Path Integral of Schrödinger’s Equation

Abstract

The path integral is a powerful tool for studying quantum mechanics because it has the merit of establishing the connection between classical mechanics and quantum mechanics. This formalism quickly gained prominence in various fields of theoretical physics, including its generalization to quantum field theory, quantum mechanics, and statistical physics. Using the Feynman propagator, we can calculate the partition function, the free energy, wave functions, and the energy spectrum of the considered physical system. Moreover, the Feynman formalism finds broad applications in geophysics and in the field of financial sciences.

1. Introduction

In this chapter, the Schrödinger solutions of potential problem have been evaluated using the Feynman path integral formulation of quantum mechanics; an appropriate space-time transformation has been applied to Green’s function associated with the problem, which made it an integrable function. Also, the energy spectrum in a non-relativistic regime with normalized wave functions for potential, is obtained using path integral formalism of quantum mechanics; the results are evaluated for any state due to the use of an approximation scheme for centrifugal term 1/r², the constructed propagator associated with the Schrödinger equation of the problem was treated by space-time transformation trick that made it integrable, and energy eigenvalues for some exceptional cases of potential were also presented to compare our solutions with those obtained in previous studies. The organization of this chapter is as follows: in Section 1, we formulate the radial propagator and its corresponding Green’s function associated with a nonrelativistic particle in the presence of a potential where we use an approximation to the centrifugal term. In Section 2, we treat Green’s function of Generalized inverse quadratic Yukawa potential by performing a nontrivial space-time transformation to pass from the actual complex problem to another already solved one, which is a Pöschl-Teller (PT) potential problem. In Section 3, energy eigenvalues and corresponding eigenfunctions are extracted from the poles and residues of the aforementioned solved Green’s function. Section 4 discusses special cases of Deng Fun potentiel, Generalized inverse quadratic Yukawa potential as Kratzer potential, Yukawa potential, inversely quadratic Yukawa potential, and Coulomb potential.

2. Propagator and Schrödinger equation

The Schrödinger equation is a fundamental equation in quantum physics that describes the behavior of quantum systems. It was formulated by Erwin Schrödinger in 1925. The Schrödinger equation describes the time evolution of the wave function of a quantum system is governed by the equation:

$$i\mathrm{\hslash }\frac{d}{\mathit{dt}}\mid \psi \left(t\right)⟩=\widehat{H}\mid \psi \left(t\right)⟩,$$E1

which integrates in the particular case of a time-independent Hamiltonian:

$$\mid \psi \left(t^{\text{'}}\right)⟩=exp\left[-\frac{i}{\mathrm{\hslash }}\widehat{H}\left(t^{\text{'}}-t\right)\right]\mid \psi \left(t\right)⟩,$$E2

where $\widehat{U}\left(t^{\text{'}},t\right)=exp\left[-\frac{i}{\mathrm{\hslash }}\widehat{H}\left(t^{\text{'}}-t\right)\right]$ represents the evolution operator. Let us consider a particle in a potential, the Hamiltonian is written as:

$$\widehat{H}=\frac{{\widehat{P}}^2}{2m}+V\left(\widehat{x}\right).$$E3

In the position representation x, the evolution equation becomes:

$$⟨x^{\text{'}}|\psi \left(t^{\text{'}}\right)⟩=⟨x^{\text{'}}|exp\left(\frac{-i}{\mathrm{\hslash }}\widehat{H}\left(t^{\text{'}}-t\right)\right)|\psi \left(t\right)⟩.$$E4

Let us use the position closure relation x

$$\int \mathit{dx}\left|x\right.⟩\left.⟨x\right|=1.$$E5

Eq. (4) becomes:

$$⟨x^{\text{'}}|\psi \left(t^{\text{'}}\right)⟩=\int \mathit{dx}⟨x^{\text{'}}|exp\left(\frac{-i}{\mathrm{\hslash }}\widehat{H}\left(t^{\text{'}}-t\right)\right)|\psi \left(t\right)⟩.$$E6

Thus, it can be written as

$$\psi \left(x^{\text{'}},t^{\text{'}}\right)=\int K\left(x^{\text{'}},t^{\text{'}};x,t\right)\psi \left(x,t\right)\mathit{dx}.$$E7

The propagator $K\left(x^{\text{'}},t^{\text{'}};x,t\right)=⟨x^{\text{'}}|exp\left(\frac{-i}{\mathrm{\hslash }}\widehat{H}\left(t^{\text{'}}-t\right)\right)|x⟩$ allows us to evaluate the transition amplitude between the two states. Let us consider an initial state localized at $x_0$

$$\psi \left(x,t_0\right)=\delta \left(x-x_0\right),$$E8

then,

$$\psi \left(x^{\text{'}},t^{\text{'}}\right)=K\left(x^{\text{'}},t^{\text{'}};x_0,t_0\right).$$E9

the probability amplitude of finding the particle at position $x^{\text{'}}$ at time $t^{\text{'}}.$

3. Transition from propagator to Green’s function

As we saw earlier, the propagator can be expressed in terms of the time-evolution operator as follows [1, 2]:

$$K\left(x^{"},t^{"};x^{\text{'}},t^{\text{'}}\right)=⟨x^{"}|U\left(t^{"},t^{\text{'}}\right)|x^{\text{'}}⟩,$$E10

where

$$U\left(t^{"},t^{\text{'}}\right)=exp\left[\frac{-i}{\mathrm{\hslash }}\widehat{H}\left(t^{"}-t^{\text{'}}\right)\right],$$E11

with $T=t^{"}-t^{\text{'}}.$ Moreover, it is possible to extract the energy spectrum as well as the wave function corresponding to a given physical system, from Green’s function, the latter being none other than the Fourier transform of the propagator. In effect [3],

$$G\left(x^{"},x^{\text{'}};E\right)=\frac{i}{\mathrm{\hslash }}{\int }_0^{\infty }{\mathit{dTe}}^{i\left(E+\mathit{i\epsilon }\right)T/\mathrm{\hslash }}K\left(x^{"},x^{\text{'}};T\right),$$E12

where $\epsilon$, is a positive constant

$$\left(\widehat{H}-E\right)G\left(x^{"},x^{\text{'}};E\right)=\delta \left(x^{"}-x^{\text{'}}\right).$$E13

Formula (10) allows us to write:

$$G\left(x^{"},x^{\text{'}};E\right)=⟨x^{"}|\frac{1}{\widehat{H}-E-\mathit{i\epsilon }}|x^{\text{'}}⟩.$$E14

By introducing the closure relation on position $\mid n,\mathrm{\ell }⟩$, we can express the probability amplitude (11) as follows:

$$G\left(x^{"},x^{\text{'}};E\right)=\frac{i}{\mathrm{\hslash }}{\int }_0^{\infty }\sum _{n,\mathrm{\ell }}\mathit{dT}⟨x^{"}|n,\mathrm{\ell }⟩⟨n,\mathrm{\ell }|e^{\frac{i\left(E+\mathit{i\epsilon }\right)T}{\mathrm{\hslash }}}{e}^{-\frac{\mathit{iHT}}{\mathrm{\hslash }}}|x^{\text{'}}⟩,$$E15

or alternatively,

$$G\left(x^{"},x^{\text{'}};E\right)=\sum _{n,\mathrm{\ell }}^{\infty }\frac{{\chi }_{n,\mathrm{\ell }}^{\ast }\left(x^{\text{'}}\right){\chi }_{n,\mathrm{\ell }}\left(x^{"}\right)}{E_{n,\mathrm{\ell }}-E-\mathit{i\epsilon }}.$$E16

where ${\chi }_{n,\mathrm{\ell }}\left(x\right)$ is the wave function corresponding to the eigenenergy $E_{n,\mathrm{\ell }}$, and it is also possible to arrive at ${\chi }_{n,l}\left(x\right)$ and $E_{n,\mathrm{\ell }}$ from the relation.

$$K\left(x^{"},t^{"};x^{\text{'}},t^{\text{'}}\right)=\sum _{n,\mathrm{\ell }}^{\infty }{\chi }_{n,\mathrm{\ell }}^{\ast }\left(x^{\text{'}}\right){\chi }_{n,\mathrm{\ell }}\left(x^{"}\right)e^{-\frac{{\mathit{iE}}_{n,\mathrm{\ell }}T}{\mathrm{\hslash }}}.$$E17

4. Path integral in spherical coordinates

In quantum mechanics, rotational symmetry is crucial in finding the wave functions and corresponding energies of physical systems. Spherical coordinates transform from the Schrödinger equation of rotational symmetry. Therefore, we can separate this equation into an angular part expressed in terms of spherical harmonics, whose solutions are known, and a radial part that contains specific information about the dynamical systems.

In the path integral, this coordinate transformation is possible, but initially, things become complicated. One of these complexities arises when studying the presence of a centrifugal barrier, which eliminates the possibility of “time slicing.”

The following relation represents the formula for the three-dimensional (3D) propagator [4, 5]:

$$\begin{array}{lll}K\left(\overrightarrow{r^{"}}\text{, }{t}^{"}\text{; }\overrightarrow{r^{\text{'}}}\text{, }{t}^{\text{'}}\right)\hfill & =\hfill & {\displaystyle {\int }_{\overrightarrow{r^{\text{'}}}}^{\overrightarrow{r^{"}}}D\overrightarrow{r}\left(t\right)\exp {{\displaystyle \int }}^{\text{}}\frac{i}{\hslash }\left(\frac{m}{2}{\left(\text{Δ}\overrightarrow{r_j}\right)}^2-V\left(\overrightarrow{r}\right)\right)dt}\hfill \\ \hfill & =\hfill & \underset{N\to \infty }{\lim }\stackrel{N+1}{{\displaystyle \prod }_{j=1}}{\left(\frac{m}{2i\pi \hslash \epsilon }\right)}^{\frac{1}{2}}\left[\stackrel{N}{{\displaystyle \prod }_{i=1}}\underset{R^3}{{\displaystyle \int }}d{\overrightarrow{r}}_j\right]\exp \left[\frac{i}{\hslash }{S}_N\right],\hfill \end{array}$$E18

with

$$\begin{array}{l}{t}^{"}=t_{N+1};t^{\text{'}}=t_0\\ \overrightarrow{r^{"}}={\overrightarrow{r}}_{N+1};\overrightarrow{r^{\text{'}}}={\overrightarrow{r}}_0\end{array}$$

and the total action:

$$S_N=\underset{j=1}{\sum ^{N+1}}{S}_j=\underset{j=1}{\sum ^{N+1}}\left(\frac{m}{2\epsilon }\left(r_j^2+r_{j-1}^2-2{\overrightarrow{r}}_j\cdot {\overrightarrow{r}}_{j-1}\right)-\mathit{\epsilon V}\left(\overrightarrow{r_j}\right)\right).$$E19

Using the spherical coordinate system $\left(r,\theta ,\phi \right)$ defined as:

$$\left\{\begin{array}{c}x=rsin\theta cos\phi \\ y=rsin\theta sin\phi ,\\ z=rcos\theta \end{array}\right.$$E20

with $r>0;0\le \theta <\pi$ and $0\le \phi <2\pi .$

Where the volume element is expressed in spherical coordinates as:

$$d\overrightarrow{r_j}=r_j^{2}sin{\theta }_j{\mathit{dr}}_{j}d{\theta }_{j}d{\phi }_j,$$E21

the propagator (18) can be rewritten in spherical coordinates as:

$$\begin{array}{lll}K\left(\overrightarrow{r^{"}},t^{"};\overrightarrow{r^{\text{'}}},t^{\text{'}}\right)& =& \underset{N\to \infty }{lim}\stackrel{N+1}{\prod _{j=1}}{\left(\frac{m}{2\mathit{i\pi }\mathrm{\hslash }\epsilon }\right)}^{\frac{3}{2}}\underset{j=1}{\prod ^N}\left[{\int }_0^{\infty }{\int }_0^{\pi }{\int }_0^{2\pi }{r}_j^{2}sin{\theta }_j{\mathit{dr}}_{j}d{\theta }_{j}d{\phi }_j\right]\\ & \times & \stackrel{N+1}{\prod _{j=1}}\left[exp\left(\frac{i}{\mathrm{\hslash }}{S}_j\right)\right],\end{array}$$E22

The elemental action is:

$$S_j=\frac{m}{2\epsilon }\left(r_j^2+r_{j-1}^2-2r_j{r}_{j-1}cos{\mathrm{\Theta }}_{j,j-1}\right)-\mathit{\epsilon V}\left(r_j\right),$$E23

where ${\mathrm{\Theta }}_{j,j-1}=\left({\overrightarrow{r}}_j,{\overrightarrow{r}}_{j-1}\right),$

with the angle between two vectors in spherical coordinates being:

$$cos{\mathrm{\Theta }}_{j,j-1}=cos{\theta }_{j}cos{\theta }_{j-1}+sin{\theta }_{j}sin{\theta }_{j-1}cos\left({\phi }_j-{\phi }_{j-1}\right),$$E24

and the measurement takes the form:

$$\underset{j=1}{\prod ^{N+1}}{\left(\frac{m}{2\mathit{i\pi }\mathrm{\hslash }\epsilon }\right)}^{\frac{3}{2}}\underset{j=1}{\prod ^N}d\overrightarrow{r}={\left(\frac{m}{2\mathit{i\pi }\mathrm{\hslash }\epsilon }\right)}^{\frac{3}{2}\left(N+1\right)}\underset{j=1}{\prod ^N}\left[{\int }_0^{\infty }{\int }_0^{\pi }{\int }_0^{2\pi }{r}_j^{2}sin{\theta }_j{\mathit{dr}}_{j}d{\theta }_{j}d{\phi }_j\right].$$E25

The previous expression of the propagator is not appropriate for integration due to the presence of the term $\left(\frac{-i}{\mathrm{\hslash }}\frac{m}{\epsilon }{r}_j{r}_{j-1}cos{\mathrm{\Theta }}_j{,}_{j-1}\right),$ and this term is separable into a radial part and an angular part.

For an explicit evolution of the angular part of the propagator, we will use the following formula [6]:

$$\begin{array}{lll}exp\left(\mathit{iz}cos\phi \right)& =& 2^{\frac{1}{2}}\mathrm{\Gamma }\left(\frac{1}{2}\right)\stackrel{+\infty }{\sum _{k=0}}\left(k+\frac{1}{2}\right)i^k{\left(z\right)}^{\frac{-1}{2}}{J}_{k+\frac{1}{2}}\left(z\right)P_k\left(cos\phi \right)\\ & =& \sqrt{\frac{\pi }{2z}}\stackrel{+\infty }{\sum _{k=0}}\left(2k+1\right)i^k{J}_{k+\frac{1}{2}}{P}_k\left(cos\phi \right).\end{array}$$E26

$J_n\left(x\right)$ is the Bessel function, which is given by:

$$J_n\left(x\right)=\stackrel{+\infty }{\sum _{P=0}}\frac{{\left(-1\right)}^P}{P!\left(n+P\right)!}{\left(\frac{x}{2}\right)}^{2P+n},$$E27

if

$$x=-U,$$E28

then

$$\begin{array}{lll}{J}_n\left(-U\right)& =& \stackrel{+\infty }{\sum _{P=0}}\frac{{\left(-1\right)}^P}{P!\left(n+P\right)!}{\left(\frac{-U}{2}\right)}^{2P+n}\\ & =& \stackrel{+\infty }{\sum _{P=0}}\frac{{\left(-1\right)}^P}{P!\left(n+P\right)!}{\left(\frac{U}{2}\right)}^{2P+n}{\left(-1\right)}^n.\end{array}$$E29

we have

$$J_n\left(\mathit{ix}\right)=i^n{I}_n,$$E30

where $I_n$ is the modified Bessel function.

We define

$$y=\mathit{iz},$$E31

According to (26) and (30), we can deduce that

$$\begin{array}{lll}{e}^{ycos\phi }& =& \sqrt{\frac{-\pi }{2\mathit{iy}}}\stackrel{+\infty }{\sum _{k=0}}\left(2k+1\right)i^k{J}_{k+\frac{1}{2}}{P}_k\left(cos\phi \right)\\ & =& {\left(-1\right)}^{\frac{1}{2}}\sqrt{\frac{\pi }{2\mathit{iy}}}\stackrel{+\infty }{\sum _{k=0}}\left(2k+1\right)i^k{\left(-1\right)}^{k+\frac{1}{2}}{J}_{k+\frac{1}{2}}\left(\mathit{iy}\right)P_k\left(cos\phi \right)\\ & =& {\left(-1\right)}^{\frac{1}{2}}\sqrt{\frac{\pi }{2\mathit{iy}}}\stackrel{+\infty }{\sum _{k=0}}\left(2k+1\right)i^k{\left(-1\right)}^{k+\frac{1}{2}}{\left(i\right)}^{k+\frac{1}{2}}{I}_{k+\frac{1}{2}}\left(y\right)P_k\left(cos\phi \right)\\ & =& \sqrt{\frac{\pi }{2y}}\stackrel{+\infty }{\sum _{k=0}}\left(2k+1\right)I_{k+\frac{1}{2}}\left(y\right)P_k\left(cos\phi \right),\end{array}$$E32

where $P_k\left(cos\phi \right)$ are the Legendre polynomials.

We arrive at the following expression for the propagator, by substituting formula (32) in (22):

$$\begin{array}{lll}K\left(\overrightarrow{r^{"}},t^{"};\overrightarrow{r^{\text{'}}},t^{\text{'}}\right)& =& \underset{N\to \infty }{lim}\stackrel{N+1}{\prod _{j=1}}{\left(\frac{m}{2\mathit{i\pi }\mathrm{\hslash }\epsilon }\right)}^{\frac{3}{2}}\underset{i=1}{\prod ^N}\left[{\int }_0^{\infty }{\int }_0^{\pi }{\int }_0^{2\pi }{r}_j^{2}sin{\theta }_j{\mathit{dr}}_{j}d{\theta }_{j}d{\phi }_j\right]\\ & \times & \stackrel{N+1}{\prod _{j=1}}\left[exp\left[\frac{i}{\mathrm{\hslash }}\left[\frac{m}{2\epsilon }\left(r_j^2+r_{j-1}^2\right)-\mathit{\epsilon V}\left(r_j\right)\right]\right]\right]\\ & \times & \underset{j=1}{\prod ^{N+1}}{\left(\frac{\mathit{i\pi }\mathrm{\hslash }\epsilon }{2{\mathit{mr}}_j{r}_{j-1}}\right)}^{\frac{1}{2}}\stackrel{+\infty }{\sum _{l\dots l_{N+1=0}}}\left(2l_j+1\right)\\ & \times & I_{l_j+\frac{1}{2}}\left(\frac{{\mathit{mr}}_j{r}_{j-1}}{i\mathrm{\hslash }\epsilon }\right)P_l\left(cos{\mathrm{\Theta }}_{j,j-1}\right),\end{array}$$E33

$$l_j=l,\forall j=1,\dots ,N+1$$

where else

$$\begin{array}{lll}K\left(\overrightarrow{r^{"}},t^{"};\overrightarrow{r^{\text{'}}},t^{\text{'}}\right)& =& \underset{N\to \infty }{lim}{\left(\frac{m}{2\mathit{i\pi }\mathrm{\hslash }\epsilon }\right)}^{\frac{3}{2}\left(N+1\right)}\stackrel{+\infty }{\sum _{l\dots l_{N+1=0}}}\left[\underset{j=1}{\prod ^N}{\int }_0^{\infty }{r}_j^2{\mathit{dr}}_j\right]\\ & \times & \left[\underset{j=1}{\prod ^N}{\int }_0^{\pi }{\int }_0^{2\pi }\left(2l_j+1\right)I_{l_j+\frac{1}{2}}\left(\frac{{\mathit{mr}}_j{r}_{j-1}}{i\mathrm{\hslash }\epsilon }\right)P_{l_j}\left(cos{\mathrm{\Theta }}_{j,j-1}\right)sin{\theta }_{j}d{\theta }_{j}d{\phi }_j\right]\\ & \times & \left[\left(2l_{N+1}+1\right)I_{l_{N+1}+\frac{1}{2}}\left(\frac{{\mathit{mr}}_j{r}_{j-1}}{i\mathrm{\hslash }\epsilon }\right)P_{l_{N+1}}\left(cos{\theta }_{j,j-1}\right)\right]\\ & \times & \underset{j=1}{\prod ^{N+1}}\left[exp\left[\frac{i}{\mathrm{\hslash }}\left[\frac{m}{2\epsilon }\left(r_j^2+r_{j-1}^2\right)-\mathit{\epsilon V}\left(r_j\right)\right]\right]\right]\\ & \times & \underset{j=1}{\prod ^{N+1}}{\left(\frac{\mathit{i\pi }\mathrm{\hslash }\epsilon }{2{\mathit{mr}}_j{r}_{j-1}}\right)}^{\frac{1}{2}},\end{array}$$E34

we can use the following expression

$$\underset{j=1}{\prod ^N}{r}_j=\frac{1}{\sqrt{r^{"}{r}^{\text{'}}}}\underset{j=1}{\prod ^{N+1}}{\left(r_j{r}_{j-1}\right)}^{\frac{1}{2}},$$E35

by substituting Expression (35) in (34), we obtain

$$\begin{array}{lll}K\left(\overrightarrow{r^{"}},t^{"};\overrightarrow{r^{\text{'}}},t^{\text{'}}\right)& =& \underset{N\to \infty }{lim}{\left(\frac{m}{2\mathit{i\pi }\mathrm{\hslash }\epsilon }\right)}^{\left(N+1\right)}\left(\frac{1}{\sqrt{r^{"}{r}^{\text{'}}}}\right)\stackrel{+\infty }{\sum _{\mathrm{\ell }\dots {\mathrm{\ell }}_{N+1=0}}}\left[\underset{j=1}{\prod ^N}{\int }_0^{+\infty }{r}_j{\mathit{dr}}_j\right]\\ & \times & \underset{j=1}{\prod ^{N+1}}\left[exp\left[\frac{i}{\mathrm{\hslash }}\left\{\frac{m}{2\epsilon }\left(r_j^2+r_{j-1}^2\right)-\mathit{\epsilon V}\left(r_j\right)\right\}\right]\right]\\ & \times & \underset{j=1}{\prod ^N}\left[{\int }_0^{\pi }{\int }_0^{2\pi }\left(2{\mathrm{\ell }}_j+1\right)I_{{\mathrm{\ell }}_j+\frac{1}{2}}\left(\frac{{\mathit{mr}}_j{r}_{j-1}}{i\mathrm{\hslash }\epsilon }\right)P_{{\mathrm{\ell }}_j}\left(cos{\mathrm{\Theta }}_{j,j-1}\right)sin{\theta }_j{d}_{j}d{\phi }_j\right]\\ & \times & \left[(2{\mathrm{\ell }}_{N+1}+I_{{\mathrm{\ell }}_{N+1}+\frac{1}{2}}\left(\frac{{\mathit{mr}}_j{r}_{j-1}}{i\mathrm{\hslash }\epsilon }\right)P_{{\mathrm{\ell }}_{N+1}}\left(cos{\mathrm{\Theta }}_j{,}_{j-1}\right)\right].\end{array}$$E36

The Legendre polynomials can be decomposed into spherical harmonics

$$P_{\mathrm{\ell }}\left(cos\theta \right)=\left(\frac{4\pi }{2\mathrm{\ell }+1}\right)\sum _{m=-\mathrm{\ell }}{Y}_{\mathrm{\ell },m}\left({\theta }_N,{\phi }_N\right)Y_{\mathrm{\ell },m}^{\ast }\left({\theta }_{N-1},{\phi }_{N-1}\right),$$E37

where $P_{\mathrm{\ell }}\left(cos\theta \right)$ represents the Legendre polynomial of order $\mathrm{\ell }$ and degree m, $\theta$ is the polar angle, and $\varphi$ is the azimuthal angle. $Y_{\mathrm{\ell },m}\left(\theta ,\phi \right)$ corresponds to the associated spherical harmonic of order $\mathrm{\ell }$ and degree m.

This formula establishes a connection between Legendre polynomials and spherical harmonics, providing an expansion in terms of angles for functions or phenomena with spherical symmetry,

where

$$Y_{\mathrm{\ell },m}\left(\theta ,\phi \right)={\left(-1\right)}^m\left[\frac{\left(2\mathrm{\ell }+1\right)}{4\pi }\times \frac{\left(\mathrm{\ell }-m\right)!}{\left(\mathrm{\ell }+m\right)!}\right]P_{\mathrm{\ell }}^m\left(cos\theta \right)exp\left(\mathit{im\phi }\right),$$E38

Formula (32) is as follows:

$$e^{ycos\phi }=2\pi \sqrt{\frac{2\pi }{y}}\stackrel{+\infty }{\sum _{\mathrm{\ell }=0}}\left(2\mathrm{\ell }+1\right)I_{\mathrm{\ell }+\frac{1}{2}}\left(y\right)\stackrel{\mathrm{\ell }}{\sum _{m=-\mathrm{\ell }}}{Y}_{\mathrm{\ell },m}\left({\theta }_N,{\phi }_N\right)Y_{\mathrm{\ell },m}^{\ast }\left({\theta }_{N-1},{\phi }_{N-1}\right),$$E39

By inserting the last formula into the propagator expression (36)

$$\begin{array}{lll}K\left(\overrightarrow{r^{"}},t^{"};\overrightarrow{r^{\text{'}}},t^{\text{'}}\right)& =& \underset{N\to \infty }{lim}{\left(\frac{m}{i\mathrm{\hslash }\epsilon }\right)}^{\left(N+1\right)}\left(\frac{1}{\sqrt{r^{"}{r}^{\text{'}}}}\right)\stackrel{+\infty }{\sum _{\mathrm{\ell }\dots {\mathrm{\ell }}_{N+1=0}}}\left[\underset{j=1}{\prod ^N}{\int }_0^{\infty }{r}_j{\mathit{dr}}_j\right]\\ & \times & \underset{j=1}{\prod ^{N+1}}\left[exp\left[\frac{i}{\mathrm{\hslash }}\left\{\frac{m}{2\epsilon }\left(r_j^2+r_{j-1}^2\right)-\mathit{\epsilon V}\left(r_j\right)\right\}\right]\right]\\ & \times & \underset{j=1}{\prod ^N}\left[I_{{\mathrm{\ell }}_j+\frac{1}{2}}\left(\frac{{\mathit{mr}}_j{r}_{j-1}}{i\mathrm{\hslash }\epsilon }\right)I_{{\mathrm{\ell }}_{N+1}+\frac{1}{2}}\left(\frac{{\mathit{mr}}_j{r}_{j-1}}{i\mathrm{\hslash }\epsilon }\right)\right]\\ & \times & \stackrel{{\mathrm{\ell }}_j}{\sum _{m_j=-{\mathrm{\ell }}_j}}{\int }_0^{\pi }{\int }_0^{2\pi }{Y}_{{\mathrm{\ell }}_j,m_j}\left({\theta }_j,{\phi }_j\right)Y_{{\mathrm{\ell }}_j,m_j}^{\ast }\left({\theta }_{j-1},{\phi }_{j-1}\right)sin{\theta }_{j}d{\theta }_{j}d{\phi }_j\\ & \times & \underset{m=-\mathrm{\ell }}{\sum ^{\mathrm{\ell }}}{Y}_{{\mathrm{\ell }}_{N+1},m}\left({\theta }_{N+1},{\phi }_{N+1}\right)Y_{{\mathrm{\ell }}_{N+1},m}^{\ast }\left({\theta }_N,{\phi }_N\right).\end{array}$$E40

Using the orthogonality relation of spherical harmonics, which is described by the following equation

$${\int }_0^{\pi }\mathit{d\phi }{\int }_0^{2\pi }{Y}_{\mathrm{\ell },m}\left(\theta ,\phi \right)Y_{\mathrm{\ell },m}^{\ast }\left(\theta ,\phi \right)sin\mathit{\theta d\theta }={\delta }_{\mathrm{\ell },{\mathrm{\ell }}^{\text{'}}},{\delta }_{m,m^{\text{'}}},$$E41

thus we find the following expression for the propagator

$$K\left(\overrightarrow{r^{"}},t^{"};\overrightarrow{r^{\text{'}}},t^{\text{'}}\right)=\underset{\mathrm{\ell }=0}{\sum ^{+\infty }}\frac{\left(2\mathrm{\ell }+1\right)}{4\pi }{K}_{\mathrm{\ell }}\left(r^{"},t^{"};r^{\text{'}},t^{\text{'}}\right)P_{\mathrm{\ell }}\left(cos\mathrm{\Theta }\right),$$E42

where the radial propagator $K_{\mathrm{\ell }}\left(r^{"},t^{"};r^{\text{'}},t^{\text{'}}\right)$ is also expressed as

$$\begin{array}{lll}{K}_{\mathrm{\ell }}\left(r^{"},t^{"};r^{\text{'}},t^{\text{'}}\right)& =& \underset{N\to \infty }{lim}{\left(\frac{m}{i\mathrm{\hslash }\epsilon }\right)}^{\left(N+1\right)}\left(\frac{1}{\sqrt{r^{"}{r}^{\text{'}}}}\right)\underset{N\to \infty }{lim}\left[\underset{j=1}{\prod ^N}{\int }_0^{\infty }{r}_j{\mathit{dr}}_j\right]\\ & \times & \underset{j=1}{\prod ^{N+1}}\left[I_{{\mathrm{\ell }}_j+\frac{1}{2}}\left(\frac{{\mathit{mr}}_j{r}_{j-1}}{i\mathrm{\hslash }\epsilon }\right)\right]\\ & \times & \underset{j=1}{\prod ^{N+1}}\left[exp\left[\frac{i}{\mathrm{\hslash }}\left\{\frac{m}{2\epsilon }\left(r_j^2+r_{j-1}^2\right)-\mathit{\epsilon V}\left(r_j\right)\right\}\right]\right],\end{array}$$E43

Indeed, considering the asymptotic behavior of the modified Bessel functions, [6].

$$I_j\left(\frac{z}{\epsilon }\right)\underset{\epsilon \to 0}{\to }{\left(\frac{\epsilon }{2\mathit{\pi z}}\right)}^{\frac{1}{2}}exp\left\{\frac{z}{\epsilon }-\frac{1}{2}\frac{\epsilon }{z}\left({\upsilon }^2-\frac{1}{4}\right)\right\},$$E44

then

$$I_{\mathrm{\ell }+\frac{1}{2}}\left(\frac{\frac{{\mathit{mr}}_j{r}_{j-1}}{i\mathrm{\hslash }}}{\epsilon }\right)\underset{\epsilon \to 0}{\to }{\left(\frac{\mathit{\epsilon i}\mathrm{\hslash }}{2\pi {\mathit{mr}}_j{r}_{j-1}}\right)}^{\frac{1}{2}}exp\left\{\frac{{\mathit{mr}}_j{r}_{j-1}}{\mathit{\epsilon i}\mathrm{\hslash }}-\left(\frac{\mathit{\epsilon i}\mathrm{\hslash }}{2{\mathit{mr}}_j{r}_{j-1}}\right)\left(\mathrm{\ell }\left(\mathrm{\ell }+1\right)\right)\right\},$$E45

then we arrive at the formulation of the radial propagator in spherical coordinates and as a function of the effective potential $V_{\mathit{eff}}\left(r_j\right)$:

$$\begin{array}{lll}{K}_{\mathrm{\ell }}\left(r^{"},t^{"};r^{\text{'}},t^{\text{'}}\right)& =& \left(\frac{1}{r^{"}{r}^{\text{'}}}\right)\underset{N\to \infty }{lim}{\left(\frac{m}{2\mathit{\pi i}\mathrm{\hslash }\epsilon }\right)}^{\frac{\left(N+1\right)}{2}}\times \left[\underset{j=1}{\prod ^N}{\int }_0^{\infty }{\mathit{dr}}_j\right]\\ & \times & exp\frac{i}{\mathrm{\hslash }}\stackrel{N+1}{\sum _{j=1}}\left\{\frac{m}{2\epsilon }{\left(\mathrm{\Delta }{r}_j\right)}^2-\mathit{\epsilon V}\left(r_j\right)-\frac{\mathrm{\ell }\left(\mathrm{\ell }+1\right){\mathrm{\hslash }}^2\epsilon }{2{\mathit{mr}}_j{r}_{j-1}}\right\},\end{array}$$E46

where the effective potential is defined by the following expression:

$$V_{\mathit{eff}}\left(r_j\right)=V\left(r_j\right)+\frac{{\mathrm{\hslash }}^2\mathrm{\ell }\left(\mathrm{\ell }+1\right)}{2{\mathit{mr}}_j{r}_{j-1}},$$E47

So the propagator (46) becomes:

$$\begin{array}{lll}{K}_{\mathrm{\ell }}\left(r^{"},t^{"};r^{\text{'}},t^{\text{'}}\right)& =& \left(\frac{1}{r^{"}{r}^{\text{'}}}\right)\underset{N\to \infty }{lim}{\left(\frac{m}{2\mathit{\pi i}\mathrm{\hslash }\epsilon }\right)}^{\frac{\left(N+1\right)}{2}}\times \left[\underset{j=1}{\prod ^N}{\int }_0^{\infty }{\mathit{dr}}_j\right]\\ & \times & exp\left[\frac{i}{\mathrm{\hslash }}\stackrel{N+1}{\sum _{j=1}}\left\{\frac{m}{2\epsilon }{\left(\mathrm{\Delta }{r}_j\right)}^2-\epsilon V_{\mathit{eff}}\left(r_j\right)\right\}\right].\end{array}$$E48

The specific form of the radial propagator will depend on the potential energy term V(r) in the radial Schrödinger equation, which corresponds to the particular physical system being studied. Different potential energy functions will lead to different solutions and, consequently, different forms of the radial propagator.

5. Feynman propagator

The propagator related to a central potential V(r) between two space-time points $\left(\overrightarrow{r^{\text{'}}},t^{\text{'}}\right)$ and $\left(\overrightarrow{r^{"}},t^{"}\right),$ in spherical coordinates is written as [4, 5]:

$$K\left(\overrightarrow{r^{"}},t^{"};\overrightarrow{r^{\text{'}}},t^{\text{'}}\right)=\frac{1}{4\pi r^{"}{r}^{\text{'}}}\sum _{\mathrm{\ell }=0}^{\infty }\left(2\mathrm{\ell }+1\right)\times K_{\mathrm{\ell }}\left(r^{"},t^{"};r^{\text{'}},t^{\text{'}}\right)P_{\mathrm{\ell }}\left(cos\mathrm{\Theta }\right),$$E49

where $P_{\mathrm{\ell }}\left(cos\mathrm{\Theta }\right)$ is the Legendre polynomial and $\mathrm{\Theta }\equiv \left(\overrightarrow{r^{"}},\overrightarrow{r^{\text{'}}}\right)$ with

$$K_{\mathrm{\ell }}\left(r^{"},t^{"};r^{\text{'}},t^{\text{'}}\right)=\underset{N\to \infty }{lim}\int \stackrel{N}{\underset{j=1}{\mathrm{\Pi }}}exp\left[\frac{i}{\mathrm{\hslash }}{S}_j\right]\times \stackrel{N}{\underset{j=1}{\mathrm{\Pi }}}{\left[\frac{m}{2\mathit{\pi i}\mathrm{\hslash }\epsilon }\right]}^{\frac{1}{2}}\stackrel{N-1}{\underset{j=1}{\mathrm{\Pi }}}{\mathit{dr}}_j,$$E50

where

$$S_j=\frac{m}{2\epsilon }{\left(\mathrm{\Delta }{r}_j\right)}^2-\epsilon V_{\mathit{eff}}\left(r_j\right),$$E51

here

$\mathrm{\Delta }{r}_j=r_j-r_{j-1},\epsilon =\mathrm{\Delta }{t}_j=t_j-t_{j-1},t^{\text{'}}=t_0$ and $t^{"}=t_N,$

and effective potential $V_{\mathit{eff}}$ is defined by the relation as:

$$V_{\mathit{eff}}\left(r\right)=\frac{{\mathrm{\hslash }}^2}{2m}\frac{\mathrm{\ell }\left(\mathrm{\ell }+1\right)}{r^2}+V\left(r\right),$$E52

Thus, the condensed form is given by:

$$K_{\mathrm{\ell }}\left(r^{"},t^{"};r^{\text{'}},t^{\text{'}}\right)=\mathfrak{D}r\left(t\right)exp\left[{\int }_{t^{\text{'}}}^{t^{"}}\left(\frac{m}{2}{\dot{r}}^2-V_{\mathit{eff}}\left(r\right)\right)\mathit{dt}\right],$$E53

5.1 Pöschl-Teller potential

This potential is an important diatomic molecular potential. Many applications of the analytical and approximate technique in the current literature have been made to establish eigensolutions and thermodynamic properties [5, 7]. Another example of this potential used as an effective model is as a reference potential manifested to elaborate on the reliability of the order ambiguity parameters.

In the present chapter, the Pöschl-Teller potential of hyperbolic form [5] has been used and is given by:

$$V^{\mathit{PT}}\left(r\right)=\left[\frac{A}{{sinh}^2\left(\mathit{\alpha r}\right)}-\frac{B}{{cosh}^2\left(\mathit{\alpha r}\right)}\right],r\ge 0,$$E54

and

$$\left\{\begin{array}{c}A=\frac{{\mathrm{\hslash }}^2{\alpha }^2}{2m}\eta \left(\eta -1\right)\\ B=\frac{{\mathrm{\hslash }}^2{\alpha }^2}{2m}\lambda \left(\lambda +1\right)\end{array}\right.,$$E55

where $\alpha ,\eta ,\lambda$ are positive constants.

5.1.1 s-states $\left(\mathrm{\ell }=0\right)$

For $\mathrm{\ell }=0$, taking into account (50), the propagator of the Pöschl-Teller potential (54) becomes:

$$\begin{array}{rcl}{K}_{\mathrm{\ell }}\left(r^{"},r^{\text{'}},s\right)& =& \int \mathfrak{D}r\left(s\right)exp\left[\frac{i}{\mathrm{\hslash }}{\int }_0^{s^{"}}\left(\frac{m}{2}{\dot{r}}^2-V_{\mathit{eff}}\left(r\right)\right)\mathit{ds}\right]\\ & =& \int \mathfrak{D}r\left(s\right)exp\left[\frac{i}{\mathrm{\hslash }}{\int }_0^{s^{"}}\left(\frac{m}{2}{\dot{r}}^2-\frac{{\mathrm{\hslash }}^2}{2m}\left[\frac{\eta \left(\eta -1\right)}{{sinh}^2\left(r\right)}-\frac{\lambda \left(\lambda +1\right)}{{cosh}^2\left(r\right)}\right]\right)\mathit{ds}\right]\\ & =& \int \mathfrak{D}r\left(s\right){\mu }_{\lambda \eta }\left[sinh\left(r\right),cosh\left(r\right)\right]exp\left(\frac{\mathit{im}}{2\mathrm{\hslash }}{\int }_0^{s^{"}}{\dot{r}}^2\mathit{ds}\right)\\ & =& \underset{N\to \infty }{lim}{\left(\frac{m}{2\mathit{\pi i\epsilon }}\right)}^{\frac{N}{2}}\prod _{j=1}^{N-1}\underset{0}{\overset{\infty }{\int }}{\mathit{dr}}^{\left(j\right)}\prod _{j=1}^N{\mu }_{\lambda \eta }\left[sinh{\left(r\right)}^{\left(j\right)},cosh{\left(r\right)}^{\left(j\right)}\right]exp\left(\frac{\mathit{im}}{2\epsilon \mathrm{\hslash }}{\left(r_j-r_{j-1}\right)}^2\right),\end{array}$$E56

we use the notation $\widehat{{sinh}^2{\left(\theta \right)}^{\left(j\right)}}=sinh{\left(\theta \right)}^{\left(j\right)}sinh{\left(\theta \right)}^{\left(j-1\right)},$ when the functional measure ${\mu }_{\lambda \eta }$ given by [4, 5]:

$$\begin{array}{lll}{\mu }_{\lambda \eta }\left[sinh\left(\alpha ,r\right),cosh\left(\alpha ,r\right)\right]& =& \underset{N\to \infty }{lim}\prod _{j-1}^N{\mu }_{\lambda \eta }\left[sinh{\left(\alpha ,r\right)}^{\left(j\right)},cosh{\left(\alpha ,r\right)}^{\left(j\right)}\right]\\ & =& \underset{N\to \infty }{lim}{\left(\frac{2\mathit{\pi m}}{\epsilon \mathrm{\hslash }}\right)}^N\prod _{j=1}^N\widehat{sinh{\left(\alpha ,r\right)}^{\left(j\right)}}\widehat{cosh{\left(\alpha ,r\right)}^{\left(j\right)}}\\ & \times & exp\left(\frac{-m}{\mathit{i\epsilon }\mathrm{\hslash }}\left(\widehat{{sinh}^2{\left(\alpha ,r\right)}^{\left(j\right)}}-\widehat{{cosh}^2{\left(\alpha ,r\right)}^{\left(j\right)}}\right)\right)\\ & \times & I_{\eta -\frac{1}{2}}\left(\frac{m}{\mathit{i\epsilon }\mathrm{\hslash }}{sinh}^2\left(\alpha ,r\right)\right)\times I_{\lambda -\frac{1}{2}}\left(\frac{\mathit{im}}{\epsilon \mathrm{\hslash }}{cosh}^2\left(\alpha ,r\right)\right),\end{array}$$E57

This is a known solved problem.

Adapting Frank and Wolf’s notion, the solution of the path integral reads $2S=\eta \left(\eta -1\right),-2C=\lambda \left(\lambda +1\right)$, and by introducing the numbers $k_1,k_2$ which are defined as a function of C and S [5], by setting,

$$\left\{\begin{array}{c}{k}_1=\frac{1}{2}\left[1\pm {\left(\frac{1}{4}-2C\right)}^{\frac{1}{2}}\right]\\ k_2=\frac{1}{2}\left[1\pm {\left(\frac{1}{4}+2S\right)}^{\frac{1}{2}}\right]\end{array}\right..$$E58

The propagator $K_{\mathrm{\ell }}\left(r^{"},r^{\text{'}},T\right)$ contains discrete and continuous terms, becomes:

$$\begin{array}{lll}{K}_{\mathrm{\ell }}\left(r^{"},r^{\text{'}},T\right)& =& \sum _{n=0}^{N_M}exp\left(\frac{-{\mathit{\text{is}}}^{"}{E}_{n,\mathrm{\ell }}^{\mathit{PT}}}{\mathrm{\hslash }}\right){\mathrm{\Psi }}_{\mathrm{\ell },n}^{\ast \left(k_1,k_2\right)}\left(r^{\text{'}}\right){\mathrm{\Psi }}_{\mathrm{\ell },n}^{\left(k_1,k_2\right)}\left(r^{"}\right)\\ & +& \underset{0}{\overset{\infty }{\int }}\mathit{dk}exp\left(-{\mathit{\text{is}}}^{"}\frac{\mathrm{\hslash }{k}^2}{2m}\right){\mathrm{\Psi }}_k^{\ast \left(k_1,k_2\right)}\left(r^{\text{'}}\right){\mathrm{\Psi }}_k^{\left(k_1,k_2\right)}\left(r^{"}\right),\end{array}$$E59

we have $N_M$ indicate the maximum number of states with $0,1,\dots ,n\le N_M<k_1-k_2-\frac{1}{2}.$ The signs depend on the boundary conditions for $r\to 0$ and $r\to \infty$, respectively.

The bound states are explicitly given by [4, 5]:

$$\begin{array}{lll}{\mathrm{\Psi }}_{\mathrm{\ell },n}^{\left(k_1,k_2\right)}\left(r\right)& =& N_n^{\left(k_1,k_2\right)}{\left(sinh\left(\alpha ,r\right)\right)}^{2k_2-\frac{1}{2}}{\left(cosh\left(\alpha ,r\right)\right)}^{-2k_1+\frac{3}{2}}\\ & {\times }_2& F_1\left(-k_1+k_2+k,-k_1+k_2-k+1;2k_2;-{sinh}^2\left(\alpha ,r\right)\right)\\ & =& {\left(\frac{2n!\left(2k_1-1\right)\mathrm{\Gamma }\left(2k_1-n-1\right)}{\mathrm{\Gamma }\left(2k_2+n\right)\mathrm{\Gamma }\left(2k_1-2k_2-n\right)}\right)}^{\frac{1}{2}}{\left(sinh\left(\alpha ,r\right)\right)}^{2k_2-\frac{1}{2}}{\left(cosh\left(\alpha ,r\right)\right)}^{2n-2k_1+\frac{3}{2}}\\ & \times & P_n^{\left[2k_2-1,2\left(k_1-k_2-n\right)-1\right]}\left(\frac{1-{sinh}^2\left(\alpha ,r\right)}{{cosh}^2\left(\alpha ,r\right)}\right),\end{array}$$E60

and

$$N_n^{\left(k_1,k_2\right)}=\frac{1}{\mathrm{\Gamma }\left(2k_2\right)}\left(\frac{\left(2k-1\right)\mathrm{\Gamma }\left(k_1+k_2-k\right)\mathrm{\Gamma }\left(k_1+k_2+k-1\right)}{\mathrm{\Gamma }\left(k_1-k_2+k\right)\mathrm{\Gamma }\left(k_1-k_2-k+1\right)}\right).$$E61

the energy spectrum is also obtained by:

$$E_n^{\mathit{PT}}=-\left(\frac{{\mathrm{\hslash }}^2{\alpha }^2}{2m}\right){\left(2k-1\right)}^2=-\left(\frac{{\mathrm{\hslash }}^2{\alpha }^2}{2m}\right){\left[2\left(k_1-k_2-n\right)-1\right]}^2.$$E62

5.1.2 $\mathrm{\ell }$-states $\left(\mathrm{\ell }\ne 0\right)$

Usually, we find that the effective potential is not exactly solvable for $\mathrm{\ell }$-states $\left({}^{"}\mathrm{\ell }\ne 0^{"}\right),$ To deal with the centrifugal term $\left(\frac{1}{r^2}\right)$, we need to find a better approximate expression for this term and such approximations have been proposed as a general approximation similar to the type of Pöschl-Teller potential [8]:

$$\frac{1}{r^2}\approx F\left(r\right)={\alpha }^2\left(\frac{1}{3{cosh}^2\left(\mathit{\alpha r}\right)}+\frac{1}{{sinh}^2\left(\mathit{\alpha r}\right)}\right),$$E63

Moreover, these approximations are only valid for small values of the parameter $\alpha$ and collapse for large $\alpha$. This choice is useful and allows us to treat this hyperbolic potential.

Substituting (63) into (52) we find:

$$V_{\mathit{eff}}\left(r\right)=\frac{{\mathrm{\hslash }}^2{\alpha }^2}{2m}\left[\frac{{\eta }_1\left({\eta }_1-1\right)}{{sinh}^2\left(\mathit{\alpha r}\right)}-\frac{{\lambda }_1\left({\lambda }_1+1\right)}{{cosh}^2\left(\mathit{\alpha r}\right)}\right]+C_1,$$E64

with

$$\left\{\begin{array}{l}{\eta }_1\left({\eta }_1-1\right)=\left[\frac{2{\mathrm{\hslash }}^2{\alpha }^2}{m}{d}_1\left[{\left(\mathrm{\ell }+\frac{D}{2}-1\right)}^2-\frac{1}{4}\right]+\eta \left(\eta -1\right)\right]\\ {\lambda }_1\left({\lambda }_1+1\right)=\left[-\frac{{\mathrm{\hslash }}^2{\alpha }^2}{m}{d}_0\left[{\left(\mathrm{\ell }+\frac{D}{2}-1\right)}^2-\frac{1}{4}\right]+\lambda \left(\lambda +1\right)\right],\\ C_1=\frac{2{\mathrm{\hslash }}^2{\alpha }^2}{m}{d}_2\left[{\left(\mathrm{\ell }+\frac{D}{2}-1\right)}^2-\frac{1}{4}\right]\end{array}\right.$$E65

with the bound states being explicitly given by [5]:

$$\begin{array}{lll}{\mathrm{\Psi }}_{\mathrm{\ell },n}^{\left(k_1,k_2\right)}\left(r\right)& =& N_n^{\left(k_1,k_2\right)}{\left(sinh\left(\alpha ,r\right)\right)}^{2k_2-\frac{1}{2}}{\left(cosh\left(\alpha ,r\right)\right)}^{-2k_1+\frac{3}{2}}\\ & {\times }_2& F_1\left(-k_1+k_2+k,-k_1+k_2-k+1;2k_2;-{sinh}^2\left(\alpha ,r\right)\right)\\ & =& {\left(\frac{2n!\left(2k_1-1\right)\mathrm{\Gamma }\left(2k_1-n-1\right)}{\mathrm{\Gamma }\left(2k_2+n\right)\mathrm{\Gamma }\left(2k_1-2k_2-n\right)}\right)}^{\frac{1}{2}}{\left(sinh\left(\alpha ,r\right)\right)}^{2k_2-\frac{1}{2}}{\left(cosh\left(\alpha ,r\right)\right)}^{2n-2k_1+\frac{3}{2}}\\ & \times & P_n^{\left[2k_2-1,2\left(k_1-k_2-n\right)-1\right]}\left(\frac{1-{sinh}^2\left(\alpha ,r\right)}{{cosh}^2\left(\alpha ,r\right)}\right),\end{array}$$E66

and

$$N_n^{\left(k_1,k_2\right)}=\frac{1}{\mathrm{\Gamma }\left(2k_2\right)}\left(\frac{\left(2k-1\right)\mathrm{\Gamma }\left(k_1+k_2-k\right)\mathrm{\Gamma }\left(k_1+k_2+k-1\right)}{\mathrm{\Gamma }\left(k_1-k_2+k\right)\mathrm{\Gamma }\left(k_1-k_2-k+1\right)}\right).$$E67

the energy spectrum is also obtained by:

$$E_{n,\mathrm{\ell }}^{\mathit{PT}}=-\left(\frac{{\mathrm{\hslash }}^2{\alpha }^2}{2m}\right){\left(2k-1\right)}^2=-\left(\frac{{\mathrm{\hslash }}^2{\alpha }^2}{2m}\right){\left[2\left(k_1-k_2-n\right)-1\right]}^2.$$E68

with

$$\left\{\begin{array}{c}{k}_1=\frac{1}{2}\left[1\pm {\left(\frac{1}{4}+{\lambda }_1\left({\lambda }_1+1\right)\right)}^{\frac{1}{2}}\right]\\ k_2=\frac{1}{2}\left[1\pm {\left(\frac{1}{4}+{\eta }_1\left({\eta }_1-1\right)\right)}^{\frac{1}{2}}\right]\end{array}\right..$$E69

The energy spectrum is obtained from Eq. (69), namely

$$E_{n,\mathrm{\ell }}^{\mathit{PT}}=\begin{array}{l}-\left(\frac{{\mathrm{\hslash }}^2{\alpha }^2}{2m}\right)[2\left(\begin{array}{c}\frac{1}{2}\left[1\pm {\left(\frac{1}{4}+\left(\frac{2m}{{\mathrm{\hslash }}^2{\alpha }^2}\left[-\frac{{\mathrm{\hslash }}^2{\alpha }^2}{6m}\frac{\left[{\left(l+\frac{D}{2}-1\right)}^2-\frac{1}{4}\right]}{3}+\frac{B}{q}\right]\right)\right)}^{\frac{1}{2}}\right]\\ -\frac{1}{2}\left[1\pm {\left(\frac{1}{4}+\left(\frac{2m}{{\mathrm{\hslash }}^2{\alpha }^2}\left[\frac{{\mathrm{\hslash }}^2{\alpha }^2}{2m}\left[{\left(l+\frac{D}{2}-1\right)}^2-\frac{1}{4}\right]+\frac{A}{q}\right]\right)\right)}^{\frac{1}{2}}\right]-n\end{array}\right){-1]}^2,\end{array}$$E70

6. Duru-Kleinert method

We often introduce a coordinate transformation followed by a local time transformation to make the study much more accessible.

Let us perform the following space and time changes [9]:

$$\left\{\begin{array}{c}r=f\left(q\right)\\ \mathit{dt}=f^{\text{'}2}\left(q\right)\mathit{ds}\end{array}\right.,$$E71

These transformations allow us to transform a difficult propagator to calculate into a more manageable form.

Moreover, Green’s function relative to a given propagator allows us to derive from its poles the spectrum of energies and the corresponding wave functions from the residues at the poles. This function is obtained from the Fourier transform of the propagator $K_{\mathrm{\ell }}\left(r^{"},r^{\text{'}};T\right)$ as follows:

$$K_{\mathrm{\ell }}\left(r^{"},r^{\text{'}};T\right)=\frac{1}{2\pi \mathrm{\hslash }}\int G_{\mathrm{\ell }}\left(r^{"},r^{\text{'}};E\right)exp\left(\frac{-\mathit{iET}}{\mathrm{\hslash }}\right)\mathit{dE},$$E72

with

$$G_{\mathrm{\ell }}\left(r^{"},r^{\text{'}};E\right)=\frac{i}{\mathrm{\hslash }}{\left[f^{\text{'}}\left(q^{\text{'}}\right)f^{\text{'}}\left(q^{"}\right)\right]}^{\frac{1}{2}}\underset{0}{\overset{\infty }{\int }}{\widehat{K}}_{\mathrm{\ell }}\left(q^{"},q^{\text{'}};s^{"}\right){\mathit{ds}}^{"},$$E73

and

$${\widehat{K}}_{\mathrm{\ell }}\left(q^{"},q^{\text{'}};s^{"}\right)=\underset{q^{\text{'}}}{\overset{q^{"}}{\int }}\mathfrak{D}q\left(s\right)exp\left[\frac{i}{\mathrm{\hslash }}\underset{0}{\overset{s^{"}}{\int }}\left(\frac{m}{2}{\dot{q}}^2-f^{\text{'}2}\left(q\right)\left[V_{\mathit{eff}}\left(q\right)-E\right]-\mathrm{\Delta }V\left(q\right)\right)\mathit{ds}\right],$$E74

and the quantum correction $\mathrm{\Delta }V$ is given by:

$$\mathrm{\Delta }V\left(q\right)=\frac{{\mathrm{\hslash }}^2}{8m}\left[3\frac{{\left[f^{"}\left(q\right)\right]}^2}{{\left[f^{\text{'}}\left(q\right)\right]}^2}-2\frac{f^{"\text{'}}\left(q\right)}{f^{\text{'}}\left(q\right)}\right]$$E75

7. Energy spectrum and wave functions

7.1 Shifted Deng-Fan Oscillator potential

Another important empirical potential of diatomic molecules is the Shifted Deng-Fan Oscillator potential [7]. It was proposed since more than half century ago, but has attracted much interest lately, and this potential is the form

$$V_{\mathit{SDF}}\left(r\right)=D_1{\left(1-\frac{b}{e^{\mathit{\alpha r}}-1}\right)}^2-D_2,b=e^{\alpha r_e}-1,$$E76

where $D_2$ is the dissociation energy, $r_e$ is the position of the minimum, and $\alpha$ denotes the radius of the potential.

Here, we use for this potential a different approximation obtained using a power series decomposition [10, 11].

$$\frac{1}{r^2}\simeq \frac{1}{r_e^2}\left[C_0+\frac{C_1}{e^{\mathit{\alpha r}}-1}+\frac{C_2}{{\left(e^{\mathit{\alpha r}}-1\right)}^2}\right],$$E77

where $r_e$ is the minimum of the potential (76) and

$$\left\{\begin{array}{l}{C}_0=\frac{\left(1-\frac{{\left(1-\eta \right)}^2}{u^2}\left(\frac{4u}{1-\eta }-\left(3+u\right)\right)\right)}{u^2}\\ C_1=\frac{\left(exp\left(u\right){\left(1-\eta \right)}^2\right)}{u^3}\\ C_2=\frac{\left(exp\left(2u\right){\left(1-\eta \right)}^4\right)}{u^4}\left(3+u-\frac{2u}{1-\eta }\right)\end{array}\right.,$$

where $u=2\alpha$ $r_e,$ and $\eta =exp\left(-u\right)$.

Substituting Eqs. (77) and (78) into Eq. (52), we find

$$V_{\mathit{eff}}\left(r\right)=\frac{{\mathrm{\hslash }}^2}{2m}\left[C_0+\frac{C_{1}exp\left(-2\mathit{\alpha r}\right)}{1-exp\left(-2\mathit{\alpha r}\right)}+\frac{C_{2}exp\left(-2\mathit{\alpha r}\right)}{{\left(1-exp\left(-2\mathit{\alpha r}\right)\right)}^2}\right]+D_1{\left(1-\frac{b}{exp\left(\mathit{\alpha r}\right)-1}\right)}^2-D_2,$$E78

In a more compact for me, it reads

$$V_{\mathit{eff}}\left(r\right)=-Acoth\left(\alpha r\right)+\frac{B}{{sinh}^2\left(\alpha r\right)}+C,$$E79

where

$$\left\{\begin{array}{l}A=\frac{{\mathrm{\hslash }}^2}{2m}\mathrm{\ell }\left(\mathrm{\ell }+1\right){\alpha }^2\left(\frac{C_2}{2}-\frac{C_1}{2}\right)+D_{1}b+D_1\frac{b^2}{2}\\ B=\frac{{\mathrm{\hslash }}^2}{2m}\mathrm{\ell }\left(\mathrm{\ell }+1\right){\alpha }^2\frac{C_2}{4}+D_1\frac{b^2}{4}\\ C=\frac{{\mathrm{\hslash }}^2}{2m}\mathrm{\ell }\left(\mathrm{\ell }+1\right){\alpha }^2\left(C_0-\frac{C_1}{2}+\frac{C_2}{2}\right)+D_1+D_{1}b+D_1\frac{b^2}{2}-D_2\end{array}\right.,$$E80

Thus, the condensed form is given by:

$$K_{\mathrm{\ell }}\left(r^{"},t^{"};r^{\text{'}},t^{\text{'}}\right)=\mathfrak{D}r\left(t\right)exp\left[{\int }_{t^{\text{'}}}^{t^{"}}\left(\frac{m}{2}{\dot{r}}^2-V_{\mathit{eff}}\left(r\right)\right)\mathit{dt}\right],$$E81

the potential given by (79) is similar to the Manning-Rosen, a direct path integration is not possible, the problem can be solved with the help of the folowing space-time transformation

$$\left\{\begin{array}{l}r=F\left(q\right)=\frac{1}{\alpha }\mathit{arc}coth\left(2{coth}^2\left(q\right)-1\right)\\ \mathit{dt}={\left[F^{\text{'}}\left(q\right)\right]}^{2}d\mathfrak{s}\end{array}\right.,$$E82

According to [7], the wave function is given by

$$\begin{array}{rcl}{\chi }_{n,\mathrm{\ell }}^{S.D.F\left(k_1,k_2\right)}\left(r\right)& =& \sqrt{\alpha }{N}_n^{\left(k_1,k_2\right)}{\left(1-u\right)}^{1/2-k_1+n}{\left(u\right)}^{k_1-1-s/2-n}\\ & & {\times }_2{F}_1\left(-n,2k_1-n-1;s+1;\frac{1}{1-u}\right)\\ & =& {\left[\frac{\alpha \left(2k_1-1\right)n!\mathrm{\Gamma }\left(2k_1-n-1\right)}{\mathrm{\Gamma }\left(n+s+1\right)\mathrm{\Gamma }\left(2k_1-s-n-1\right)}\right]}^{1/2}{\left(1-e^{-2r}\right)}^{k_2}exp\left[-2r\left(k_1-\frac{s}{2}-n-1\right)\right]\\ & & \times P_n^{\left(2k_2-2n-s-2,s\right)}\left(1-2e^{-2r}\right),\end{array}$$E83

where $P_n^{\left(\alpha ,\beta \right)}$ denotes the Jacobi polynomials and $u=\frac{1}{2}\left[1-tanh\left(2\mathit{\alpha r}\right)\right],$ where $k=k_1-k_2-n.$

and

$$k_1=\frac{1}{2}\left[\left(1+\frac{1}{2}\left(s+2n+1\right)\right)+\frac{2mA}{{\alpha }^2{\mathrm{\hslash }}^2\left(s+2n+1\right)}\right],$$E84

$$k_2=\frac{1}{2}\left[1+\sqrt{1+\frac{8mB}{{\alpha }^2{\mathrm{\hslash }}^2}}\right]\equiv \frac{1}{2}\left(1+s\right),$$E85

The energy spectrum is obtained from the poles of the Green function, Eq. (82), namely

$$E_{n,l}^{\mathit{SDF}}=-\left[\frac{{\alpha }^2{\mathrm{\hslash }}^2{\left(s+2n+1\right)}^2}{8m}+\frac{2m{A}^2}{{\alpha }^2{\mathrm{\hslash }}^2{\left(s+2n+1\right)}^2}\right]+C.$$E86

7.2 Generalized inverse quadratic Yukawa potential

The generalized inverse quadratic Yukawa potential extends this concept by introducing additional parameters or modifications to the potential. These modifications can include terms that account for different types of interactions or other physical phenomena, depending on the specific context or application.

The general form of Generalized Inverse Quadratic Yukawa Potential is:

$$V_{\mathit{\text{GIQY}}}\left(r\right)=-a-b\frac{e^{-\mathit{\alpha r}}}{r}-c\frac{e^{-2\mathit{\alpha r}}}{r^2},$$E87

which means that the effective potential becomes

$$V_{\mathit{eff}}\left(r\right)=-a-b\frac{e^{-\mathit{\alpha r}}}{r}-c\frac{e^{-2\mathit{\alpha r}}}{r^2}+\frac{{\mathrm{\hslash }}^2\mathrm{\ell }\left(\mathrm{\ell }+1\right)}{2r^2}.$$E88

First of all, we deal with the centrifugal terms using the approximation [10, 11].

$$\frac{1}{r^2}=\frac{4{\alpha }^2{e}^{-2\mathit{\alpha r}}}{{\left(1-e^{-2\mathit{\alpha r}}\right)}^2},$$E89

and

$$\frac{1}{r}=\frac{2\alpha e^{-\mathit{\alpha r}}}{1-e^{-2\mathit{\alpha r}}},$$E90

putting these considerations together, we find the following:

$$V_{\mathit{eff}}\left(r\right)=-b\frac{2\alpha e^{-2\mathit{\alpha r}}}{1-e^{-2\mathit{\alpha r}}}-c\frac{4{\alpha }^2{e}^{-4\mathit{\alpha r}}}{{\left(1-e^{-2\mathit{\alpha r}}\right)}^2}+\frac{{\mathrm{\hslash }}^2\mathrm{\ell }\left(\mathrm{\ell }+1\right)}{2}\frac{4{\alpha }^2{e}^{-2\mathit{\alpha r}}}{{\left(1-e^{-2\mathit{\alpha r}}\right)}^2}-a,$$E91

$V_{\mathit{eff}}\left(r\right)$ can be reformulated as

$$V_{\mathit{eff}}\left(r\right)=Acoth\left(\mathit{\alpha r}\right)+\frac{B}{{sinh}^2\left(\mathit{\alpha r}\right)}+C,$$E92

with

$$\left\{\begin{array}{c}A=\alpha \left(2\mathit{c\alpha }-b\right);\\ B={\alpha }^2\left(\frac{{\mathrm{\hslash }}^2}{2m}\mathrm{\ell }\left(\mathrm{\ell }+1\right)-c\right);\\ C=-2c{\alpha }^2+\mathit{b\alpha }-a.\end{array}\right.$$E93

Since the difficulties of doing the integration of Eq. (53) straightforwardly, we perform a space-time transformation depending on the Duru-Kleinert method [4, 9], so we do a nontrivial change of variable $\left(r\to q\right)$ followed by time local transformation $\left(t\to s\right)$

$$\left\{\begin{array}{c}r=h\left(q\right)=\frac{1}{\alpha }argcoth\left(2{coth}^2\left(q\right)-1\right);\\ t\to s\iff \mathit{dt}={\left[h^{\text{'}}\left(q\left(s\right)\right)\right]}^2\mathit{ds}.\end{array}\right.$$E94

Putting these considerations together, we find the new Green’s function

$$G_{\mathrm{\ell }}\left(q_b,q_a;E\right)=\frac{i}{\mathrm{\hslash }}{\left[h^{\text{'}}\left(q_a\right)h^{\text{'}}\left(q_b\right)\right]}^{\frac{1}{2}}{\int }_0^{\infty }{P}_{\mathrm{\ell }}\left(q_b,q_a;S\right)\mathit{dS},$$E95

where $h^{\text{'}}$ is the derivative of h with respect to q, and the new form of the promotor is

$$P_{\mathrm{\ell }}\left(q_b,q_a;S\right)=\int \mathit{Dq}\left(s\right)exp\left[\frac{i}{\mathrm{\hslash }}{\int }_0^S\left\{\frac{m}{2}{\dot{q}}^2-h^{\text{'}2}\left(V_{\mathit{eff}}\left(q\right)-E\right)-\mathrm{\Delta }V\left(q\right)\right\}\mathit{ds}\right],$$E96

the quantum correction $\mathrm{\Delta }V\left(q\right)$ [4] is given by

$$\mathrm{\Delta }V\left(q\right)=\frac{{\mathrm{\hslash }}^2}{8m}\left(3\frac{h^{"2}}{h^{\text{'}2}}-2\frac{h^{"\text{'}}}{h^{\text{'}}}\right)=\frac{{\mathrm{\hslash }}^2}{8m}\left(\frac{1}{{cosh}^2\left(q\right)}+\frac{1}{{sinh}^2\left(q\right)}\right),$$E97

and the transformed effective potential is

$$V_{\mathit{eff}}\left(q\right)=A\left(2{coth}^2\left(q\right)-1\right)+2B\left(2{coth}^2\left(q\right)-2\right){coth}^2\left(q\right)+C,$$E98

therefore

$$\begin{array}{lll}{h}^{\text{'}2}\left(V_{\mathit{eff}}\left(q\right)-E\right)+\mathrm{\Delta }V\left(q\right)& =& \frac{{\mathrm{\hslash }}^2}{2m}\left(\frac{\frac{8\mathit{mB}}{{\alpha }^2{\mathrm{\hslash }}^2}+\frac{3}{4}}{{sinh}^2\left(q\right)}+\frac{\frac{2m}{{\alpha }^2{\mathrm{\hslash }}^2}\left(E+A-C\right)+\frac{1}{4}}{{cosh}^2\left(q\right)}\right)\\ & -& \frac{1}{{\alpha }^2}\left(E-A-C\right).\end{array}$$E99

And using the following abbreviations

$$\left\{\begin{array}{c}D=\frac{1}{{\alpha }^2}\left(E-A-C\right);\\ {\eta }^2-\frac{1}{4}=\frac{8\mathit{mB}}{{\alpha }^2{\mathrm{\hslash }}^2}+\frac{3}{4};\\ {\upsilon }^2-\frac{1}{4}=-\frac{2m}{{\alpha }^2{\mathrm{\hslash }}^2}\left(E+A-C\right)-\frac{1}{4},\end{array}\right.$$E100

which means that

$$\left\{\begin{array}{c}D=\frac{1}{{\alpha }^2}\left(E-A-C\right);\\ \eta =\pm \sqrt{1+\frac{8\mathit{mB}}{{\alpha }^2{\mathrm{\hslash }}^2}};\\ \upsilon =\pm \sqrt{-\frac{2m}{{\alpha }^2{\mathrm{\hslash }}^2}\left(E+A-C\right)},\end{array}\right.$$E101

we can rewrite the promotor as follows:

$$\begin{array}{lll}{P}_{\mathrm{\ell }}\left(q_b,q_a;s\right)& =& \int \mathit{Dq}\left(s\right)exp\left[\frac{i}{\mathrm{\hslash }}\underset{0}{\overset{S}{\int }}\left\{\frac{m}{2}{\dot{q}}^2-\frac{{\mathrm{\hslash }}^2}{2m}\left(\frac{{\eta }^2-\frac{1}{4}}{{sinh}^2\left(q\right)}+\frac{{\upsilon }^2-\frac{1}{4}}{{cosh}^2\left(q\right)}\right)\right\}\mathit{ds}\right]\\ & \times & exp\left[\frac{i}{\mathrm{\hslash }}\mathit{DS}\right],\end{array}$$E102

which is nothing but a promotor formula corresponding to a system with modified Pöschl-Teller potential and energy D [12], and accordingly, the integration over time S enables us to obtain directly the radial Green’s function related to this system

$$G_{\mathrm{\ell }}\left(q_b,q_a;D\right)={\int }_0^{\infty }{P}_{\mathrm{\ell }}\left(q_b,q_a;S\right)\mathit{dS},$$E103

thus

$$\begin{array}{lll}{G}_{\mathrm{\ell }}\left(q_b,q_a;D\right)& =& {\int }_0^{\infty }\mathit{dS}exp\left[\frac{i}{\mathrm{\hslash }}\mathit{DS}\right]\\ & \times & \int \mathit{Dq}\left(s\right)exp\left[\frac{i}{\mathrm{\hslash }}\underset{0}{\overset{S}{\int }}\left\{\frac{m}{2}{\dot{q}}^2-\frac{{\mathrm{\hslash }}^2}{2m}\left(\frac{{\eta }^2-\frac{1}{4}}{{sinh}^2\left(q\right)}+\frac{{\upsilon }^2-\frac{1}{4}}{{cosh}^2\left(q\right)}\right)\right\}\mathit{ds}\right],\end{array}$$E104

The energy spectrum is obtained from the poles of Green’s function which leads us to

$$D=-\frac{{\mathrm{\hslash }}^2}{2m}{\left(2n+\eta -\upsilon +1\right)}^2,$$E105

therefore

$$\begin{array}{lll}{E}_{n,\mathrm{\ell }}& =& -\frac{{\mathrm{\hslash }}^2{\alpha }^2}{8m}{\left(2n+1+\sqrt{1+\frac{8\mathit{mB}}{{\alpha }^2{\mathrm{\hslash }}^2}}\right)}^2\\ & -& \frac{2A^2}{\frac{{\mathrm{\hslash }}^2{\alpha }^2}{m}{\left(2n+1+\sqrt{1+\frac{8\mathit{mB}}{{\alpha }^2{\mathrm{\hslash }}^2}}\right)}^2}+C,\end{array}$$

the energy spectrum is thus

$$\begin{array}{lll}{E}_{n,\mathrm{\ell }}& =& -\frac{{\mathrm{\hslash }}^2{\alpha }^2}{8m}{\left(2n+1+\sqrt{1+4\mathrm{\ell }\left(\mathrm{\ell }+1\right)-\frac{8m}{{\mathrm{\hslash }}^2}c}\right)}^2\\ & -& \frac{2{\left(2\mathit{c\alpha }-b\right)}^2}{\frac{{\mathrm{\hslash }}^2}{m}{\left(2n+1+\sqrt{1+4\mathrm{\ell }\left(\mathrm{\ell }+1\right)-\frac{8m}{{\mathrm{\hslash }}^2}c}\right)}^2}-2c{\alpha }^2+\mathit{b\alpha }-a,\end{array}$$E106

On the other hand, the associated wave functions can be displayed as

$$\begin{array}{lll}{\psi }_n\left(r\right)& =& {\left[\left(\alpha -\frac{4\mathit{mA}}{\alpha {\mathrm{\hslash }}^2{\left(\omega +2n+1\right)}^2}\right)\frac{\left(2k_1-2n-\omega -2\right)n!\mathrm{\Gamma }\left(2k_1-n-1\right)}{\mathrm{\Gamma }\left(n+\omega +1\right)\mathrm{\Gamma }\left(2k_1-\omega -n-1\right)}\right]}^{1/2}\\ & \times & {\left(1-exp\left(-2\mathit{\alpha r}\right)\right)}^{\frac{\omega +1}{2}}exp\left(k_1-\omega /2-n-1\right)\\ & \times & P_n^{\left(2k_1-2n-\omega -2,\omega \right)}\left(1-2exp\left(-2\mathit{\alpha r}\right)\right),\end{array}$$

where $P_n^{\left(2k_1-2n-\omega -2,\omega \right)}$ are Jacobi polynomials with the notations

$$\left\{\begin{array}{l}{k}_1=\frac{1}{2}\left(1+\frac{1}{2}\left(\omega +2n+1\right)-\frac{2\mathit{mA}}{{\alpha }^2{\mathrm{\hslash }}^2\left(\omega +2n+1\right)}\right);\\ k_2=\frac{1}{2}\left(1+\omega \right),\end{array}\right.$$E107

and

$$\omega =\sqrt{1+\frac{8\mathit{mB}}{{\alpha }^2{\mathrm{\hslash }}^2}}.$$E108

7.3 Modified screened Coulomb plus inversely quadratic Yukawa potential

The Modified Screened Coulomb plus Inversely Quadratic Yukawa potential (MSC-IQY) is a combined potential energy function that incorporates both the screened Coulomb potential and the inversely quadratic Yukawa potential. This modified potential is often used in various areas of physics to describe interactions between charged particles, taking into account both screening effects and long-range Coulombic interactions. For $a=0,$ the GIQY potential reduces to Modified Screened Coulomb Plus Inversely Quadratic Yukawa potential (MSC-IQY) of the form

$$V_{\mathit{\text{GIQY}}}\left(r\right)=-b\frac{e^{-\mathit{\alpha r}}}{r}-c\frac{e^{-2\mathit{\alpha r}}}{r^2},$$E109

and the associated energy eigenvalues are obtained as

$$\begin{array}{l}{E}_{n,\mathrm{\ell }}^{\mathit{\text{GIQY}}}=-\frac{{\mathrm{\hslash }}^2{\alpha }^2}{8m}{\left(2n+1+\sqrt{1+4\mathrm{\ell }\left(\mathrm{\ell }+1\right)-\frac{8m}{{\mathrm{\hslash }}^2}c}\right)}^2\\ -\frac{2{\left(2\mathit{c\alpha }-b\right)}^2}{\frac{{\mathrm{\hslash }}^2}{m}{\left(2n+1+\sqrt{1+4\mathrm{\ell }\left(\mathrm{\ell }+1\right)-\frac{8m}{{\mathrm{\hslash }}^2}c}\right)}^2}-2c{\alpha }^2+\mathit{b\alpha }.\end{array}$$E110

7.4 Kratzer potential

The Kratzer potential [13] is a mathematical model used to describe the interaction between a particle and a central force field. It is commonly employed to study molecular systems and the vibrational motion of diatomic molecules. For $\alpha =0,a=0,b=2D_e{r}_e$, and $c=D_e{r}_e^2$, the GIQY potential (87) reduces to the Kratzer potential of the form

$$V_K\left(r\right)=-2D_e\left(\frac{r_e}{r}-\frac{1}{2}\frac{r_e^2}{r^2}\right),$$E111

where $r_e$ is the equilibrium bond length and $D_e$ is the dissociation energy.

The energy eigenvalues of the Kratzer potential are obtained as

$$E_{n,\mathrm{\ell }}^K=-\frac{b^2}{\frac{{\mathrm{\hslash }}^2}{2m}{\left(2n+1+\sqrt{1+4\mathrm{\ell }\left(\mathrm{\ell }+1\right)-\frac{8m}{{\mathrm{\hslash }}^2}c}\right)}^2},$$E112

thus

$$E_{n,\mathrm{\ell }}^K=-\frac{{\left(2D_e{r}_e\right)}^2}{\frac{{\mathrm{\hslash }}^2}{2m}{\left(2n+1+\sqrt{1+4\mathrm{\ell }\left(\mathrm{\ell }+1\right)-\frac{8m}{{\mathrm{\hslash }}^2}{D}_e{r}_e^2}\right)}^2}.$$E113

7.5 Yukawa potential

The Yukawa potential, also known as the screened Coulomb potential or the Debye-Hückel potential, is a mathematical model used to describe the interaction between charged particles with an exponential decay due to screening effects. It is commonly employed in physics to study phenomena such as electromagnetic interactions, nuclear forces, and scattering processes.

The Yukawa potential is given by the following equation (setting $a=0$ and $c=0$, Eq. (88) takes the form)

$$V_Y\left(r\right)=-b\frac{e^{-\mathit{\alpha r}}}{r},$$E114

which is known as Yukawa potential, its corresponding energy eigenvalues achieved are

$$\begin{array}{lll}{E}_{n,\mathrm{\ell }}^Y& =& -\frac{{\mathrm{\hslash }}^2{\alpha }^2}{8m}{\left(2n+1+\sqrt{1+4\mathrm{\ell }\left(\mathrm{\ell }+1\right)}\right)}^2\\ & -& \frac{2b^2}{\frac{{\mathrm{\hslash }}^2}{m}{\left(2n+1+\sqrt{1+4\mathrm{\ell }\left(\mathrm{\ell }+1\right)}\right)}^2}+\mathit{b\alpha },\end{array}$$E115

or equivalently

$$E_{n,\mathrm{\ell }}^Y=-\frac{{\mathrm{\hslash }}^2{\alpha }^2}{2m}{\left(n+1+\mathrm{\ell }\right)}^2-\frac{b^2}{\frac{{\mathrm{\hslash }}^2}{m}2{\left(n+1+\mathrm{\ell }\right)}^2}+\mathit{b\alpha },$$E116

7.6 Inversely quadratic Yukawa potential

The Inversely Quadratic Yukawa potential (IQY) is a modified version of the Yukawa potential that takes into account an additional inverse square term. As $a=0$ and $b=0$, Eq. (88) reduces to the Inversely Quadratic Yukawa potential (IQY) of the form

$$V_{\mathit{IQY}}\left(r\right)=-c\frac{e^{-2\mathit{\alpha r}}}{r^2},$$E117

the energy eigenvalue equation becomes

$$\begin{array}{lll}{E}_{n,\mathrm{\ell }}^{\mathit{IQY}}& =& -\frac{{\mathrm{\hslash }}^2{\alpha }^2}{8m}{\left(2n+1+\sqrt{1+4\mathrm{\ell }\left(\mathrm{\ell }+1\right)-\frac{8m}{{\mathrm{\hslash }}^2}c}\right)}^2\\ & -& \frac{2{\left(2\mathit{c\alpha }\right)}^2}{\frac{{\mathrm{\hslash }}^2}{m}{\left(2n+1+\sqrt{1+4\mathrm{\ell }\left(\mathrm{\ell }+1\right)-\frac{8m}{{\mathrm{\hslash }}^2}c}\right)}^2}-2c{\alpha }^2.\end{array}$$E118

7.7 Coulomb potential

The Coulomb potential is used to calculate important properties such as the electric potential, electric field, and electrostatic forces in systems involving charged particles. It forms the basis for understanding phenomena such as the behavior of ions in solutions, the interaction between charged particles in plasmas, and the structure of atoms and molecules.

When $a=0,\alpha =0$, and $c=0$, Eq. (88) reduces to the Coulomb potential of the form

$$V_C\left(r\right)=-\frac{b}{r},$$E119

the energy eigenvalues of the Coulomb potential are obtained as

$$E_{n,\mathrm{\ell }}^C=-\frac{b^2}{\frac{{\mathrm{\hslash }}^2}{2m}{\left(2n+1+\sqrt{1+4\mathrm{\ell }\left(\mathrm{\ell }+1\right)}\right)}^2},$$E120

hence

$$E_{n,\mathrm{\ell }}^C=-\frac{2m}{{\mathrm{\hslash }}^2}\frac{b^2}{2{\left(n+\mathrm{\ell }+1\right)}^2},$$E121

8. Conclusions

We have presented a rigorous treatment using the path integral approach of Feynman. We affirm that this formalism is an efficient and powerful tool for finding the propagator associated with several problems in quantum physics, particularly nonrelativistic problems. Most of these problems cannot be treated exactly, and practically no physical system can be studied without approximation methods.

In this chapter, we have adopted a two-step approach to study exponentially shaped potentials. In the first step, by introducing a judicious approximation to handle the centrifugal term, we were able to transition from solving a problem related to $\mathrm{\ell }$-states to that of the s-state. The other step involves adapting a spatio-temporal transformation by Duru-Kleinert. The use of this transformation was revisited in the reasoning process to reduce the unsolvable relative propagator to the effective potential of several potentials, specifically to the modified Pöschl-Teller potential. This problem is well known and was previously addressed within the framework of the Schrödinger formulation and the path integral. The energy spectrum and wave functions are determined in this case.

In conclusion, our method is effective in solving this type of potential. We hope to continue developing the path integral formalism not only for exponential-type potentials but also for other types and more general forms, and in other domains of physics.

Acknowledgments

The authors would like to thank the LESI laboratory of the University of Khemis Miliana for its help in carrying out this study.