# Bound state solutions of the two–dimensional Schrödinger equation with
Kratzer–type potentials

Roman Ya. Kezerashvili1,2,3, Jianning Luo1, and Claudio R. Malvino1 1New York
City College of Technology, The City University of New York, Brooklyn, USA
2The Graduate School and University Center, The City University of New York,
New York, USA
3Long Island University, New York, USA

###### Abstract

Exactly solvable models play an extremely important role in many fields of
quantum physics. In this study, the Schrödinger equation is applied for a
solution of a two–dimensional (2D) problem for two particles interacting via
Kratzer, and modified Kratzer potentials. We found the exact bound state
solutions of the two–dimensional Schrödinger equation with Kratzer–type
potentials and present analytical expressions for the eigenvalues and
eigenfunctions. The eigenfunctions are given in terms of the associated
Laguerre polynomials.

##  I Introduction

The investigation and understanding of a quantum mechanical system is
dependent on garnering solutions to the Schrödinger equation. Solutions to the
equation provides the eigenvalues and eigenfunctions of the system, which
defines the fundamental information needed to understand the system. There are
only several potentials for which there are explicit solutions for all partial
waves or an exact solution of the $s-$wave Schrödinger equation in three-
dimensional (3D) space Landau ; Schiff2014 ; Davydov1975 ; Sakurai1985 ; Fluge
; Robinett2006 ; Griffiths2018 ; Suslov2023 .

As scientific advances continue to be made in the fields of transistors,
material sciences, and semiconductors, it has become increasingly valuable to
conduct quantum mechanics in two–dimensional space Gugiuzza2017 ; Wachter2017
; Robinett2006 . The most prominent field of study for the use of 2D quantum
mechanics are monolayer materials. Two decades since the Noble prize winning
discovery of monocrystalline graphitic films, by Novoselov and Geim in 2004
Novoselov2004 , the study of two–dimensional monolayer materials have expanded
far beyond monoatomic materials, such as graphene.

Although transition metal dichalcogenides have been known and studied in bulk
form since the 1960’s Frindt1966 , it was not until 2010, that the full
spectrum of applications for 2D monolayer transition metal dichalcogenides
were realized Mak2010 . It was then that the application of transitional metal
dichalcogenides, such as WSe2, WS2, MoSe2, MoS2, TeSe2, and TeS2, as
semiconductors became established due to breakthroughs in understanding their
properties at the monolayer scale (Dai2016, ; Avouris2017, ). These molecular
layered 2D systems received considerable attention (Mak2010, ; Cheng2014, ;
Lee2014, ; Cotlet2016, ; Manzeli2017, ; Berman2016, ; Berman2017, ; Wang2018,
; Brunetti2018, ; RKAS2021b, ). We cite these works, but the recent literature
on the subject is not limited by them. Most of the research was focused on
optical phenomena, Bose-Einstein condensation of exciton, magnetic properties
of excitons, et al. One can consider the application of the Kratzer–type
potentials for descriptions of vibrational and rotational energies for the
two–dimensional molecules of WSe2, WS2, MoSe2, MoS2, TeSe2, and TeS2 in 2D
semiconductors. The Kratzer Kratzer1926  and modified Kratzer–type LeRoy1970 ;
Berkdemir2006  potentials is mostly applied in atomic physics, molecular
physics, and quantum chemistry. They are used to describe the interactions of
molecular structure in 3D quantum mechanics. These potential are reliable in
terms of obtaining eigenvalues and eigenfunctions for vibrational and
rotational energies. The Kratzer potential is known to approach infinity when
the internuclear distance in molecules approaches zero, due to the repulsion
that exist between the molecules. As the internuclear molecular distance
approaches infinity, the potential approaches to zero. In Ref. Molas2019  was
suggested the modified Kratzer potentional for description of energy levels of
the Rydberg 2D excitons.

In Ref. Suslov2023  using the Nikiforov-Uvarov (NU) method NikUvarov  are
given the summary for analytical solutions of Schrödinger equations in 3D
space for $s-$ and arbitrary $l$ states for basic potentials employed in
non–relativistic quantum mechanics. Solutions of two–dimensional Schrödinger
equation for two particles are obtained in the close form Kezerashvili2023
within the framework the NU paradigm NikUvarov . The decrease of the
dimensionality from 3D to 2D decreases the degree of freedom by one and,
hence, the kinetic energy of the particles. Thus, the reduction of
dimensionality decreases the kinetic energy of two particles in 2D
configuration space due to the decrease of the degrees of freedom from three
to two kezFB2019 ; Kezerashvili2020FewBody . The latter leads to the change of
the energy spectrum for two interacting molecules in 2D space. Using the
approach Fluge  applied in 3D space, below we are finding the closed solution
for the 2D Schrödinger equation for the Kratzer and modified Kratzer
potentials.

##  II Kratzer and Kratzer–type potentials

To study the rotation-vibration spectrum of a diatomic molecule Kratzer in
1920 introduced the following central symmetry potential Kratzer1926

$V(r)=-2D_{0}\left(\frac{r_{0}}{r}\ -\frac{1}{2}\ \frac{r_{0}^{2}}{r^{2}}\
\right),$ (1)

where $D_{0}$ is the chemical dissociation energy of the lowest vibrational
level which determines the depth of the potential and the parameter $r_{0}>0$
is the equilibrium internuclear separation. $D_{0}$ differs slightly from the
electronic (or spectroscopic) dissociation energy of the diatomic molecule
$D_{e}$ used in some works with potential (1). This potential has a minimum
$V(r_{0})=-D_{0}$ and contains both a repulsive part and a long-range
attraction. For the Kratzer potential, as $r\rightarrow 0$,
$V(r)\rightarrow\infty$ due to the internuclear repulsion and as
$r\rightarrow\infty$, $V(r)\rightarrow 0$ i.e., the molecule decomposes.

To study the rotation-vibration spectrum of diatomic molecules the central
symmetry Kratzer potential Kratzer1926  was modified by a long-range
interatomic potential Berkdemir2006  that has wide applications in chemical
physics

$V(r)=D_{0}\left(\frac{r-r_{0}}{r}\right)^{2}.$ (2)

This potential presents the modified Kratzer potential and is shifted in
amount of $D_{0}$. In Refs. Berkdemir2006 ; Doma2023  two-particle problem
with the modified Kratzer potential (2) is solved in 3D space using the NU
method.

Improvements in semiconductor growth techniques over the subsequent decades,
which enabled the manufacture of effectively two–dimensional structures, led
to a resurgence of excitons in 2D materials. An exciton is the bound state of
an electron and hole in bulk, two–dimensional and even in one–dimensional
materials. The Schrödinger equation were applied to problems of excitons in
two dimensions. It was suggested the modified Kratzer potential Molas2019  for
a formation of the Rydberg excitons in 2D materials

$V(r)=-\mathfrak{K}\frac{e^{2}}{\rho}\left(\frac{r_{0}}{r}\ -g^{2}\
\frac{r_{0}^{2}}{r^{2}}\ \right).$ (3)

In Eq. (3) $\mathfrak{K}=\frac{1}{4\pi\varepsilon_{0}}=9\times
10^{9}Nm^{2}/C^{2}$ with the dielectric permittivity of vacuum
$\varepsilon_{0}$, $\rho=r_{0}\varepsilon$ is the screening length and
$\varepsilon$ is the dielectric constant of the bulk material. When $r_{0}=1$
and $g=0$ this potential becomes the Coulomb potential, while if
$\mathfrak{K}\frac{e^{2}}{\rho}=2D$ and $g^{2}=1/2$ it coincides with the
Kratzer potential (1).

##  III Kratzer potential

After separation of the center-of-mass and relative motions, for the relative
motion of two interacting particles we obtain the following equation

$\left[-\frac{\hbar^{2}}{2\mu}\Delta_{\mathbf{r}}+V(r)\right]\Psi(\mathbf{r})=E\Psi(\mathbf{r}),\
\ $ (4)

where $\mu=\frac{m_{1}m_{2}}{m_{1}+m_{2}}$ is the two-particle reduced mass,
$V(r)=V(\left|\mathbf{r}_{1}-\mathbf{r}_{2}\right|)$, $\Psi(\mathbf{r})$ and
$E$ are two particles’ eigenfunction and eigenenergy, respectively, and
$\Delta_{\mathbf{r}}$ is the Laplace operator in 2D space with respect to the
coordinate of the relative motion $\mathbf{r.}$

Let find the eigenfunctions and eigenenergies of two particles interacting via
the Kratzer potential in 2D configuration space. Due to the central symmetry
of the potential $V(r)$ it is convenient to write the 2D Laplace operators in
the polar coordinates Morse1953 . Following the standard procedure
Robinett2006 ; Zaslow1967  to separate the radial $r$ and angular variables in
2D space, one can write the radial Schrödinger equation in 2D space with a
cental symmetry potential as

$\frac{d^{2}\phi(r)}{dr^{2}}+\frac{1}{r}\frac{d\phi(r)}{dr}-\frac{m^{2}}{r^{2}}\phi(r)+\frac{2\mu}{\hbar^{2}}\left[E-V(r)\right]\phi(r)=0.$
(5)

In Eq. (5) $m$ is the azimutal quantum number and $\mu$ the reduced mass of
two particles. The 2D radial Schrödinger equation (5) with potential (1) reads

$\left(\frac{d^{2}}{dr^{2}}+\frac{1}{r}\frac{d}{dr}-\frac{m^{2}}{r^{2}}\right)\phi(r)+\frac{2\mu}{\hslash^{2}}\left[E+2D_{0}\left(\frac{r_{0}}{r}\
-\frac{1}{2}\ \frac{r_{0}^{2}}{r^{2}}\ \right)\right]\phi(r)=0.\ \ $ (6)

Let introduce notations

$k^{2}=-\frac{2\mu r_{0}^{2}}{\hslash^{2}}E,\text{ \ }\gamma^{2}=\frac{2\mu
r_{0}^{2}}{\hslash^{2}}D_{0}.\ $ (7)

For the bound state $k>0$ and $\gamma>0$. Using a new variable
$z=\frac{r}{r_{0}}$, so $r=r_{0}z$ and notations (7) we can rewrite (6) in the
following form

$\frac{d^{2}\phi(z)}{dz^{2}}+\frac{1}{z}\frac{d\phi(z)}{dz}+\frac{-k^{2}z^{2}+2\gamma^{2}z-\gamma^{2}-m^{2}}{z^{2}}\phi(z)=0.\
$ (8)

The differential equation (8) has a singularity at $z=0$ and an irregular
singularity when $z\rightarrow\infty$. One can use the characteristic equation

$\alpha^{2}-m^{2}-\gamma^{2}=0\ \ $ (9)

so

$\alpha=\sqrt{m^{2}+\gamma^{2}}\ \ $ (10)

which leads to $\phi(z)\propto z^{\alpha}$ at $z\rightarrow 0,$ when
$\alpha>0$, and, therefore, the radial wave function vanishes at $z=0$.
Rewriting Eq. (8) as

$z^{2}\frac{d^{2}\phi(z)}{dz^{2}}+z\frac{d\phi(z)}{dz}+\left(2\gamma^{2}z-k^{2}z^{2}-\alpha^{2}\right)\phi(z)=0\
\ $ (11)

and using substitution

$\phi(z)=z^{\alpha}e^{-kz}u(z),\ $ (12)

after lengthily calculation Eq. (11) becomes

$z\frac{d^{2}u(z)}{dz^{2}}+(1+2\alpha-2kz)\frac{du(z)}{dz}-\left(k+2k\alpha-2\gamma^{2}\right)u(z)=0.\
\ $ (13)

Such type equation as (13) has an analytical solution in the class of
hypergeometric functions Polyanin2018 .

After introducing in (13) a new variable $x=2kz$ this equation can be written
as Polyanin2018

$x\frac{d^{2}u(x)}{dx^{2}}+(b-x)\frac{du(x)}{dx}-au(x)=0,$ (14)

where $a=\frac{1}{2}+\alpha-\frac{\gamma^{2}}{k}$ and $b=1+2\alpha$. Equation
(14) is the confluent hypergeometric equation, and it is often called Kummer’s
equation Gradshteyn ; Abramowitz ; Polyanin2018 . This equation has two
linearly independent solutions Polyanin2018 .

$f(x)=C_{1}\Phi(a,b;x)+C_{2}\Psi(a-b+1,2-b;x)\ \ $ (15)

where $C_{1}$ and $C_{2}$ are the normalization constants. One solution
$\Phi(a,b;z)$ is regular at the origin $z=0$ and is a confluent hypergeometric
function. The second solution $\Psi(a-b+1,2-b;z)$ is diverging. Equation (14)
is the degenerate hypergeometric equation which is the general type of
Kummer’s equation of the confluent hypergeometric series. Because $b$ is not
integer the Kummer series is a particular solution

$u(x)=C_{1}\text{
}_{1}F_{1}\left(\frac{1}{2}+\alpha-\frac{\gamma^{2}}{k},1+2\alpha;x\right).$
(16)

This leads to the radial wave function

$\phi(r)=C_{0}\left(\frac{r}{r_{0}}\right)^{\alpha}e^{-k\frac{r}{r_{0}}}\,\,_{1}F_{1}(\frac{1}{2}+\alpha-\frac{\gamma^{2}}{k},\,\,1+2\alpha;\,\,2k\frac{r}{r_{0}}),$
(17)

where $C_{0}$ is the normalization constant. The function (17) is normalizable
if $\phi(r\rightarrow\infty)\rightarrow 0$. This requires that

$\frac{1}{2}+\alpha-\frac{\gamma^{2}}{k}=-n,\ n=0,1,2,3,...\ $ (18)

Using values for $k$ and $\alpha$ from Eqs. (7) and (10) we obtain

$E_{nm}=-\frac{\hslash^{2}}{2\mu
r_{0}^{2}}\frac{\gamma^{4}}{\left(\frac{1}{2}+n+\sqrt{m^{2}+\gamma^{2}}\
\right)^{2}\ }\ ,\text{ \ }n=0,1,2,3,...\text{ \ }m=0,1,2,3,..$ (19)

where $\gamma^{2}$ is defined in Eq. (7). Thus, the energy levels $E_{nm}$ are
discrete and depend on radial and azimuthal quantum numbers.

A representation of special functions in terms of a confluent hypergeometric
function allows a connection of the confluent hypergeometric function (16)
with the associated Laguerre polynomials $L_{n}^{\alpha}(x)$, $\alpha\in R$,
Gradshteyn

$L_{n}^{\alpha}(x)=\frac{\Gamma(n+\alpha+1)}{\left(n+1\right)!\Gamma(\alpha+1)}\text{
}_{1}F_{1}(-n,1+\alpha;x),$ (20)

where $\Gamma$ is a gamma function. Therefore, the confluent hypergeometric
function is proportional to the associated Laguerre polynomial and the radial
wave function (17) can be expressed in terms of $L_{n}^{\alpha}(x)$ as:

$\phi(r)=C\left(\frac{r}{r_{0}}\right)^{\alpha}e^{-k\frac{r}{r_{0}}}\,\,L_{n}^{2\alpha}\left(2k\frac{r}{r_{0}}\right).$
(21)

The normalization constant $C$ can be found through the evaluation of the
integral

$\int\limits_{0}^{\infty}x^{\alpha+1}e^{-x}\left[L_{n}^{\alpha}(x)\right]^{2}dx,\text{
when }\alpha\in R\text{ (integer or real). }$ (22)

To evaluate this integral one can use a formula for recurrence in the index
$n$ of $L_{n}^{\alpha}(x)$ and orthogonality relation for $L_{n}^{\alpha}(x)$
Gradshteyn

$xL_{n}^{\alpha}(x)=\left(2n+\alpha+1\right)L_{n}^{\alpha}(x)-\left(n+1\right)L_{n+1}^{\alpha}(x)-(n+\alpha)L_{n-1}^{\alpha}(x),$
(23)

$\int\limits_{0}^{\infty}x^{\alpha}e^{-x}L_{n}^{\alpha}(x)L_{m}^{\alpha}(x)dx=\left\\{\begin{array}[]{c}0,\text{\
\ \ \ \ \ \ \ \ \ \ }n\neq m\\\ \frac{\Gamma(n+\alpha+1)}{n!},\text{
}n=m\end{array}\right..$ (24)

The consideration of (23) and (24) leads to the normalization coefficient of
the radial wave function
$C=\frac{\left(2k\right)^{\alpha+1}}{r_{0}}\sqrt{\frac{n!}{(2n+2\alpha+1)\Gamma(n+2\alpha+1)}}$.
Finally, the eigenfunctions for two particles in two–dimensional space
interacting via the Kratzer potential (1) are:

$\Psi(r,\varphi)=\frac{\left(2k\right)^{\alpha+1}}{r_{0}}\sqrt{\frac{n!}{2\pi(2n+2\alpha+1)\Gamma(n+2\alpha+1)}}\left(\frac{r}{r_{0}}\right)^{\alpha}e^{-k\frac{r}{r_{0}}}\,\,L_{n}^{2\alpha}\left(2k\frac{r}{r_{0}}\right)e^{im\varphi}$
(25)

##  IV Modified Kratzer’s potential 1

Below we find eigenfunctions and eigenenergies of two particles interacting
via (2) potential in 2D configuration space. The radial Schrödinger equation
in 2D space with the potential (2) reads

$\left(\frac{d^{2}}{dr^{2}}+\frac{1}{r}\frac{d}{dr}-\frac{m^{2}}{r^{2}}\right)\phi(r)+\frac{2\mu}{\hslash^{2}}\left[E-D_{0}\left(1-2\frac{r_{0}}{r}+\frac{r_{0}^{2}}{r^{2}}\right)\right]\phi(r)=0.$
(26)

Considering notations (7) and (10) and the same variable $z,$ one can rewrite
Eq. (26) in the following form

$z^{2}\frac{d^{2}\phi(z)}{dz^{2}}+z\frac{d\phi(z)}{dz}+\left[-\left(k^{2}+\gamma^{2}\right)z^{2}+2\gamma^{2}z-\alpha^{2}\right]\phi(r)=0.$
(27)

Following the same procedure as for the Kratzer potential (1), Eq. (27) can be
simplified using the substitution $\phi(z)=z^{\alpha}e^{-\beta z}u(z)$ with
$\beta=\sqrt{k^{2}+\gamma^{2}}$. Applying this substitution, Eq. (27) takes
the form

$z\frac{d^{2}u(z)}{dz^{2}}+\left(1+2\alpha-2\beta
z\right)\frac{du(z)}{dz}-\left[\beta+2\alpha\beta-2\gamma^{2}\right]u(z)=0.$
(28)

Introducing a new variable $x=2\beta z$ in Eq. (28) leads to

$x\frac{d^{2}u(x)}{dx^{2}}+\left(1+2\alpha-x\right)\frac{du(x)}{dx}-\left[\frac{1}{2}+\alpha-\frac{\gamma^{2}}{\beta}\right]u(x)=0.$
(29)

The latter equation is the Kummer’s equation of the confluent hypergeometric
series, where $1+2\alpha=1+2\left(m^{2}+\gamma^{2}\right)^{1/2}$ is not an
integer due to $\gamma^{2}.$ Hence, (28) has a particular solution

$u(2\beta z)\text{
=}_{1}F_{1}(\frac{1}{2}+\alpha-\frac{\gamma^{2}}{\beta},1+2\alpha,2\beta z).$
(30)

Finally, the radial wave function in terms of the associated Laguerre
polynomials becomes

$\phi(r)=C\left(\frac{r}{r_{0}}\right)^{\alpha}\text{e}^{-\beta\frac{r}{r_{0}}}\text{
}L_{n}^{2\alpha}\left(2\beta\frac{r}{r_{0}}\right).$ (31)

The comparison of (31) and (3) shows that while the radial wave funcions have
the same analytical funtional dependence, they are different because
$k<\beta=\sqrt{k^{2}+\gamma^{2}}.$ The requirement for the normalizability of
the radial wave function leads to the quantization condition

$\frac{1}{2}+\alpha-\frac{\gamma^{2}}{\beta}=-n,\ n=0,1,2,3,...\ $ (32)

and normalization constant
$C=\frac{(2\beta)^{\alpha+1}}{r_{0}}\sqrt{\frac{n!}{(2n+2\alpha+1)\Gamma(n+2\alpha+1)}}.$
This constant coincides with the normalization constant for the solution with
the Kratzer potential only when $\beta=k$.

Using values for $\gamma$ from Eq. (7), $\alpha=\sqrt{m^{2}+\gamma^{2}},$
$\beta=\sqrt{k^{2}+\gamma^{2}}$ in Eq. (32) and solving it for the energy $E$
we obtain

$E_{nm}=-\frac{\hslash^{2}}{2\mu
r_{0}^{2}}\frac{\gamma^{4}}{\left(\frac{1}{2}+n+\sqrt{m^{2}+\gamma^{2}}\
\right)^{2}\ }\ -D_{0},\text{ \ }n=0,1,2,3,...\text{ \ }m=0,1,2,3,..$ (33)

Thus, the energy levels $E_{nm}$ depend on radial and azimuthal quantum
numbers. The comparison of energy spectra (19) and (33) shows that they have
the same dependence on the quantum numbers $n$ and $m$, but the potential (2)
leads to the shift of the energy levels by the dissociation energy $D_{0}$.
The total eigenfunction for two particle interacting via (2) potential in
terms of the associated Laguerre polynomials becomes

$\Psi(r,\varphi)=\frac{(2\beta)^{\alpha+1}}{r_{0}}\sqrt{\frac{n!}{2\pi(2n+2\alpha+1)\Gamma(n+2\alpha+1)}}\left(\frac{r}{r_{0}}\right)^{\alpha}\text{e}^{-\beta\frac{r}{r_{0}}}\text{
}L_{n}^{2\alpha}\left(2\beta\frac{r}{r_{0}}\right)e^{im\varphi}$ (34)

##  V Modified Kratzer’s potential 2

By substituting (3) into the radial Schrödinger equation in 2D space (5) we
obtain

$\left(\frac{d^{2}}{dr^{2}}+\frac{1}{r}\frac{d}{dr}-\frac{m^{2}}{r^{2}}\right)\phi(r)+\frac{2\mu}{\hslash^{2}}\left[E+\frac{e^{2}}{4\pi\rho}\left(\frac{r_{0}}{r}\
-g^{2}\frac{r_{0}^{2}}{r^{2}}\ \right)\right]\phi(r)=0.\ \ $ (35)

One can start from (35), introduce a variable $r=r_{0}z,$ notation

$\text{ \ }\delta^{2}=\mathfrak{K}\frac{\mu
r_{0}^{2}}{\hslash^{2}}\frac{e^{2}}{\rho},\ $ (36)

repeat the procedure presented for the Kratzer potential (1) using the
substitution $\phi(z)=z^{\eta}e^{-kz}u(z)$ with $\eta$
$=\sqrt{2g^{2}\delta^{2}+m^{2}}$, and finally obtains

$z\frac{d^{2}u(z)}{dz^{2}}+(1+2\eta-2kz)\frac{du(z)}{dz}-(k+2k\eta-2\delta^{2})u(z)=0.$
(37)

Therefore, (37) has an analytical solution that is a confluent hypergeometric
function:
$u(z)=\,_{1}F_{1}\left(\frac{1}{2}+\eta-\frac{\delta^{2}}{k},1+2\eta,2kz\right)$.
In terms of the associated Laguerre polynomials the total radial wave function
of the exciton bound via (3) potential reads

$u(r)=C\left(\frac{r}{r_{0}}\right)^{\eta}e^{-k\frac{r}{r_{0}}}\,\,L_{n}^{2\eta}\left(2k\frac{r}{r_{0}}\right).$
(38)

The radial function (38) can be normalized only if
$\frac{1}{2}-\frac{\delta^{2}}{k}+\eta=-n;\,\,\,\,\,\,\,\,n=0,1,2,...$ The
latter condition with the values of $k$, $\delta$, and $\eta$ gives the
following quantized energy levels

$E_{nm}=-\mathfrak{K}^{2}\frac{e^{4}\mu}{2\hbar^{2}}\frac{1}{\left(n+\frac{1}{2}+\sqrt{2g^{2}\delta^{2}+m^{2}}\right)^{2}}.$
(39)

Note that when $g=0$, since $n$ and $m$ are any positive integers included 0,
a new quantum number in Eq. (39) is $n+m=N$. Hence, the energy levels (39)
coincide with the spectra for the Coulomb potential Zaslow1967 ; Yang1991 ;
Portnoi2002 ; Kezerashvili2023 :

$E_{n}=-\mathfrak{K}^{2}\frac{e^{4}\mu}{2\hslash^{2}}\frac{1}{\left(n+\frac{1}{2}\right)^{2}},\text{
\ \ }n=0,1,2,3,...$ (40)

While when $g=1/2$ the energy spectrum (39) coincides with (19) if
$2D=\mathfrak{K}\frac{e^{2}}{\rho}$. Each energy level (40) is ($2n+1$)-fold
degenerate, the so-called accidental degeneracy. Notably, (40) does not
contain explicitly the azimuthal quantum number $m$, which enters the radial
equation. From (40) one can conclude that the spectrum of energy is changes
from $E_{n}\sim n^{-2}$ (Rydberg series) in 3D space Landau , to
$E_{n}\sim\left(n+1/2\right)^{-2}$ in 2D space. Therefore, for example, the
ground state energy increases by a factor of 4 in 2D space. Thus, the
reduction of dimensionality decreases the kinetic energy of two particles in
2D space interacting via the Coulomb potential due to the decrease of the
degrees of freedom from three to two.

The normalization of the radial function (39) when
$-\frac{1}{2}-\eta+\frac{\delta^{2}}{k}=n$ gives

$C=\frac{(2k)^{\eta+1}}{r_{0}}\sqrt{\frac{n!}{(2n+2\alpha+1)\Gamma(n+2\alpha+1)}}.$
(41)

Hence, the total eigenfunctions related to the eigenenergies (39) in terms of
the Laguerre polynomials is

$\Psi(r,\varphi)=\frac{(2k)^{\eta+1}}{r_{0}}\sqrt{\frac{n!}{2\pi(2n+2\alpha+1)\Gamma(n+2\alpha+1)}}\left(\frac{r}{r_{0}}\right)^{\eta}e^{-k\frac{r}{r_{0}}}\,\,L_{n}^{2\eta}\left(2k\frac{r}{r_{0}}\right)e^{im\varphi}.$
(42)

noindent A careful examination of Eqs. (25), (34), and (42) shows that the
total eigenfunctions obtained for the Kratzer (1), modified Kratzer’s (2), and
(3) potentials have the same analytical dependences with different $\alpha$,
$\beta$, $\eta$, and $k$. Nevertheless, the most remarkable feature is that
coordinate-space probability distributions for two particles calculated with
the eigenfuctions (25), (34), and (42) are entirely different. The probability
distributions for two particles calculated with the eigenfuctions (25), (34),
and (42) are presented in Fig. 1. Obviously, the eigenfunction (42) is the
same as (25) when $\eta=\alpha$ and (42) coincides with (34) when
$\eta=\alpha$ and $\beta=k$. Under these conditions the probability
distributions for two particles interacting via (1), (2), and (3) potentials
are the same.

Figure 1: (Color online) The probability distribution for two particles
calculated with ($a$) the Kratzer potential (1), ($b$) the modified Kratzer’s
potential (2) for interactions of diatomic molecules, ($c$) the modified
Kratzer’s potential (3) for an electron-hole interaction. Calculations are
performed for $n=3$ and $m=1$.

##  VI Conclusion

One of the challenging tasks of quantum mechanics in two-dimension is to find
the exact analytical solution of the Schrödinger equation for a given
potential and any arbitrary value $m$ in 2D configuration spaces. This
solution can be further used to define the observables of the system.

In this study, we found the exact bound state solutions of the two–dimensional
Schrödinger equation with Kratzer–type potentials and present analytical
expressions for the eigenvalues and eigenfunctions. The eigenfunctions are
given in terms of the associated Laguerre polynomials.

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