# Novel Outlook on the Eigenvalue Problem for the Orbital Angular Momentum
Operator

G. S. Japaridze  Clark Atlanta University, Atlanta, GA 30314, USA  Institute
of High Energy Physics, Tbilisi State University, Tbilisi 0186, Georgia  A. A.
Khelashvili  Institute of High Energy Physics, Tbilisi State University,
Tbilisi 0186, Georgia  K. Sh. Turashvili  Institute of High Energy Physics,
Tbilisi State University, Tbilisi 0186, Georgia

###### Abstract

Based on the novel prescription for the power function, $(x+iy)^{m}$, a new
expression for $\Psi(x,y|m)$, the eigenfunction of the operator of the third
component of the angular momentum, $\hat{M}_{z}$, is presented. These
functions are normalizable, single valued and, distinct to the traditional
presentation, $(x+iy)^{m}=\rho^{m}e^{im\phi}$, are invariant under the
rotations at $2\pi$ for any, not necessarily integer, $m$ \- the eigenvalue of
$\hat{M}_{z}$. For any real $m$ the functions $\Psi(x,y|m)$ form an
orthonormal set, therefore they may serve as a quantum mechanical
eigenfunction of $\hat{M}_{z}$. The eigenfunctions and eigenvalues of the
angular momentum operator squared, $\hat{M}^{2}$, derived for the two
different prescriptions for the square root, $(m^{2})^{1/2}$,
$(m^{2})^{1/2}=|m|$ and $(m^{2})^{1/2}=\pm m$ are reported. The normalizable
eigenfunctions of $\hat{M}^{2}$ are presented in terms of hypergeometric
functions, admitting integer as well as non-integer eigenvalues. It is shown
that the purely integer spectrum is not the most general solution but is just
the artifact of a particular choice of the Legendre functions as the pair of
linearly independent solutions of the eigenvalue problem for the
$\hat{M}^{2}$.

##  I 1\. Introduction

It is commonly accepted that, from theoretical quantum mechanics, it follows
that the spectrum of the eigenvalues of the angular momentum operator is
discrete and is comprised of the integer values only; see, e.g., Bohm ; Schiff
; Fock ; LL .

Non-integer values of angular momentum do not contradict the principles of
quantum theory and were considered few times from different viewpoints. Back
in 1932, Majorana noted that in the framework of relativistic quantum
mechanics, the general formulation of a one-particle equation admits a
solution with an arbitrary angular momentum, a predecessor of the theory of
infinite dimensional representations of the Lorentz group M . Working on the
analytical properties of the scattering amplitude, Regge Regge  considered
angular momentum as a continuous complex variable, and derived the
singularities in the plane of the complex angular momentum that became
universally known as Regge poles. Götte et al. GG , exploiting the freedom in
fixing the orientation of phase discontinuity, introduced states with non-
integer angular momentum and applied formalism of the propagation of light
modes with the fractional angular momentum in the paraxial and non-paraxial
regime. Exploring polar solutions for the harmonic oscillator, Land Land
discovered that the Fock space equivalent to the Hilbert space wave functions,
found by solving the Schr̈odinger equation in spherical coordinates is
realized by acting with the creation and annihilation operators, allowing
states with both integer and non-integer angular momentum.

In JKT1 ; JKT2 , we argued that if only physical conditions are imposed, what
can be derived from the principles of quantum mechanics is that the spectrum
is discrete with the only condition that the difference $L-|m|$ is integer
while $L$ and $m$ could be integer as well as non-integer. Throughout,
$L(L+1)$ is the eigenvalue of the angular momentum operator squared and $m$ is
the eigenvalue of the operator of the third component of the angular momentum.

In this paper, a solution of the eigenvalue problem for the quantum-mechanical
orbital angular momentum (hereafter referred to as angular momentum) operator
is reported obtained when only the physical requirement is imposed on the
eigenfunction and is shown that in the framework of theoretical quantum
mechanics, the eigenfunctions with both integer and non integer eigenvalues
are allowed.

The paper is organized as follows. In Section 2, the multivaluedness and
periodicity of the eigenfunctions of the operator of the third component of
the angular momentum, $\hat{M}_{z}$, are discussed. A new prescription for the
power of a complex variable, differing from the Euler–de Moivre prescription,
$(x+iy)^{m}=\rho^{m}\,e^{im\phi}$, used in quantum mechanics, is presented.
Based on this prescription, the eigenfunction of $\hat{M}_{z}$ in terms of
Gauss’s hypergeometric series is given. This wave function is normalizable and
distinct from the traditional eigenfunction being proportional to
$e^{im\phi}$, is single valued and invariant under the rotations at $2\pi$ for
any, not necessarily integer $m$. In other words, the requirement of single-
valuedness of the wave function does not necessarily lead to the solution with
only integer $m$. This eigenfunction satisfies the physical requirement of
orthonormality, and, therefore, it can be considered as the wave function
describing the physical state with the eigenvalue $m$, being not necessarily
integer. In Section 3, it is discussed how the different prescriptions for the
power function alter the eigenfunctions and the spectrum of the angular
momentum operator squared $\hat{M}^{2}$. The eigenfunction of $\hat{M}^{2}$ is
found which is normalizable and satisfies physical requirements for an integer
as well as non-integer $L$; to a fixed value of $L$ corresponds a discrete
spectrum of $m$, defined by the relation $|m|=L-k,\,k=\\{0,1,\cdots,[L]\\}$,
where $[L]$ is an integer part of $L$. It is shown that the statement that the
spectrum of eigenvalues consists of only integer $L$ (see, e.g., Bohm ; Schiff
; Fock ; LL ) is just an artifact of choosing the Legendre function.
$P^{m}_{L}$, as an eigenfunction of $\hat{M}^{2}$. Results are discussed in
Section 4.

##  II 2\. Eigenfunctions of $\hat{M}_{z}$ that Are Single-Valued and Periodic
for Integer as well as for Non-Integer Eigenvalues

$\Psi(x,y|m)$, the eigenfunction of the operator of the third component of the
angular momentum, $\hat{M}_{z}$, is defined as the solution of the following
eigenvalue equation:

$\hat{M}_{z}\Psi(x,y|m)=i\left(y{d\over dx}-x{d\over
dy}\right)\Psi(x,y|m)=m\,\Psi(x,y|m),$ (1)

where $m$ is the eigenvalue and throughout the reduced Planck constant
$\hbar=1$. Solving Equation (1) for the complex
$\Psi(x,y|m)=\Psi_{R}(x,y|m)+i\Psi_{I}(x,y|m)$ is equivalent to solve the
following system of the two coupled equations for the real and imaginary
parts:

$\displaystyle\left(y\frac{d}{dx}-x\frac{d}{dy}\right)\Psi_{R}(x,y|m)=-m\Psi_{I}(x,y|m),\quad\quad\left(y\frac{d}{dx}-x\frac{d}{dy}\right)\Psi_{I}(x,y|m)=m\Psi_{R}(x,y|m).$
(2)

Acting on Equation (2) with the operator $(y\,d/dx-x\,d/dy)$ results into the
one and the same equation for both $\Psi_{R}$ and $\Psi_{I}$:

$\displaystyle\left(x\,{d\over dy}-y\,{d\over
dx}\right)^{2}\Psi_{R,\,I}(x,y|m)=-m^{2}\,\Psi_{R,\,I}(x,y|m).$ (3)

The two linearly independent solutions of the homogeneous differential
Equation (3) can be presented as $\Psi_{1}(x,y|m)=C_{1}(x;y|m)F_{1}(x;y|m)$
and $\Psi_{2}=C_{2}(x;y|m)F_{2}(x;y|m)$, where $F_{1,\,2}$ are linearly
independent particular solutions of Equation (3) and $C_{1,\,2}$ satisfy the
condition, $(yd/dx-xd/dy)C_{1,\,2}=0$. If one chooses those $F_{1,\,2}$ that
satisfy Equation (2), then $C_{1}=C_{2}$ and the general solution of the
eigenvalue Equation (1) is $\Psi(x,y|m)=C(x;y|m)[F_{1}(x;y|m)+iF_{2}(x;y|m)]$,
where $C(x;y|m)$ is a complex function with an absolute value, fixed by the
physical requirement of normalizability and the phase remains undetermined.

After $(x,\,y)$ is transformed to another set of independent variables,
$(f(x^{2}+y^{2})$, $\,\zeta(x,y))$, where $f$ is any differentiable function
of $x^{2}+y^{2}$, Equation (1) turns into an equation with the only variable,
$\zeta$. This technique of separating variables is used below but first let us
quote and discuss the function that is cited in textbooks as a solution of
Equation (1) Bohm ; Schiff ; Fock ; LL :

$F(x,y|m)\sim(x+iy)^{m}.$ (4)

If $m$ is non-integer, $F(x;y|m)$ is undetermined, since for non-integer
exponents, the power function is multivalued. In order for this function to be
a solution of Equation (1) it must be defined as a differentiable function of
$x$ and $y$. This may be achieved using, e.g., the Euler–de Moivre
prescription for the power of a complex number ww :

$\displaystyle(x+iy)^{m}=(\rho
e^{i\phi})^{m}=\rho^{m}e^{im\phi}=\rho^{m}(\cos\phi+i\sin\phi)^{m}=\rho^{m}(\cos
m\phi+i\sin m\phi),$ (5)

where

$\displaystyle\rho=|(x^{2}+y^{2})^{1/2}|,\
\sin\phi=\frac{y}{|(x^{2}+y^{2})^{1/2}|},\
\cos\phi=\frac{x}{|(x^{2}+y^{2})^{1/2}|},$ (6)

and $|z|$ stands for the absolute value of $z$.

Note that in the chain of Equation (5) rotational symmetry of the original
expression $(x+iy)^{m}$ is violated when $m$ is non-integer. Indeed, due to
the invariance of the Cartesian coordinates, $x,\,y$, under the
$2\pi$-rotation, $(x+iy)^{m}$ is formally rotationally invariant for any $m$.
Expressions $(\rho e^{i\phi})^{m}$ and $\rho^{m}(\cos\phi+i\sin\phi)^{m}$ are
invariant with respect to $\phi\to\phi+2\pi$ for any $m$, while
$\rho^{m}e^{im\phi}$ and $\rho^{m}(\cos m\phi+i\sin m\phi)$ violate rotational
symmetry for a non-integer $m$.

If one requires all expressions in Equation (5) to be invariant with respect
to $\phi\to\phi+2\pi$, then, $m$ must be an integer. Then, $(x+iy)^{m}$ is a
single valued function of $x$ and $y$. This connection between the rotational
invariance and the single valuedness of $(x+iy)^{m}$ caused the following
assertion: if one requires the invariance of the wave function
$(x+iy)^{m}\to\rho^{m}e^{im\phi}$ with respect to $\phi\to\phi+2\pi$, this is
equivalent to the single valuedness of this function Bohm ; Schiff ; Fock ; LL
. Both these conditions are satisfied when $m$ is integer and that is a reason
why, based on the requirements of single valuedness or/and periodicity, it was
declared that $m$ can only be integer and these requirements were formalized
in theoretical quantum mechanics as follows Bohm ; Schiff ; Fock ; LL :

$\displaystyle\Psi(\rho,\phi|m)=\Psi(\rho,\phi+2\pi k|m),$ (7)

where the polar coordinates $\rho$ and $\phi$ are given by Equation (6) and
$k$ is integer.

Imposing these physical conditions of single valuedness and rotational
invariance on the wave function that is not observable has been criticized by
Pauli Pauli  (see also in merz ). We agree with Pauli’s criticism and
emphasize that the purely integer spectrum of the eigenvalues is obtained only
when the conditions of single valuedness and/or periodicity are realized in
the framework of the Euler–de Moivre prescription (5). In fact, there are
other possible prescriptions for determining $(x+iy)^{m}$ and it turns out
that for one of these prescriptions, the eigenfunction $\Psi(x,y|m)$ will be
single valued, differentiable with respect to its variables, invariant with
respect to $\phi\to\phi+2\pi k$ for any $m$, integer as well as non-integer.
In other words, even if one imposes the requirement of single
valuedness/rotational invariance on wave function, this still does not
necessarily lead to a purely integer spectrum.

To demonstrate this, in Equation (3), the technique of separating variables is
used transforming from $(x,\,y)$ to $(\rho,\,\zeta)$, where
$\rho=|(x^{2}+y^{2})^{1/2}|$. Now the equation for $F(x,y|m)$ depends only on
$\zeta$ and $\zeta$ is chosen so that Equation (3) takes a form of an equation
solutions of which are well documented. Defining
$\zeta=[1/2-x/(2|(x^{2}+y^{2})^{1/2}|)]=[1/2-x/(2\rho)]$ turns Equation (3)
into the Gauss hypergeometric equation:

$\left[\zeta(1-\zeta)\,\frac{\rm d^{2}}{{\rm
d}\zeta^{2}}+\left(\frac{1}{2}-\zeta\right)\,\frac{\rm d}{{\rm
d}\zeta}+m^{2}\right]F(\zeta|m)=0.$ (8)

As known, any pair from the Kummer’s 24 solutions can be chosen as a set of
linearly independent solutions to the Gauss equation ww ; here the pair,

$\displaystyle F_{1}(\zeta|m)$ $\displaystyle=$
$\displaystyle{}_{2}F_{1}\left(m,-m;\frac{1}{2};\zeta\right)=(1-\zeta)^{\frac{1}{2}}\,{}_{2}F_{1}\left(\frac{1}{2}+m,\frac{1}{2}-m;\frac{1}{2};\zeta\right),$
$\displaystyle F_{2}(\zeta|m)$ $\displaystyle=$
$\displaystyle\zeta^{\frac{1}{2}}\,{}_{2}F_{1}\left(\frac{1}{2}+m,\frac{1}{2}-m;\frac{3}{2};\zeta\right)=\zeta^{\frac{1}{2}}(1-\zeta)^{\frac{1}{2}}\,{}_{2}F_{1}\left(1+m,1-m;\frac{3}{2};\zeta\right),$
(9)

is chosen, where ${}_{2}F_{1}(a,b;c;\zeta)$ is the Gauss’s hypergeometric
function ww ; Abramowitz ; Br . Finally, $\Psi(x,y|m)=C(F_{1}+iF_{2})$, the
eigenfunction of the operator of the third component of the angular momentum,
is given by

$\displaystyle\Psi(x,y|m)$ $\displaystyle=$ $\displaystyle
C(\rho|m)\left[{}_{2}F_{1}\left(m,-m;{1\over 2};{1\over 2}-{x\over
2\rho}\right)\right.$ (10) $\displaystyle-im{y\over\rho}\left({1\over
2}+{x\over 2\rho}\right)^{-1/2}\left.{}_{2}F_{1}\left({1\over 2}+m,{1\over
2}-m;{3\over 2};{1\over 2}-{x\over 2\rho}\right)\right],$

where the square root is determined via prescription
$(f^{2}(x))^{1/2}=|f(x)|$.

It is straightforward to verify that the eigenfunction (10), as a function of
$x,\,y$, is single-valued, invariant under the rotation at $2\pi k$, is
continuous and has continuous derivatives of all orders up to infinity for any
real, not necessarily integer $m$. Though $\Psi(x,y|m)$ contains a square
root, it is infinitely differentiable. This is guaranteed as soon as
$d(x/|x|)/dx=0$ and $d(y/|y|)/dy=0$ are satisfied which is readily
demonstrated using Equation (6). Indeed, from
$\cos\phi(x,y)=x/|(x^{2}+y^{2})^{1/2}|$, i.e., $\cos\phi(x,y)|_{y=0}=x/|x|$ it
follows that

$\displaystyle{d\cos\phi(x,y)\over
dx}|_{y=0}=\frac{y^{2}}{|(x^{2}+y^{2})^{3/2}|}|_{y=0}=0\quad\to\quad{d(x/|x|)\over
dx}=0.$ (11)

Similar to that of Equation (11), from $d\sin\phi(x,y)/dy|_{x=0}$:
$d(y/|y|)/dy=0$.

Let us consider particular values of $m$. Let us start with the integer $m=\pm
N$, $\,N=1,2,3\cdots$. The corresponding hypergeometric functions
${}_{2}F_{1}(N,-N;1/2;z)$ and ${}_{2}F_{1}(1/2+N,1/2-N;3/2;z)$ are tabulated
(see, e.g., Abramowitz ). Then, Equation (10) reads:

$\displaystyle\Psi(x,y|\pm N)={C(\rho|\pm N)\over\rho^{N}}\,(x\mp iy)^{N}.$
(12)

So, for integer $m$, $\Psi(x,y|N)$ reproduces, up to the factor $C(\rho|\pm
N)/\rho^{N}$, solution (4), $(x+iy)^{N}$.

Let us consider next the half-integer values of $m$. Explicit expressions are
lengthy and involved; results for $m=\pm 1/2$ and $m=\pm 3/2$ are:

$\displaystyle\Psi\left(x,y|\pm\frac{1}{2}\right)$ $\displaystyle=$
$\displaystyle
C\left(\rho|\pm\frac{1}{2}\right)\left[\left(\frac{1}{2}+\frac{x}{2\rho}\right)^{\frac{1}{2}}\mp
i\frac{y}{2\rho}\left(\frac{1}{2}+\frac{x}{2\rho}\right)^{-\frac{1}{2}}\right],$
(13) $\displaystyle\Psi\left(x,y|\pm\frac{3}{2}\right)$ $\displaystyle=$
$\displaystyle
C\left(\rho|\pm\frac{3}{2}\right)\left[\left(\frac{1}{2}+\frac{x}{2\rho}\right)^{\frac{1}{2}}\left(\frac{2x}{\rho}-1\right)\mp
i\frac{y}{2\rho}\left(\frac{1}{2}+\frac{x}{2\rho}\right)^{-\frac{1}{2}}\left(1+\frac{2x}{\rho}\right)\right].$

A relation between the wave functions for the integer and half-integer $m$ is
found being verified for $m=1/2,3/2,5/2$:

$\displaystyle\left[{\Psi\left(x,y|\pm\frac{N}{2}\right)\over
C\left(\rho|\pm\frac{N}{2}\right)}\right]^{2}={\Psi(x,y|\pm N)\over
C\left(\rho|\pm N\right)},$ (14)

i.e., wave function for the half-integer $m$, $\Psi(x,y|\pm N/2)$, satisfies
$\Psi^{2}(x,y|\pm N/2)\sim\Psi(x,y|\pm N)$, a relation similar to that which
holds for Equation (4), $((x\pm iy)^{N/2})^{2}=(x\pm iy)^{N}$. This result,
along with Equation (12), indicates that the eigenfunction $\Psi(x,y|m)$,
given by Equation (10), presents one possible prescription for the power
function $(x+iy)^{m}$.

Let us demonstrate with the example of the half-integer $m$ the importance of
choosing prescription for the square root as $(f^{2}(x))^{1/2}=|f(x)|$. To
this end, one moves from Cartesian to polar coordinates,
$(x,y)\to(\rho,\phi)$, see Equation (6). In polar coordinates, the argument,
$(1/2-x/2\rho)$, of the hypergeometric functions reads:
$(1-\cos\phi)/2=\sin^{2}(\phi/2)$. First, the prescription,
$(f^{2}(x))^{1/2}=f(x)$ is used. Using in Equation (13) the known relation
${}_{2}F_{1}(a,b;b;z)=(1-z)^{-a}$ Abramowitz ; Br , one obtains:

$\displaystyle\Psi\left(x,y|\pm\frac{1}{2}\right)$ $\displaystyle\to$
$\displaystyle\Psi\left(\rho,\phi|\pm\frac{1}{2}\right)=C\left(\rho|\pm\frac{1}{2}\right)\left[\left(\cos^{2}\frac{\phi}{2}\right)^{\frac{1}{2}}\mp\frac{i}{2}\,\sin\phi\left(\cos^{2}\frac{\phi}{2}\right)^{-\frac{1}{2}}\right]=$
(17) $\displaystyle
C\left(\rho|\pm\frac{1}{2}\right)\left(\cos\frac{\phi}{2}\mp
i\,\sin\frac{\phi}{2}\right)=C\left(\rho|\pm\frac{1}{2}\right)\,e^{\mp
i\frac{\phi}{2}}.$

Equation (17), apart from the normalization factor, is the known $e^{im\phi}$,
$m=\pm 1/2$, originated by the Euler–de Moivre prescription for the
$(x+iy)^{1/2}$ and presented as a standard expression for the eigenfunction of
$\hat{M}_{z}$ Bohm ; Schiff ; Fock ; LL . Obviously, because of the rotational
invariance of Cartesian coordinates,
$x(\phi)=x(\phi+2k\pi),\,y(\phi)=y(\phi+2k\pi)$, the left hand side of
Equation (17), $\Psi(x,y|\pm 1/2)$, given by Equation (13), is invariant under
the rotation $\phi\to\phi+2\pi k$. On the other hand, the right-hand side
(r.h.s.) of Equation (17) is invariant under the translations
$\phi\to\phi+4\pi k$, but not under $\phi\to\phi+2\pi k$. This inconsistency
stems from the prescription, $(f^{2}(x))^{1/2}=f(x)$ while deriving Equation
(17). For example, $\cos(\phi/2)$ appeared in the real part of Equation (17)
because for $(\cos^{2}(z))^{1/2}$ we used $\cos(z)$:

${}_{2}F_{1}\left(-{1\over 2},{1\over 2};{1\over 2};\sin^{2}\left({\phi\over
2}\right)\right)=\left[1-\sin^{2}\left({\phi\over
2}\right)\right]^{1/2}=\left[\cos^{2}\left({\phi\over
2}\right)\right]^{1/2}=\cos\left({\phi\over 2}\right).$ (18)

If $\phi=2\pi k$, the left-hand side (l.h.s.) of this relation is unity,
${}_{2}F_{1}(-1/2,1/2;1/2;0)=+1$, while for r.h.s. one gets $\cos\pi k$,
which, depending on $k$, can be either $+1$ or $-1$. If $\phi=\pi(2k+1)$, both
l.h.s. and r.h.s. of Equation (18) vanish. This means that Equation (18) is
valid only for $\phi$ with $\cos(\phi/2)\geq 0$; this condition, lacking from
Abramowitz ; Br , is also noted in Ref. NIST . Meantime, both l.h.s. and
r.h.s. of Equation (18) exist and are well defined for all values of $\phi$
and that calls for the question of how relation (18) has to be interpreted
when $\cos(\phi/2)<0$. Note that the inconsistency does not arise if, as an
alternative, instead of Equation (18), the following relation,

${}_{2}F_{1}\left(-{1\over 2},{1\over 2};{1\over 2};\sin^{2}\left({\phi\over
2}\right)\right)=\left|\cos\left({\phi\over 2}\right)\right|,$ (19)

is used, i.e., if one applies the prescription
$(\cos^{2}(\phi/2))^{1/2}=|\cos(\phi/2)|$, but not
$(\cos^{2}(\phi/2))^{1/2}=\cos(\phi/2)$.

If, instead of the prescription, $(f^{2}(x))^{1/2}=f(x)$,
$(f^{2}(x))^{1/2}=|f(x)|$ is applied, one obtains:

$\displaystyle\Psi\left(x,y|\pm\frac{1}{2}\right)\to\Psi\left(\rho,\phi|\pm\frac{1}{2}\right)=C\left(\rho|\pm\frac{1}{2}\right)\left(\left|\cos\frac{\phi}{2}\right|\mp{i\over
2}\,\sin\phi\left|\cos\frac{\phi}{2}\right|^{-1}\right)\ .$ (20)

Expression (20) is well defined for all values of $\phi$ and, most
importantly, it is invariant under translations, $\phi\to\phi+2\pi k$; the
above-mentioned inconsistency disappears. Similarly, using, for the case of
$m=3/2$, the prescription, $(f^{2}(x))^{1/2}=|f(x)|$, one obtains the same
result, $\Psi(\rho,\phi|\pm 3/2)=\Psi(\rho,\phi+2k\pi|\pm 3/2)$ but for the
prescription, $(f^{2}(x))^{1/2}=f(x)$, the resulting wave function is no
longer invariant under $\phi\to\phi+2\pi$.

From Equation (1) and its conjugation, using the properties of the Gauss
hypergeometric functions (see, e.g., ww ), one finds that $|\Psi(x,y|m)|^{2}$
only depends on $\rho=|(x^{2}+y^{2})^{1/2}|$ and the real and imaginary parts
of $\Psi(x,y|m)$ satisfy the relation resembling the trigonometric identity,
$\cos^{2}x+\sin^{2}x=1$:

$\displaystyle{\Psi^{2}_{R}(x,y|m)+\Psi^{2}_{I}(x,y|m)\over|C(\rho|m)|^{2}}=1.$
(21)

Particular examples of the general result (21) are cases of integer $m=N$,
when Equation (21) reduces to $\cos^{2}N\phi+\sin^{2}N\phi=1$, and of half-
integer $m$, quoted here for $m=1/2$:
$(\Psi^{2}_{R}(x,y|1/2)+\Psi^{2}_{I}(x,y|1/2))/|C(\rho|1/2)|^{2}=|\cos(\phi/2)|^{2}+\sin^{2}(\phi)/(4|\cos(\phi/2)|^{2})=1$.
Relation (21) is another indication that the functions (4) and (10) belong to
the same class since for any $m$, both Equations (4) and (10) satisfy relation
$|\Psi/C|^{2}=1$.

The physical requirement the solution should satisfy is that $\Psi(x,y|m)$
must be orthonormal. To verify normalizability, let us we use relation (21).
Normalizability condition,

$\displaystyle\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\rm d}x\,{\rm
d}y\,|\Psi(x,y|m)|^{2}=\pi\int_{0}^{\infty}{\rm
d}\rho\,|C(\rho|m)|^{2}<\infty$ (22)

can be readily realized by the appropriate choice of $C$. It suffices to
choose $|C|^{2}\sim\rho^{\gamma}$ with $\gamma<-1$ for $\rho\to\infty$ and $C$
finite for $\rho\to 0$.

Orthogonality follows from the relation which is obtained from Equation (1):

$\displaystyle i(m^{\prime}-m)\int\int_{-\infty}^{\infty}{\rm d}x\,{\rm
d}y\,\Psi^{*}(x,y|m^{\prime})\Psi(x,y|m)$
$\displaystyle\quad=\int_{-\infty}^{\infty}{\rm}{\rm
d}y\,y\Psi^{*}(x,y|m^{\prime})[\Psi(x,y|m)|_{x=\infty}-\Psi(x,y|m)|_{x=-\infty}]$
$\displaystyle\quad-\int_{-\infty}^{\infty}{\rm}{\rm
d}x\,x\Psi^{*}(x,y|m^{\prime})[\Psi(x,y|m)|_{y=\infty}-\Psi(x,y|m)|_{y=-\infty}].$
(23)

Using the above-mentioned constraints on $C(\rho|m)$ one obtains
$\Psi(x,y|m)|_{x,y=\infty}-\Psi(x,y|m)|_{x,y=-\infty}=0$, from which follows
that, if the integral in l.h.s. of Equation (23) exists, then, for
$(m^{\prime}-m)\neq 0$, the integral is zero being the condition of
orthogonality. Therefore, $\Psi(x,y|m)$ from Equation (10) fulfills every
physical requirement which the eigenfunction of the third component of the
quantum mechanical angular momentum operator should satisfy.

Using a certain prescription for the power function may lead to an expression
of the wave function that is not invariant under the translations at $2k\pi$
for a non-integer eigenvalues. This case is realized by the Euler–de Moivre
prescription, $(\rho e^{i\phi})^{m}=\rho^{m}e^{im\phi}$. On the other hand, if
another prescription is applied, this may result in a wave function that is
invariant under the translations at $2k\pi$ for integer as well as for non-
integer eigenvalues. This case is realized by the eigenfunction (10), where
the prescription, $(f^{2}(x))^{1/2}=|f(x)|$, is used. Prescription for the
power function affects not only the features of eigenfunctions of
$\hat{M}_{z}$, but also the features of the eigenvalues of the operator of the
angular momentum squared, as described in Section III below.

##  III 3\. Eigenfunctions and Eigenvalues of $\hat{M}^{2}$

The eigenvalue equation for
$\hat{M}^{2}=\hat{M}_{x}^{2}+\hat{M}_{y}^{2}+\hat{M}_{z}^{2}$, the operator of
the angular momentum squared,

$\hat{M}^{2}\Psi_{M}(L,m|\theta)=\left(\sin^{2}\theta\,{d^{2}\over
d\cos^{2}\theta}-2\cos\theta{d\over
d\cos\theta}-\frac{m^{2}}{\sin^{2}\theta}\right)\Psi_{M}(L,m|\theta)=L(L+1)\Psi_{M}(L,m|\theta),$
(24)

where $\theta$ is the polar angle and $L>0$ is the eigenvalue, reduces to the
equation for the Gauss’s hypergeometric series, solutions of which can be
represented by various linearly independent pairs of functions. A possible
pair is $\Psi_{M(1)},\;\Psi_{M(2)}$ JKT1 :

$\displaystyle\Psi_{M(1)}(L,m|\theta)$ $\displaystyle=$ $\displaystyle
f_{1}(m|\theta)\,\Phi_{1}(L,m|\theta),\quad
f_{1}(m|\theta)=(\sin^{2}\theta)^{{(m^{2})^{1/2}\over 2}},$
$\displaystyle\Phi_{1}(L,m|\theta)={}_{2}F_{1}\left({1\over
2}+{(m^{2})^{1/2}\over 2}+{L\over 2},{(m^{2})^{1/2}\over 2}-{L\over 2};{1\over
2};\cos^{2}\theta\right);$ $\displaystyle\Psi_{M(2)}(L,m|\theta)$
$\displaystyle=$ $\displaystyle f_{2}(m|\theta)\,\Phi_{2}(L,m|\theta),\quad
f_{2}(m|\theta)=\cos\theta\,(\sin^{2}\theta)^{{(m^{2})^{1/2}\over 2}},$
$\displaystyle\Phi_{2}(L,m|\theta)={}_{2}F_{1}\left(1+{(m^{2})^{1/2}\over
2}+{L\over 2},{1\over 2}+{(m^{2})^{1/2}\over 2}-{L\over 2};{3\over
2};\cos^{2}\theta\right).$

Any linear superposition of $\Psi_{M(1)}(L,m|\theta)$ and
$\Psi_{M(2)}(L,m|\theta)$ is also a solution of Equation (24).

The only physical requirement for the functions $\Psi_{M(1,\,2)}(L,m|\theta)$
is the normalizability. The necessary condition for the normalizability is
that $\Psi_{M}(L,m|\theta)$, presented as a product, $\Psi=f\Phi$, must be a
regular function. Unfortunately it is not known how to realize such a
condition for the product and the only option left is that normalizability can
be achieved when the factors $f_{j}(m|\theta)$ and $\Phi_{j}(L,m|\theta)$ are
regular for all values of their arguments which leads to a regular product
$\Psi=f\Phi$.

Functions, $f_{j}(m|\theta)\sim(\sin^{2}\theta)^{(m^{2})^{1/2}/2}$, are
regular for any $\theta$ when $(m^{2})^{1/2}\geq 0$. Hypergeometric series,
${}_{2}F_{1}(a,b;c;\cos^{2}\theta)$, converge for $\cos^{2}\theta<1$; for
$\cos^{2}\theta=1$ the series converge only if $a+b-c<0$ ww . This condition
for ${}_{2}F_{1}$ from Equation (III) reads: $a+b-c=(m^{2})^{1/2}<0$, which is
opposite to the condition of the regularity of $f_{j}(m|\theta)$,
$(m^{2})^{1/2}\geq 0$. Therefore, $f_{j}(m|\theta)$ and infinite series
${}_{2}F_{1}$ cannot simultaneously be regular, and in order
$\Psi_{M(j)}(L,m|\theta)$ to be regular, the infinite hypergeometric series
${}_{2}F_{1}$ should terminate resulting in polynomials; hereafter, this
truncation of infinite hypergeometric series is referred to as
“polynomialization.” As known, the infinite hypergeometric series
${}_{2}F_{1}(a;b;c;z)$ are polynomials if either $a$ or $b$ is a non-positive
integer ww . Since the parameters of ${}_{2}F_{1}(a,\,b;\,c;\;\cos^{2}\theta)$
depend on $L,\,m$ (see Equation (III)), setting $a$ or $b$ to a non-positive
integer results into constraints on $L,\,m$. It is obvious that a different
prescription for $(m^{2})^{1/2}$ generates different restrictions on
eigenvalues of the angular momentum.

Let us report results for the eigenvalue problem of the operator of the
angular momentum squared; for explicit but somewhat lengthy calculations, see
JKT1 . It is essential to specify which prescription is used for the
$(m^{2})^{1/2}$ appearing in $\Psi_{M(j)}(L,m|\theta)$, given by Equation
(III), since, as shown below, after the normalizability is required, different
prescriptions lead to the different results for the spectrum.

There are two possible prescriptions, namely, $(m^{2})^{1/2}=|m|$ and
$(m^{2})^{1/2}=\pm m$. First, the case when the prescription is
$(m^{2})^{1/2}=|m|$ is considered. Equating parameters $a$ and $b$ of the two
hypergeometric functions from Equation (III) to non-positive integers, $-k$,
results in four conditions, two conditions for $\Psi_{M(1)}$ and two
conditions for $\Psi_{M(2)}$. One condition out of the two for $\Psi_{M(1)}$
generates singular functions and, thus, to be dropped; the same is true for
$\Psi_{M(2)}$ and, finally, one remains with only two conditions of
polynomialization, generating regular eigenfunctions JKT1 .

The spectrum of eigenvalues corresponding to the two left regular
eigenfunctions is obtained from the following polynomialization conditions:

$\displaystyle\left|{m_{(1)}\over 2}\right|-{L\over 2}=-k_{1};\
L-2\left[{L\over 2}\right]\leq|m_{(1)}|\leq L,$ (26)
$\displaystyle\left|{m_{(2)}\over 2}\right|-{L-1\over 2}=-k_{2};\
(L-1)-2\left[{L-1\over 2}\right]\leq|m_{(2)}|\leq(L-1).$ (27)

Here, $[X]$ stands for the integer part of $X$ satisfying $X-[X]\geq 0$,
$k_{1}=0,1,2,\cdots[L/2]$ and $k_{2}=0,1,2,\cdots[(L-1)/2]$. The sets (26) and
(27) are comprised of the numeric sequence of positive and negative elements
$m_{(j),k},\;j=1,2$ with the step size 2, e.g., $|m_{(1)}|=L-2k_{1}$. From
Equations (26) and (27) it follows that the spectrum is discrete with the only
condition that $L-|m|$ is necessarily integer, while there are no constraints
on $L$ and $m$ separately; the solution (III) is regular for integer as well
as for non-integer $L,\,m$.

The sets of the eigenvalues, $\\{m_{(1)}\\}$ and $\\{m_{(2)}\\}$ (and their
corresponding eigenfunctions), can be formally combined into one set,
comprised of the positive and negative elements, $m_{k}$, with the step size
1: $m_{k}-m_{k-1}=\pm 1$. Using numerical ordering from the smallest to the
largest value, the combined set of all possible eigenvalues reads as follows:

$\displaystyle\\{m\\}|_{(m^{2})^{1/2}=|m|}$ $\displaystyle=$
$\displaystyle\\{-L,-L+1,-L+2,..,-m_{0};m_{0},..,L-2,L-1,L\\},$ (28)

where, depending on a numeric value of $L$, $m_{0}$ is either $(L-2[L/2])$,
the minimal positive value from the set (26), or $(L-1-2[(L-1)/2])$, the
minimal positive value from the set (27) JKT1 .

Starting from the subset of Equation (28) with $m$ positive, applying
$\hat{M}_{-}=\hat{M}_{x}-i\hat{M}_{y}$ leads to a subset with the negative $m$
and vice versa,
$\hat{M}_{+}\Psi(m<0)=(\hat{M}_{x}+i\hat{M}_{y})\Psi(m<0)\rightarrow\Psi(m>0)$
only when $L$ is either integer or half-integer. Acting by $\hat{M}_{-}$ on
the regular functions with $m$ positive leads to the regular functions with
$m$ negative, $\hat{M}_{-}\Psi_{\mathrm{reg}}(m>0)\to\Psi_{\mathrm{reg}}(m<0)$
only when $L$ is integer. When $L$ is half-integer, acting by $\hat{M}_{-}$ on
the regular functions with $m$ positive leads to the singular functions with
$m$ negative,
$\hat{M}_{-}\Psi_{\mathrm{reg}}(m>0)\to\Psi_{\mathrm{sing}}(m<0)$. A symmetric
result is valid when applying the rising operator:
$\hat{M}_{+}\Psi_{\mathrm{reg}}(m<0)\to\Psi_{\mathrm{reg}}(m>0)$ only when $L$
is integer and when $L$ is half-integer,
$\hat{M}_{+}\Psi_{\mathrm{reg}}(m<0)\to\Psi_{\mathrm{sing}}(m>0)$ JKT1 .

Evidently, if it is required that when moving with the step size 1, starting
from the wave function with $m=(-L)$, one should arrive at the wave function
with $m=+L$ and vice versa, this will be possible only when $m$ is either
integer or half-integer. In this case, no analysis of the eigenvalue problem
is necessary since the spectrum is already predefined to consist of only
integer or half-integer $m$. The requirement that, starting from the state
with $m=\mp L$ one arrives, moving with step size 1, to the state with $m=\pm
L$, is postulated in the method of commutator algebra of the angular momentum
operators Bohm ; Schiff ; Fock ; LL .

This requirement, customarily taken for granted to be a physical postulate, is
actually a mathematical condition, imposed by hand which filters out possible
non-integer and non-half-integer $m$ from the spectrum, similarly to how
imposing the non physical condition of periodicity on $e^{im\phi}$, filters
out non integer $m$ from the spectrum of $\hat{M}_{z}$. When $L$ is non-
integer, the requirement that starting from the state with $m=\mp L$ one
arrives at the state with $m=\pm L$, cannot be satisfied. Indeed, e.g., for
$L=1.7$, acting with the lowering operator
$\hat{M}_{-}=\hat{M}_{x}-i\hat{M}_{y}$ on $\Psi(1.7,1.7|\theta)$ would never
result in $\Psi(1.7,-1.7|\theta)$ and then terminate; instead one gets:
$\Psi(1.7,1.7|\theta)\rightarrow\Psi(1.7,0.7|\theta)\rightarrow\Psi(1.7,-0.3|\theta)\rightarrow\Psi(1.7,-1.3|\theta)\rightarrow\Psi(1.7,-2.3|\theta)\rightarrow\cdots$.
Let us recall that as an alternative to an unphysical requirement of single
valuedness of the wave function, Pauli suggested that acting by the rising and
lowering operators $\hat{M}_{x}\pm i\hat{M}_{y}$ on regular wave functions one
should find:
$\Psi_{M}(L,\,-L|\theta)\leftrightarrow\Psi_{M}(L,\,-L+1|\theta)\leftrightarrow\cdots\leftrightarrow\Psi_{M}(L,\,-1+L|\theta)\leftrightarrow\Psi_{M}(L,\,L|\theta)$.
Pauli justified this by postulating that as a result of acting on regular wave
functions by $\hat{M}_{x}\pm i\hat{M}_{y}$, no singular functions appear Pauli
; merz . In the case of the prescription $(m^{2})^{1/2}=|m|$, moving up and
down in spectrum with step size 2, indeed no singular functions are generated
for any, integer or non-integer, $m$, as follows from the conditions of
polynomialization JKT1 .

However, singular functions appear if instead of $(m^{2})^{1/2}=|m|$ the
prescription $(m^{2})^{1/2}=\pm m$ is used. In this case, the operators,
$\hat{M}_{\pm}=\hat{M}_{x}\pm i\hat{M}_{y}$, connecting wave functions and
$m\to m\pm 1$, can be defined and, acting by $\hat{M}_{\pm}$, results in a set
of eigenfunctions with the eigenvalues JKT1 ,

$\displaystyle\\{m_{1}\\}\downarrow_{(m^{2})^{1/2}=\pm m}$ $\displaystyle=$
$\displaystyle(L-2k_{1})=\\{L;L-2;\ldots;L-2[L/2];L-2[L/2]-2;\ldots;-\infty\\},$
$\displaystyle\\{m_{1}\\}\uparrow_{(m^{2})^{1/2}=\pm m}$ $\displaystyle=$
$\displaystyle(-L+2k_{2})=\\{-L;-L+2;\ldots;-L+2[L/2];-L+2[L/2]+2;\ldots;\infty\\},$
$\displaystyle\\{m_{2}\\}\downarrow_{(m^{2})^{1/2}=\pm m}$ $\displaystyle=$
$\displaystyle(L-1-2k_{3})=\left\\{L-1;L-3;\ldots;\right.$
$\displaystyle\left.L-1-2[(L-1)/2];L-1-2[(L-1)/2]-2;\ldots;-\infty\right\\},$
$\displaystyle\\{m_{2}\\}\uparrow_{(m^{2})^{1/2}=\pm m}$ $\displaystyle=$
$\displaystyle(-L+1+2k_{4})=\left\\{-L+1;-L+3;\ldots;\right.$ (29)
$\displaystyle\left.-L+1+2[(L-1)/2];-L+1+2[(L-1)/2]+2;\ldots;\infty\right\\},$

where $k_{1},k_{2},k_{3},k_{4}$ are positive integers and the elements of the
sets $\\{m_{j}\\}$ can be any real number, not necessarily integer or half-
integer. For $m$s from Equation (29), the corresponding hypergeometric
functions, ${}_{2}F_{1}$, are regular, some
$f(m|\theta)\sim(\sin^{2}\theta)^{(m^{2})^{1/2}/2}=(\sin\theta)^{\pm m}$ are
singular and, therefore, some $\Psi_{M(1,2)}(L,m|\theta)$ also are singular;
for explicit calculations and technicalities, see JKT1 . For the case of
$(m^{2})^{1/2}=\pm m$, one obtains, similar to the spectrum, resulting from
Equations (26) and (27), that the spectrum is discrete with the only condition
that $L-|m|$ is necessarily integer, while $L$ and $m$ can be integer as well
as non-integer each. If $L$ is integer, the sequences (29) do not extend to
$\pm\infty$ but truncate at $\pm L$ and reproduce the set of eigenvalues (28),
obtained using prescription, $(m^{2})^{1/2}=|m|$.

Let us note that in the group theoretical framework, the eigenfunctions,
corresponding to the eigenvalue spectrum (29), form an irreducible
representation of $SO(3)$, the three dimensional rotation group; see, e.g.,
group . As mentioned just above, when $L$ and $m$ are integer, infinite
sequences (29) truncate into a finite set of eigenvalues, $-L\leq m\leq L$,
and the corresponding eigenfunctions are regular and form a finite set. In
terms of the representation theory, these are the finite dimensional
irreducible representations of the $SO(3)$ group. When $L$ and $m$ are non
integer (half-integers included), the sequences (29) do not truncate and
remain infinite. Then, the corresponding infinite set of eigenfunctions is
formed of both singular and regular functions. In terms of the representation
theory, these are the infinite dimensional irreducible representations of the
$SO(3)$ group. Using prescription $(m^{2})^{1/2}=\pm m$ results in an infinite
set of eigenfunctions (containing both singular and regular functions),
corresponding to an infinite dimensional representation of the rotation group.
Using $(m^{2})^{1/2}=|m|$, the finite set of eigenvalues $-L\leq m\leq L$,
symmetric with respect to $m\rightarrow\,-m$, is filtered out from the
infinite set (29) because of $|m\geq 0$. The set of corresponding
eigenfunctions is comprised of regular functions only. In other words, the set
of regular eigenfunctions is being filtered out from a general infinite set of
eigenfunctions exactly the same way, similar to the case of integer $L$ and
$m$.

So, depending on a prescription for $(m^{2})^{1/2}$, eigenfunctions of the
operator of angular momentum square could be regular or singular and the
eigenvalues could be given either by Equations (28) or (29). When the
prescription, $(m^{2})^{1/2}=|m|$, is used, all eigenfunctions are regular and
the eigenvalue spectrum is given by Equation (28). When the prescription is
$(m^{2})^{1/2}=\pm m$, some eigenfunctions are regular, while some are
singular and the eigenvalue spectrum is given by Equation (29).

Finally, let us discuss what could cause the statement that the eigenvalue
problem for the $\hat{M}^{2}$ admits normalizable solutions only when $L$ is
integer Bohm ; Schiff ; Fock ; LL . The eigenfunctions and the spectrum, e.g.,
the set of regular functions and eigenvalues (28), are obtained by requiring
normalizability of a solution that is presented in terms of a specific pair of
linearly independent functions, $\Psi_{M(1)}$ and $\Psi_{M(2)}$. Quite a
different picture arises when the normalization condition is applied to
another pair of linearly independent functions, e.g., to the Legendre
functions, $P^{m}_{L}(\theta)$ and $Q^{m}_{L}(\theta)$, which were, from the
early days of quantum mechanics, considered as eigenfunctions of the operator
of the angular momentum squared Bohm ; Schiff ; Fock ; LL .

Certainly, both pairs, $\Psi_{M(1)}(L,m|\theta)$, $\Psi_{M(2)}(L,m|\theta)$
and $P^{m}_{L}(\theta)$, $Q^{m}_{L}(\theta)$, are solutions of the eigenvalue
equation (24). Legendre functions can be written as linear combinations of
hypergeometric functions, $\Psi_{M(1,2)}(L,m|\theta)$ ww :

$\displaystyle P^{m}_{L}(\theta)$ $\displaystyle=$ $\displaystyle
C_{11}\Psi_{M(1)}(L,m|\theta)+C_{12}\Psi_{M(2)}(L,m|\theta);$ $\displaystyle
Q^{m}_{L}(\theta)$ $\displaystyle=$ $\displaystyle
C_{21}\Psi_{M(1)}(L,m|\theta)+C_{22}\Psi_{M(2)}(L,m|\theta).$ (30)

Using just a polynomialization, it is not sufficient to normalize both
$P^{m}_{L}(\theta)$ and $Q^{m}_{L}(\theta)$ simultaneously. The reason is that
the sets $\\{m_{(1)}\\}$ and $\\{m_{(2)}\\}$, generated by the two conditions
of polynomialization (26) and (27), have no common element, and, thus, it is
impossible to satisfy these conditions simultaneously. As soon as the
polynomialization conditions are applied, either $\Psi_{M(1)}(L;m|\theta)$ or
$\Psi_{M(2)}(L;m|\theta)$ is singular JKT1 . Therefore, in order to carry out
the normalizability of $P^{m}_{L}(\theta)$ and $Q^{m}_{L}(\theta)$, only
polynomialization would not suffice and an additional requirement to filter
out singular parts of $P^{m}_{L}(\theta)$ and $Q^{m}_{L}(\theta)$ is
necessary. This can be achieved by choosing the coefficients, $C_{ij}$ in
Equation (30). Namely, as soon as the polynomialization condition (26) is
satisfied, what leads to regular $\Psi_{M(1)}(L;m|\theta)$ and singular
$\Psi_{M(2)}(L;m|\theta)$, the coefficients, $C_{12}$ and $C_{22}$, of
$\Psi_{M(2)}(L;m|\theta)$ must vanish. Similarly, as soon as the
polynomialization condition (27) is satisfied what leads to regular
$\Psi_{M(2)}(L;m|\theta)$ and singular $\Psi_{M(1)}(L;m|\theta)$, the
coefficients, $C_{11}$ and $C_{21}$, of $\Psi_{M(1)}(L;m|\theta)$ must vanish.

Coefficients $C_{ij}$ are calculated in JKT1 :

$\displaystyle C_{11}^{-1}\sim\Gamma\left({1\over 2}-{L\over 2}+{|m|\over
2}\right)\,\Gamma\left(1+{L\over 2}+{|m|\over 2}\right),\quad
C_{12}^{-1}\sim\Gamma\left({1\over 2}+{L\over 2}+{|m|\over
2}\right)\Gamma\left(-{L\over 2}+{|m|\over 2}\right),$
$C_{21}\sim{\Gamma\left({1\over 2}+{L\over 2}-{|m|\over
2}\right)\over\Gamma\left(1+{L\over 2}+{|m|\over 2}\right)},\quad
C_{22}\sim{\Gamma\left(1+{L\over 2}-{|m|\over
2}\right)\over\Gamma\left({1\over 2}+{L\over 2}+{|m|\over 2}\right)}.$ (31)

Here $\Gamma$ is the Euler Gamma function.

It is straightforward to show that, after applying the polynomialization
conditions (26) and (27) and using the property of the $\Gamma$ function such
that $1/\Gamma(\mathrm{nonpositive}\;\;\mathrm{integer})=0$ ww ; Abramowitz ,
one obtains that $C_{11}=0$ and $C_{12}=0$ can be realized only for integer
$L$ and $m$ and that $C_{21}=0$ and $C_{22}=0$ can never be satisfied JKT1 .
Therefore, $Q^{m}_{L}(\theta)$ has to be excluded, and only
$P^{m}_{L}(\theta)$ remains as the quantum mechanical eigenfunction.
Consequently, for the pair $P^{m}_{L}(\theta)$ and $Q^{m}_{L}(\theta)$,
normalizability is achieved only when $L$ and $m$ are integer. This is not a
general result; one explicit example of the eigenfunction, normalizable for
integer as well as for non-integer eigenvalues, is presented by the pair
$\Psi_{M(1)}(L,m|\theta)$ and $\Psi_{M(2)}(L,m|\theta)$, given in Equation
(III) and leading to spectra (28) or (29), depending on the prescription for
the power function. Hence, the statement that from theoretical quantum
mechanics it follows that the eigenvalue spectrum of $\hat{M}^{2}$ is
comprised of only integers is not necessarily correct in sense that it
corresponds to a special case when $P^{m}_{L}(\theta)$ and $Q^{m}_{L}(\theta)$
are chosen as a pair of linearly independent solutions of the eigenvalue
equation (24).

##  IV 4\. Conclusions

In this paper, the eigenvalue problem for the operator of the angular momentum
is studied in the framework of nonrelativistic quantum mechanics. The general
result for the spectrum is that it is discrete, namely, $|m|=L-k$ with $k$
been integer, $k=\\{0,1,\cdots,[L]\\}$, where $[L]$ is the integer part of
$L$. $L$ and $m$ can be integer as well as non-integer and this does not
contradict any physical principle.

The above is in stark contrast with the known statement that from theoretical
quantum mechanics it follows that $m$ and $L$ can only be integer or half-
integer Bohm ; Schiff ; Fock ; LL . An explanation of the contradiction is
that the result, obtained here, does not impose a non-physical requirements of
either periodicity of the wave function or postulating that, moving with the
step size 1 and starting from a state with $m=-L$, one should arrive at the
state with $m=+L$ and vice versa. The discreteness condition, $|m|=L-k$, does
not require that moving with the step size 1 from $\Psi_{M}(L,-L|\theta)$ one
should end with $\Psi_{M}(L,+L|\theta)$.

Using the Legendre functions, $P^{m}_{L}(\theta)$ and $Q^{m}_{L}(\theta)$, as
a pair of linearly independent solutions for the eigenvalue equation,
$\hat{M}^{2}\Psi=L(L+1)\Psi$, is a specific choice that does not encompass the
most general case. When $P^{m}_{L}(\theta)$ and $Q^{m}_{L}(\theta)$ are used,
the normalizability requirement filters out non-integer $L$ and $m$, but the
eigenvalue equation solution that is normalizable may exist for any real
eigenvalues, integer and non-integer. Another solution, presented here,
Equation (III), satisfies the necessary physical requirement of the
normalizability for integer and non-integer $L$ and $m$.

Imposing the condition of the single valuedness on the eigenfunction of the
third component of the angular momentum $(x+iy)^{m}$ does not necessarily lead
to the violation of the rotational invariance. Indeed, it is found such a
representation of the power function, Equation (10), which for any $m$ is a
single valued function of ($x,\,y$) and is invariant under the $2\pi k$
rotation. Two results indicate that $\Psi(x,y|m)$ from Equation (10) and
$(x+iy)^{m}$ belong to the same class of functions. First, according to
Equation (12), solution $\Psi(x,y|m)$, coincide, up to the normalization
factor, with $(x+iy)^{m}$ when $m$ is integer. Second, for the half-integer
values of $m=N/2$, one obtaines $(\Psi(x,y|N/2))^{2}\sim\Psi(x,y|N)$ (Equation
(14)), relation mirroring the one valid for the function (4),
$((x+iy)^{N/2})^{2}=(x+iy)^{N}$. For the rational $m=p/n$, it is not shown
that $(\Psi(x,y|p/n))^{n}\sim\Psi(x,y|p)$ but particular cases of integer and
half-integer $m$ are indications that
$(\Psi(x,y|\alpha))^{\beta}\sim\Psi(x,y|\alpha\beta)$ may well enough be true
for any $\alpha$ and $\beta$. Another important property, indicating that
$(x+iy)^{m}$ and $\Psi(x,y|m)$ belong to the same class, is that for any $m$,
both the functions satisfy the relation, $|\Psi/C|^{2}=1$; see Equation (21).
To summarize, the wave function (10), solution of the eigenvalue equation
$\hat{M}_{z}\Psi(x,y|m)=m\,\Psi(x,y|m)$, represents a possible prescription
for the power function, $(x+iy)^{m}$.

In the general case, when the only condition imposed on a wave function is the
physical requirement of the normalizability, i.e., when the periodicity
requirement for a wave function is lifted or when the different pair of
linearly independent functions is chosen, there is no constraint on $L$ and
$m$ to be integer only. From the physics point of view, the only self-
consistent approach is to drop all non-physical conditions and consider the
problem in the presence of only the physical requirements. This is what is
done in this paper and, as a result, a new quantum-mechanical solution of the
eigenvalue problem for the angular momentum operator is obtained.

To conclude, the main result of this paper is that from the framework of
theoretical quantum mechanics it does not follow that the eigenvalues of the
angular momentum operator should only be integer.

Surely, the spectrum of the angular momentum cannot be defined from
theoretical quantum mechanics alone but has to be established by comparing
theoretical calculations with experiments. However, this is not a goal of the
current paper which seeks to analyze the eigenvalue problem for the angular
momentum operator from the purely theoretical viewpoint.

###### Acknowledgements.

We are indebted with C. M. Bender, E. E. Boos, O. Daalhuis, J. T. Gegelia and
V. A. Petrov for illuminating discussions.

## References

  * (1) Bohm, D. Quantum Theory; Dover Publications, Inc.: New York, NY, USA, 1989.
  * (2) Schiff, L.I. Quantum Mechanics; McGraw-Hill Book Company: New York, NY, USA, 1968.
  * (3) Fock, V.A. Fundamentals of Quantum Mechanics; Mir Publishers: Moscow, USSR, 1970.
  * (4) Landau, L.D.; Lifshits, E.M. Quantum Mechanics: Non-Relativistic Theory; Butterworth-Heinemann: Oxford, UK, 1991.
  * (5) Majorana, E. Teoria relativistica di particelle con momento intrinseco arbitrario; Nuovo Cimento 1932, 9, 335–344.
https://doi.org/10.1007/BF02959557

  * (6) de Alfaro, V.; and Regge, T. Potential Scattering; North-Holland Publishing Company: Amsterdam, North Holland, 1965\.
  * (7) G$\ddot{o}$tte, J. B.; Franke-Arnold, S.; Zambrini, R.; Barnett, S.M. Quantum formulation of fractional orbital angular momentum; J. Mod. Opt. 2007, 54, 1723–1728.
https://doi.org/10.1080/09500340601156827

  * (8) Land, M. Harmonic oscillator states with integer and non-integer orbital angular momentum; J. Phys. Conf. Ser. 2011, 330, 012014.
https://doi.org/10.1088/1742-6596/330/1/012014

  * (9) Japaridze, G.S.; Khelashvili, A.A.; Turashvili, K.S. Critical comments on quantization of the angular momentum: I. Analysis based on the physical requirement on eigenfunctions and on the commutation relations; arXiv:1912.08042.
https://doi.org/10.48550/arXiv.1912.08042

  * (10) Japaridze, G.S.; Khelashvili, A.A.; Turashvili, K.S. Critical comments on the quantization of the angular momentum: II. Analysis based on the requirement that the eigenfunction of the third component of the operator of the angular momentum must be a single valued periodic function.; arXiv:2004.10673.
https://doi.org/10.48550/arXiv.2004.10673

  * (11) Whittaker, E.T.; Watson, G.N. A Course of Modern Analysis; Cambridge University Press: Cambridge, UK, 1996.
https://doi.org/10.1017/CBO9780511608759

  * (12) Pauli, W. Über ein Kriterium für Ein- oder Zweiwertigkeit der Eigenfunktionen in der Wellenmechanik. Helv. Phys. Acta 1939, 12, 147–168,
available online:
https://www.e-periodica.ch/digbib/view?pid=hpa-001%3A1939%3A12#157 ;
reprinted in Collected Scientific Scientific Papers by W.Pauli, in two
Volumes”, Ed. By R. Kronig and V.F. Weisskopf, Interscience Publishers, John
Wiley and Sons, Inc. New York, NY, USA, 1964.

  * (13) Merzbacher, E. Single valuedness of wave functions; Am. J. Phys. 1962, 30, 237–247.
https://doi.org/10.1119/1.1941984

  * (14) Abramowitz, M.; Stegun, I.A., Eds. Handbook of Mathematical Functions; Dover Publications, Inc.: New York, NY, USA, 1970.
  * (15) Prudnikov, A.P.; Brychkov, Y.A.; Marichev, O.I. Integrals and Series. Volume 3: More Special Functions; Gordon and Breach Science Publishers: London, UK 1992\.
  * (16) Olver, F.W.J.; Lozier, D.W.; Boisvert, R.F.; Clark, C.W., Eds. NIST Digital Library of Mathematical Functions. Comments to Formula 15.4.12. Available online:
https://dlmf.nist.gov (accessed on 15 May 2022).

  * (17) Vilenkin, N.Ja. Special Functions and the Theory of Group Representations; American Mathematical Society: Providence, RI, USA 1968\. Volume 22.
https://doi.org/10.1090/mmono/022