Schr¨odinger Equation with the Potential V (r) = Ar 4 + Br 3 + Cr 2 + Dr 1 9 − − − − 9 9 1 n Shi-Hai Dong ∗ a J 5 Institute ofHighEnergyPhysics,P.O.Box918(4), Beijing100039, People’sRepublicofChina 1 1 v Zhong-Qi Ma 6 3 0 1 ChinaCenter forAdvancedScienceandTechnology (WorldLaboratory),P.O.Box8730,Beijing100080 0 9 andInstitute ofHighEnergyPhysics,P.O.Box918(4), Beijing100039, People’sRepublicofChina 9 / h p - t n Abstract a u q Making use of an ansatz for the eigenfunction, we obtain exact closed form : v i solutions to the Schro¨dinger equation with the inverse-power potential, V(r) = X r Ar−4 +Br−3+Cr−2+Dr−1 both in three dimensions and in two dimensions, a where the parameters of the potential A,B,C,D satisfy some constraints. PACS numbers: 03. 65. Ge. ∗ Electronic address: DONGSH@BEPC4. IHEP. AC. CN 1 1. Introduction The exact solutions to the fundamental dynamical equations play an important role inthedifferentfieldsofphysics. AsfarastheSchr¨odingerequationconcerned, theexact solutions are possible only for the several potentials and some approximation meth- ods are frequently used to arrive at the solutions. The problem of the inverse-power potential, 1/rn, has been widely carried out on the different fields of classic mechan- ics as well as on the quantum mechanics. For instance, the interatomic interaction potential in molecular physics [1-2], the inverse-power potentials V(r) = Z2α/r4 [3] − (interaction between an ion and a neutral atom) and V(r) = d d /r3 [4] (interaction 1 2 − between a dipole d and another dipole d ) are often applied to explain the interaction 1 2 between one matter and another one. The interaction in one-electron atoms, muonic and hadronic and Rydberg atoms also requires considering the inverse-power potentials [5]. Indeed, the interaction potentials mentioned above are only special cases of the inverse-power potential when some parameters of the potential vanishes. The reason we write this paper is as follows. On the one hand, O¨zcelik and Simsek discussed this potential in the three-dimensional spaces [6]. They obtained the eigen- values and eigenfunctions for the arbitrary node. Simultaneously, the corresponding constraints on the parameters of the potential were obtained. Unfortunately, they did not find that it is impossible to discuss the higher order excited state except for the ground state. In the later discussion, we will draw this conclusion and find some essen- tial mistakes occurred in their calculations even for the ground state. We recalculate the solutions to the Schr¨odinger equation with this potential in three dimensions fol- lowing their idea and correct their mistakes. On the other hand, with the advent of growth technique for the realization of the semiconductor quantum wells, the quantum mechanics of low-dimensional systems has become a major research field. Almost all of the computational techniques developed for the three-dimensional problems have al- ready been extended to lower dimensions. Therefore, we generalize this method to the two-dimensionalSchr¨odinger equationbecauseofthewideinterest inlower-dimensional fields theory. Besides, we has succeeded in dealing with the Sch¨odinger equation with 2 the anharmonic potentials, such as singular potential both in two dimensions and in three dimensions[9, 10], the sextic potential [11], the octic potential [12] and the Mie- type potential [13] by this method. We now attempt to study the Sch¨odinger equation with the inverse-power potential by the same way both in three dimensions and in two dimensions. This paper is organized as follows. In section 2, we study the three-dimensional Schr¨odinger equation with this potential using an ansatz for the eigenfunctions. The study of the two-dimensional Schr¨odinger equation with this potential will be discussed in section 3. The figures for the unnormalized radial functions are plotted in the last section. 2. Solutions in three dimensions Throughout this paper the natural unit h¯ = 1 and µ = 1/2 are employed. Consider the Schr¨odinger equation 2ψ +V(r)ψ = Eψ, (1) −∇ where here and hereafter the potential V(r) = Ar−4 +Br−3 +Cr−2 +Dr−1, A > 0, D < 0. (2) Let ψ(r,θ,ϕ) = r−1R (r)Y (θ,ϕ), (3) ℓ ℓm where ℓ and E denote the angular momentum and the energy, respectively, and the radial wave function R (r) satisfies ℓ d2R (r) ℓ(ℓ+1) ℓ + E V(r) R (r) = 0. (4) dr2 " − − r2 # ℓ O¨zcelik and Simsek [6] make an ansatz for the ground state R0(r) = exp[g(r)], (5) ℓ where a g(r) = +br +clnr, a < 0, b < 0. (6) r 3 After calculating, one can obtain the following equation d2R0(r) d2g(r) dg(r) 2 ℓ + R0(r) = 0. (7) dr2 − dr2 dr !  ℓ   Compare Eq. (7) with Eq. (4) and obtain the following sets of equations a2 = A, b2 = E, (8a) − 2bc = D, 2a(1 c) = B, (8b) − 1 C +ℓ(ℓ+1) = c2 2ba c. (8c) − 4 − − It is not difficult to obtain the value of the parameter a from Eq. (8a) written as a = √A. In order to retain the well-behaved solution at r 0 and at r , they ± → → ∞ choose negative sign in a, i. e. a = √A. According to this choice, they arrive at a − constraint on the parameters of the potential from Eq. (8c) written as B2 B 2AD C = + + ℓ(ℓ+1). (9) 4A 2√A B +2√A − Then the energy is read as 1 2 ± E = C +ℓ(ℓ+1) [C +ℓ(ℓ+1)]2 2BD . (10) 0 −16A ± − (cid:26) q (cid:27) It is readily to find that Eq. (10) is a wrong result. From Eqs. (6) and (8b), as we know, since the parameter b is negative, when we calculate the energy E from Eq. (8a), we only take the b as a negative value, so that Eq. (10) only takes the negative sign. Actually, it is not difficult to obtain the corresponding values of the parameters for the g(r) from Eq. (8), i. e. B +2√A D√A c = , b = . (11) 2√A B +2√A The eigenvalue E, however, will be simply expressed as from Eq. (8a) AD2 E = . (12) −B2 +4A+4B√A The corresponding eigenfunction Eq. (5) can now be read as 1 R0 = N rcexp a+br , (13) ℓ 0 r (cid:20) (cid:21) 4 where N is the normalized constant and here and hereafter the parameters a, b and c 0 are given above. After their discussing the ground state, O¨zcelik and Simsek continue to study the first excited state. They make the ansatz for the first excited state, R1(r) = f(r)exp[g(r)], (14) ℓ where g(r) is the same as Eq. (6) and f(r) = r α , where α is a constant. For 1 1 − short, it is readily to find from Eq. (14) that the radial wave function R1(r) satisfies ℓ the following equation ′′ ′ ′ f(r) +2g(r)f(r) R1(r)′′ g(r)′′ +(g(r)′)2 + R1(r) = 0, (15) ℓ −" f(r) !# ℓ where the prime denotes the derivative of the radial wave function with respect to the variable r. Compare Eq. (15) with Eq. (4) and obtain the following sets of equations 2b 2bc+D+b2α +eα = 0, (16a) 1 1 − − b2 E = 0, a2α Aα = 0, (16b) 1 1 − − − a2 +A+2aα Bα 2acα = 0, (16c) 1 1 1 − − − 2ab c c2 +C +ℓ(ℓ+1)+2bcα Dα = 0, (16d) 1 1 − − − B +2ac 2abα cα +c2α Cα ℓα ℓ2α = 0, (16e) 1 1 1 1 1 1 − − − − − it is not hard to obtain the following sets of equations from Eqs. (16a-16c) B +2√A E = b2, a2 = A, c = , (17a) − 2√A D√A b = , (17b) B +4√A where the constant α = 0 and it is determined by Eqs. (16d) and (16e). Furthermore, 1 6 it is evident to find that Eq. (17b) does not coincide with Eq. (11) with respect to the same parameter b, which will lead to the their wrong calculation for the first excited state. In fact, they obtained two different relations during their calculation through 5 the compared equation, i. e. D = 2bc (see Eq. (9) in [6]) and D = 2b(c+1) (see Eq. (16) in [6]). The parameter D does not exist if the parameter b is not equal to zero. It is another main mistaken that arises their wrong result, that’s to say, it is impossible to discuss the first excited state for the Schr¨odinger equation by this method. We only discuss the ground state by this simpler ansatz method as mentioned above. As a matter of fact, the normalized constants N can be calculated in principle from 0 the normalized relation ∞ R0 2dr = 1. (18) | ℓ| Z0 In the course of calculation, making use of the standard integral [14](Reλ > 0, Reλ > 1 2 0 and Reν > 0) ∞ λ ν/2 rν−1exp[ (λ r +λ r−1)]dr = 2 2 K (2 λ λ ), (19) 1 2 ν 1 2 Z0 − λ1! q which implies 1 N = , (20) 0 2(ab)2c2+1K2c+1(4√ab)   where the values of the parameters b,c and a are given by Eq. (11) and √A, respec- − tively. The figure 1 for the unnormalized radial eigenfunction in three dimensions is plotted in the last section. 3. Solutions in tow dimensions We now generalize this method to the two-dimensional Schr¨odinger equation. Con- sider Schr¨odinger equation with a potential V(r) that depends only on the distance r from the origin 1 ∂ ∂ 1 ∂2 Hψ = r + ψ +V(r)ψ = Eψ. (21) r∂r ∂r r2∂ϕ2! Let ψ(r,ϕ) = r−1/2R (r)e±imϕ, m = 0,1,2,..., (22) m where the radial wave function R (r) satisfies the following radial equation m d2R (r) m2 1/4 m + E V(r) − R (r) = 0, (23) dr2 " − − r2 # m 6 where m and E denote the angular momentum and energy, respectively. For the solution of Eq. (23), we make an ansatz [6-13] for the ground state R0 (r) = exp[g (r)], (24) m m where a 1 g (r) = +b r+c lnr. (25) m 1 1 r After calculating, we arrive at the following equation d2R0 (r) d2g (r) dg (r) 2 m m + m R0 (r) = 0. (26) dr2 − dr2 dr !  m   Compare Eq. (26) with Eq. (23) and obtain the following sets of equations a2 = A, b2 = E, (27a) 1 1 − 2b c = D, 2a (1 c ) = B, (27b) 1 1 1 1 − 1 C +m2 = c2 2b a c . (27c) − 4 1 − 1 1 − 1 It is not difficult to obtain the values of the parameters a from Eq. (27a) written 1 as a = √A. Likely, in order to retain the well-behaved solution at r 0 and at 1 ± → r , we choose negative sign in a , i. e. a = √A. According to this choice, Eq. 1 1 → ∞ − (27b) will give the other parameter values as B +2√A D√A c = , b = . (28) 1 1 2√A B +2√A Besides, it is readily to obtain from Eq. (27c) that B2 B 2AD C = + + (m2 1/4), (29) 4A 2√A B +2√A − − which istheconstraint ontheparameters forthetwo-dimensional Schr¨odinger equation with the inverse-power potential. The eigenvalue E, however, will be given by Eq. (27a) as AD2 E = . (30) −B2 +4A+4B√A 7 The corresponding eigenfunction Eq. (24) can now be read as 1 R0 = Nrc1exp a +b r , (31) m r 1 1 (cid:20) (cid:21) Similarly, the normalized constants N can be calculated in principle from the nor- malized relation ∞ R0 2dr = 1. (32) | m| Z0 According to Eq. (19), we can obtain 1 N = , (33) 2(ab11)2c12+1K2c1+1(4√a1b1)   where the values of the parameters a ,b and c are given above. The figure 2 for the 1 1 1 unnormalized radial eigenfunction in two dimensions is plotted in the last section. Considering the values of the parameters of the potential, we fix them as follows. The values of parameters A,C,D are first fixed, for example A = 4.0,C = 2.0 and D = 2.0, the value of the parameter B is given by Eq. (10) and Eq. (29) for the − cases both in three dimensions and two dimensions for ℓ = 0 and m = 0, respectively. By this way, the parameter B turns out to B = 5.87 in three dimensions and B = 5.65 in two dimensions, respectively. The ground state energy corresponding to these values are obtained as E = 0.164 for the case in three dimensions and E = 0.172 for the − − case in two dimensions. Actually, when we study the properties of the ground state, as we know, the unnormalized radial wave functions do not affect the main features of the wave functions. We have plotted the unnormalized radial wave functions in figures 1 and 2 for the cases both in three dimensions and in two dimensions, respectively. With respect to figures 1 and 2, it is easy to find that they are similar to each other, which stems from the same values of the angular momentum ℓ = 0 and m = 0. They will be different if we take the different values of the angular momentum in the course of calculations. In conclusion, we obtain the exact analytic solutions to the Schr¨odinger equation with the inverse-power potential V(r) = Ar−4 +Br−3 +Cr−2 +Dr−1 using a simpler ansatz for the eigenfunction both in three dimensions and in two dimensions, and 8 simultaneously the constrains on the parameters of the potential are arrived at from the compared equations. Finally, we remark that this simple and intuitive method can be generalized to other potential. The study of the Schro¨dinger equation with the asymmetric potential is in progress. Acknowledgments. This work was supported by the National Natural Science Foundation of China and Grant No. LWTZ-1298 from the Chinese Academy of Sci- ences. References [1] G.C.Maitland, M.Rigby, E. B.Smith andW.A.Wakeham, Intermolecular forces (Oxford Univ. Press, Oxford, 1987. [2] R. J. LeRoy and W. Lam, Chem. Phys. Lett. 71, 544 (1970); R. J. LeRoy and R. B. Bernstein, J. Chem. Phys. 52, 3869(1970). [3] E. Vogt and G. H. Wannier, Phys. Rev. 95, 1190(1954). [4] L. D. Laudau and E. M. Lifshitz, Quantum mechanics, Vol. 3, 3rd Ed. (Pergamon, Oxford, 1977); D. R. Bates and I. Esterman, Advances in atomic and molecular physics, Vol. 6 (Academic Press, New York, 1970). [5] B. H. Bransden and C. J. Joachain, Physics of atomics and molecules(Longman, London, 1983). [6] S. O¨zcelik and M. Simsek, Phys. Lett. 152, 145(1991). [7] R. S. Kaushal and D. Parashar, Phys. Lett. A 170, 335(1992). [8] R. S. Kaushal, Ann. Phys. (N. Y. )206, 90(1991). [9] Shi-Hai Dong and Zhong-Qi Ma, Schr¨odinger Equation with the Potential V(r) = ar2 +br−4 +cr−6 in Two Dimensions, accepted by J. Phys. A (in press). 9 [10] Shi-Hai Dong, Xi-Wen Hou and Zhong-Qi Ma, Schr¨odinger Equation with the Potential V(r) = ar2 +br−4 +cr−6 , submited to J. Phys. A. [11] Shi-HaiDongandZhong-QiMa, Exact solutionstotheSchr¨odinger Equationwith the Sextic Potential in Two Dimensions, submitted to J. Phys. A. [12] Shi-Hai Dong and Zhong-Qi Ma, An Exact Solution to the Schr¨odinger Equation with the Octic Potential in two dimensions, submitted to Il Nuovo Cimento B. [13] Shi-Hai Dong and Zhong-Qi Ma, Exact Solutions to the Schr¨odinger Equation with the Mie-type Potential in two dimensions, submitted to Il Nuovo Cimento B. [14] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series and products (Aca- demic Press, New York, 1965) p. 342. 10