The search for exact solutions of the radial Schrö dinger equation constitutes a very useful research area and is of general interest. Many efforts have been made to study the scattering and bound states of relativistic and non-relativistic cases with some physical potential models.^{[1– 4]} It is well known that exact solutions for the s-state (angular momentum ℓ = 0) cannot be generalized to the case of the state with arbitrary angular momentum. In this case, the radial Schrö dinger equation as well as relativistic Dirac and Klein– Gordon equations are solved either numerically^[5] or by approximation methods.^{[6– 12]}

Among the potentials under consideration, we mention the class of exponential-type potentials, such as the Morse, Woods– Saxon, Hulthé n, and Manning– Rosen potentials. For these potentials the Schrö dinger equation admits an exact solution for ℓ = 0.^{[13– 15]}

In the case of nonzero angular momentum states, accurate eigenvalues are reached by using an appropriate approximation scheme for the centrifugal term. Different methods have been used in this sense, such as the standard method, ^[16] the asymptotic iteration method, ^[17] supersymmetric quantum mechanics, ^[18] and the exact quantization rule method.^[8]

In this paper, we study another exponential-type potential, namely an attractive hyperbolical potential of the form

where D, α , and σ ₀ are positive parameters.

Studying the energy eigenvalues of diatomic molecules was first suggested by Schiö berg in 1986^[19] as an alternative to the Morse potential. He suggested that the potential function  (1) is more accurate than the Morse potential function in fitting the experimental Rydberg– Klein– Rees (RKR) potential energy curves and the rotational– vibrating levels for some diatomic molecules. Wang et al.^[20] showed that a simple version of the Schiö berg empirical potential function and the well-known Manning– Rosen potential energy function are the same empirical function, and the Wei potential energy function and the Tietz potential energy functions are also the same solvable empirical function.^[21]

Using the dissociation energy and the equilibrium bond length for a diatomic molecule as explicit parameters, Wang et al.^[22] generated improved expressions for two versions of the Schiö berg potential energy function. Both versions of the Schiö berg potential function are Rosen– Morse and Manning– Rosen potential functions. By choosing the experimental values of the dissociation energy, equilibrium bond length, and equilibrium harmonic vibrational frequency as inputs, they calculated the average deviations of energies calculated with the potential model from the experimental data for five diatomic molecules and found that none of the six three-parameter empirical potential energy functions is superior to the other potentials in fitting the experimental data for all of the molecules examined.

The solutions of the Schrö dinger equation for the Schiö berg potential have been obtained for the s-wave case.^[23] Motivated by the success of the approximate bound state solutions of the Feynman propagator with the Manning– Rosen and Rosen– Morse potentials, ^{[24, 25]} we attempt here to solve the general ℓ -wave propagator of the Schiö berg potential by using an improved approximation scheme of the centrifugal term that was proposed by Ikhdair et al.^[26] The Eckart potential, which can be related to the Schiö berg potential by a change of parameters, is also treated. This is used in physics^[27] as well as in chemical physics.^{[28, 29]} We recall that the Feynman Path Integrals formalism only applies to the s-wave and we are faced with difficulties for nonzero ℓ -states similar to the ones encountered with the Schrö dinger equation.

The organization of this paper is as follows. In Section  2, we look for the most relevant approximation scheme of the centrifugal barrier to be used in the formulation of the path integral for the Schiö berg potential. In Section  3, using the Duru– Kleinert method, we determine the ℓ -wave eigenvalues for the Schiö berg potential. In Section  4, we study the Eckart potential. In Section  5 we present and discuss our results for both potentials. Our conclusions are summarized in Section  6.

To deal with the centrifugal term some authors have used the approximation proposed by Greene and Aldrich, ^[11] which provides analytical results for arbitrary ℓ -wave bound states, scattering states of the Schrö dinger equation, and relativistic wave equations for different forms of exponential potentials.^{[16, 30, 31]} In this approximation scheme, here labeled as F₁, the centrifugal term is replaced by

As expected, this schematic approximation is more relevant when α is weaker. Other approximations have been proposed. For instance, Qiang^[32] has suggested:

which we label as F₂. A third scheme is due to Ikhdair in Ref.  [26], which is labeled as F₃

where the parameter C₀ = 0.0823058167837972 is a shift determined by the expansion procedures. This value of C₀ is an approximation. Indeed, the exact value of C₀ is 1/12, as reported in Ref.  [33]. This has been obtained using a power series decomposition (see Eq.  (6) of Ref.  [34]). For bound state solutions of the Schrö dinger equation with the Manning– Rosen potential and the Eckart potential, the numerical experiments showed that the analytical results obtained by employing the improved Greene– Aldrich approximation scheme are in better agreement with those obtained by using the numerical integration approach than the analytical results obtained by using the original Greene– Aldrich approximation scheme to deal with the centrifugal term.^{[35, 36]}

Moreover, in order to compare the degree of validity of the three approximations, we display in Fig.  1 the graphs of the three expressions. This clearly shows that the approximation F₃ is most suitable for the low values of α that are considered in this work. Consequently, this approximation F₃ will be used throughout the present study.

Fig.  1.

	Figure Option ViewDownloadNew Window
	Fig.  1. The plot of r2Fi, i = 1, 2, 3, with α = 0.05.

In spherical coordinates, the time-sliced representation of the Feynman propagator takes the form^[37]

P_ℓ (cos θ ) denotes the Legendre polynomials with θ ≡ (r″ , r′ ) and

with

Here, Δ r_j = r_j – r_{j– 1}, ɛ = t_j – t_{j– 1}, t′ = t₀, and t″ = t_N, and

To solve Eq.  (6) for nonzero angular momentum states, we apply the approximation F₃ (included in S_j via ɛ V_eff in Eq.  (7)). Therefore, equation  (7) becomes

In a more compact form, it reads

where

As mentioned above, we also consider the Eckart potential

which is often used in chemical physics. The Eckart potential has been investigated, especially by Dong, ^[45] Zhang et al.^[46] by means of the Nikiforov Uvarov (N.U.) method, ^[26] and Stanek, ^[47] who has used an improved approximation scheme to the centrifugal term. Here, the Eckart potential is combined with the approximation F₃, to derive the corresponding path integral.

It should be noted that the Eckart potential reduces to the Schiö berg potential by setting α ′ = 4Dσ ₀, , and a = 1/2α . Consequently, for these parameter values, the energy spectrum as well as the associated wavefunctions are written as

and

The parameters appearing in Eqs.  (32) and (33) have been defined in Eqs.  (10) and (28) and depend on ℓ .

Energies calculated by using the path integral method that is described above are reported in the following tables. The energy eigenvalues E_{n′ , ℓ} have been calculated for principal quantum numbers n′ = n+ ℓ + 1 and various angular quantum number ℓ , as in Ref.  [47]. Two different values of the parameter σ ₀ and a set of parameters α are considered. The present results for the Schiö berg potential Eq.  (30) are displayed in Tables  1 and 2, and compared to the values given by Ikhdair and Sever, ^[26] who used the N.U. method with the same approximation of the barrier. All of the results are checked against the numerical data of Lucha et al., ^[5] who calculated the energies by using MATHEMATICA. The method of Ref.  [5] gives the harmonic oscillator energies with a precision of 10^{− 5}. We found that our calculated energy eigenvalues are improved as compared to the values of Ikhdair and Sever^[26] and closer to the numerical results of Lucha et al.^[5] At higher values of α both approaches are of similar quality. The relative error between our calculations and the numerical results is of about 2 × 10^{− 4} for all values of ℓ .

Table 1.

Table 1.

Table 1. Eigenvalues – En′ , ℓ of the Schiö berg potential in atomic units (ħ = m = 1) with D = 10 and σ 0 = 0.1. States α Present Ikhdair et al.[26] Lucha et al.[5] 2p 0.1 2.61889 2.61874 2.61935   0.15 3.90579 3.90544 3.90645   0.2 5.00395 5.00331 5.00457   0.25 5.88694 5.88594 5.88725 3p 0.1 4.73556 4.73540 4.73638   0.15 6.04578 6.04543 6.04649   0.2 6.91727 6.91663 6.91733   0.25 7.48500 7.48400 7.48358 3d 0.1 3.62746 3.62699 3.62769   0.15 5.29512 5.29404 5.29510   0.2 6.47684 6.47492 6.47598 4p 0.1 6.00302 6.00287 6.00390   0.15 7.11562 7.11526 7.11589   0.2 7.71967 7.71903 7.71826 4d 0.1 5.33176 5.33129 5.33216   0.15 6.73691 6.73583 6.73642   0.2 7.54672 7.54480 7.54331 4f 0.1 4.69061 4.68965 4.69058   0.15 6.43208 6.42992 6.43112   0.2 7.43781 7.43397 7.43334 5p 0.1 6.80360 6.80345 6.80432 5d 0.1 6.37810 6.37762 6.37842 5f 0.1 5.98134 5.98063 5.98147 5g 0.1 5.62964 5.62805 5.62926 6p 0.1 7.32432 7.32416 7.32476 6d 0.1 7.04871 7.04824 7.04873 6f 0.1 6.79574 6.79479 6.79528 6g 0.1 6.57537 6.57377 6.57452

Table 1. Eigenvalues – En′ , ℓ of the Schiö berg potential in atomic units (ħ = m = 1) with D = 10 and σ 0 = 0.1.

Table 2.

Table 2.

Table 2. Eigenvalues – En′ , ℓ of the Schiö berg potential in atomic units (ħ = m = 1) with D = 10 and σ 0 = 0.2. States α Present Ikhdair et al.[26] Lucha et al.[5] 2p 0.1 1.20892 1.20876 1.20903   0.15 1.86671 1.86636 1.86689   0.2 2.52064 2.52000 2.52080   0.25 3.14766 3.14666 3.14766 3p 0.1 2.68323 2.68308 2.68358   0.15 3.67163 3.67127 3.67198   0.2 4.46579 4.46516 4.46579   0.25 5.09330 5.09231 5.09235 3d 0.1 1.57920 1.57873 1.57920   0.15 2.54880 2.54773 2.54859   0.2 3.48311 3.48119 3.48228 4p 0.1 3.75708 3.75692 3.75758   0.15 4.81251 4.81215 4.81274   0.2 5.53175 5.53111 5.53087 4d 0.1 2.95305 2.95257 2.95317   0.15 4.10518 4.10410 4.10470   0.2 5.00371 5.00179 5.00137 4f 0.1 2.07438 2.07342 2.07417   0.15 3.35838 3.35622 3.35742   0.2 4.47792 4.47408 4.47486 5p 0.1 4.54961 4.54946 4.55015 5d 0.1 3.95725 3.95677 3.95740 5f 0.1 3.31592 3.31497 3.31567 5g 0.1 2.64177 2.64017 2.64124 6p 0.1 5.13778 5.13763 5.13824 6d 0.1 4.69977 4.69929 4.69979 6f 0.1 4.22750 4.22654 4.22706 6g 0.1 3.73461 3.73301 3.73378

Table 2. Eigenvalues – En′ , ℓ of the Schiö berg potential in atomic units (ħ = m = 1) with D = 10 and σ 0 = 0.2.

For the Eckart potential, the results, Eq.  (32) are displayed in Table  3 and 4, where a comparison is made with the calculations of Stanek, ^[47] who analytically solved the Schrö dinger equation, but using another approach for the centrifugal barrier. All of the results are compared to the numerical data.^[5] Our results are almost indistinguishable from the numerical ones.^[5] Nevertheless, except in a few cases, we get results closer to the numerical estimates compared to the method of Stanek.^[47] This is due to our choice of the centrifugal barrier.

Table 3.

Table 3.

Table 3. Eigenvalues – En′ , ℓ of the Eckart potential in atomic units (ħ = m = 1) with α ′ = 1/a and β ′ = 0.0001. States α = 1/(2a) Present Stanek[47] Lucha et al.[5] 2p 0.025 0.1008358 0.1008306 0.1008358   0.050 0.0978350 0.0978345 0.0978358   0.075 0.0884143 0.0884174 0.0884183   0.100 0.0783726 0.0783776 0.0783854   0.150 0.0590391 0.0590651 0.0591059 3p 0.025 0.0401247 0.0401232 0.0401250   0.050 0.0322432 0.0322435 0.0322482   0.075 0.0235301 0.0235330 0.0235553   0.100 0.0157766 0.0157843 0.0158588   0.150 0.0039335 0.0039725 0.0044091 3d 0.025 0.0413635 0.0413638 0.0413642   0.050 0.0321870 0.0321936 0.0321973   0.075 0.0227453 0.0227792 0.0227991   0.100 0.0141878 0.0142976 0.0143675 4p 0.025 0.0184621 0.0184616 0.0184632   0.050 0.0106990 0.0106995 0.0107159   0.075 0.0044141 0.0044170 0.0045059   0.100 0.0003795 0.0003883 0.0007212 4d 0.025 0.0189189 0.0189196 0.0189216   0.050 0.0104172 0.0104280 0.0104603   0.075 0.0035230 0.0035778 0.0037658 4f 0.025 0.0190182 0.0190212 0.0189216   0.050 0.0098510 0.0098979 0.0104603   0.075 0.0021369 0.0023963 0.0037658 5p 0.025 0.0086822 0.0086820 0.0086850 5d 0.025 0.0088503 0.0088512 0.0088576 5f 0.025 0.0088178 0.0088225 0.0088297 5g 0.025 0.0086797 0.0086919 0.0086943

Table 3. Eigenvalues – En′ , ℓ of the Eckart potential in atomic units (ħ = m = 1) with α ′ = 1/a and β ′ = 0.0001.

Table 4.

Table 4.

Table 4. Eigenvalues – En′ , ℓ of the Eckart potential in atomic units (ħ = m = 1) with α ′ = 1/a and β ′ = 5 × 10– 5. States α = 1/(2a) Present Stanek[47] Lucha et al.[5] 2p 0.025 0.1064736 0.1064616 0.1064737   0.050 0.0994153 0.0994116 0.0994162   0.075 0.0891243 0.0891252 0.0891284   0.100 0.0787681 0.0787732 0.0787809   0.150 0.0592067 0.0592326 0.0592734 3p 0.025 0.0418397 0.0418361 0.0418400   0.050 0.0326962 0.0326956 0.0327011   0.075 0.0237215 0.0237237 0.0237464   0.100 0.0158740 0.0158817 0.0159559   0.150 0.0039632 0.0040020 0.0044376 3d 0.025 0.0424581 0.0424587 0.0424588   0.050 0.0324634 0.0324700 0.0324736   0.075 0.0228610 0.0228947 0.0229146   0.100 0.0142465 0.0143561 0.0144257 4p 0.025 0.0191776 0.0191762 0.0191787   0.050 0.0108685 0.0108687 0.0108852   0.075 0.0044723 0.0044750 0.0045636   0.100 0.0003980 0.0004067 0.0007380 4d 0.025 0.0193727 0.0193735 0.0193753   0.050 0.0105205 0.0105312 0.0105633   0.075 0.0035583 0.0036127 0.0038001 4f 0.025 0.0193488 0.0193516 0.0193525   0.050 0.0099252 0.0099719 0.0099876   0.075 0.0021621 0.0024207 0.0025321 5p 0.025 0.0090293 0.0090287 0.0090320 5d 0.025 0.0090698 0.0090707 0.0090768 5f 0.025 0.0089774 0.0089820 0.0089892 5g 0.025 0.0088049 0.0088169 0.0088194

Table 4. Eigenvalues – En′ , ℓ of the Eckart potential in atomic units (ħ = m = 1) with α ′ = 1/a and β ′ = 5 × 10– 5.

In this paper, path integral formalism was used to solve two exponential-type potentials. We have calculated the energy levels of the Feynman kernel with the Schiö berg and the Eckart potentials for angular momentum ℓ ≠ 0, approximating the centrifugal term. First, we have shown that the most appropriate approximation scheme of the centrifugal barrier for this type of potential is the one given by Eq.  (4). Then, the Duru– Kleinert transformation allows us to convert a nonsolvable propagator to another solvable one, for which the energy spectrum and the wave functions are known. This, allows us to determine analytically the energy spectrum of the starting potential.

Our results show that the obtained eigenvalues are in very good agreement with those given by other approximate methods and turn out to be in good agreement with the numerical data. The quality of our results proves that we can provide analytical eigenenergies and corresponding eigensolutions of the spectrum of the potential because the treatment of the barrier allows a tractable calculation leading to analytical results.

When the barrier approximation is fixed, we note that the Feynman integral method gives results closer to a numerical method than solving the Schrö dinger equation via the N.U. method.

To conclude, our method is efficient in solving this type of potential. We intend in the near future to expand the path integral method to a more general form of potentials and relativistic cases.