2\pi}}e^{\displaystyle{i(l+1)(\theta-\bar{\theta})}}\,\,\,\,,\,\,\,\,\mbox{or}\,\,\,\,,\,\,\,\,\frac{1}{\sqrt{2\pi}}e^{\displaystyle{il(\theta-\bar{\theta})}}\,\,,$
(23)

in the case $1/2<\epsilon<1$ or in the case $1/2<\epsilon<1$, respectively,
with $\Delta L=0$ and the limiting value of $\Delta\theta$ given by
$\pi/\sqrt{3}$.

It is now clear from Eqs.(21) and (22) with $\epsilon=1/2$ that exact half-
integer values of $\bar{l}=l+1/2$, where $l$ is a positive integer, play a
very special role, at least for sufficiently small $\lambda$. The asymptotic
form of $\psi(\theta)$ is then a superposition of the $l+1$ and $l$ angular
momentum eigenstates in Eq.(23), i.e.,

$\displaystyle\psi(\theta)=\frac{1}{\sqrt{4\pi}}\left(e^{\displaystyle{il(\theta-\bar{\theta})}}+e^{\displaystyle{i(l+1)(\theta-\bar{\theta})}}\right)\,\,.$
(24)

For sufficiently small values of $\lambda$, and according to Eq.(21) or using
Eq.(24), we then find that $\langle L\rangle=l+1/2$ and $\Delta L=1/2$ as well
as the limiting value $(\Delta\theta)^{2}=\pi^{2}/3-2$. Here we notice another
unique feature in the case of $\epsilon=1/2$ namely that $\langle
L\rangle=l+1/2$ is actually exact and valid for all values of $\lambda>0$.
This can be verified by making use of the exact form of $\langle L\rangle$
according to Eq.(11) and is due a simple but remarkable cancellation of
contributing terms and by making use of the exact form of the normalization
constant $N$ in Eq.(9).

For half-integer mean values of $\langle L\rangle$, and for
$\langle\theta\rangle=0$, it also follows that the lower limit in uncertainty
relation Eq.(18) is such that equality in $\Delta\theta\Delta L\geq 1/2$
implies a minimum uncertainty state. This is due to the fact that in this case
it can be verified $|(\psi(\theta),\psi(\theta))|=0$ for all values of
$\lambda$ if $\theta=\pi$. These remarkable features are illustrated in Figs.1
and 2 for various values of the parameter $\bar{l}$. One can now verify that
the state $\psi(\theta)$ with the parameters as in Fig.1 is not a minimum
uncertainty state.

It is of importance to notice that for any real value of $\bar{l}$ arbitrarily
close to a half-integer rational number we always obtain the branching to
angular momentum $l$ or $l+1$ eigenstates for sufficiently small values of
$\lambda$ as illustrated in Fig.2. The well-defined quantum state as
constructed in the present work therefore provides for an explicit example
with observables that exhibit exponential sensitivity to the difference
between rational numbers and real numbers in quantum mechanics. It would, of
course, be of great interest if the state $\psi(\theta)$ according to Eq.(3)
could be prepared and investigated in a concrete physical situation revealing
the fundamental role of real numbers in quantum physics [27] perhaps by
extending the experimental $\bar{l}=0$ procedure of Ref.[1].

In summary we have studied a well-defined quantum state $\psi(\theta)$ which
exhibits some remarkable properties. For $\langle L\rangle$ arbitrarily close
to a half-integer we have, e.g., shown that the state $\psi(\theta)$ leads to
two orthogonal eigenstate of $L$ in the limit as $\Delta\theta$ approaches the
conventional $\pi/\sqrt{3}$. In the case of an exact half-integer mean value
of the angular momentum $L$ the uncertainty $\Delta\theta$ has, furthermore,
an upper precise bound $\sqrt{\pi^{2}/3-2}$ which is smaller than conventional
bound $\pi/\sqrt{3}$ and $\Delta L\geq 1/2$. We have also exhibited an
explicit exponential sensitivity to the difference of rational and real
numbers in terms of an allowed and well-defined quantum state.

ACKNOWLEDGMENT

The research by B.-S. Skagerstam (B.-S. S.) has been supported in by NTNU and
Molde University College. B.-S.S. is grateful to J.R. Klauder for discussions
and for informing us about Ref.[33]. The research by P.K. Rekdal has been
supported by Molde University College.

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