ARTICLE

Bound and scattering state solutions of a hyperbolic-type potentialAli Ghoumaid, Farid Benamira, and Larbi Guechi

Abstract: A hyperbolic-type potential with a centrifugal term is solved approximately using the path integral approach. Theradial Green's function is expressed in closed form, fromwhich the energy spectrum and the suitably normalizedwave functions

of bound and scattering states are extracted for (1/2) − �(1/4) � (�2�2/2�D)l(l � 1) < � < (1/2) + �(1/4) � (�2�2/2�D)l(l � 1). Besides,the phase shift and the scattering function Sl for each angular momentum l are deduced. The particular cases corresponding tothe s-waves (l = 0) and the barrier potential (� = 1) are also analyzed.

PACS Nos.: 03.65.Ca, 03.65.Db.

Résumé :Utilisant l'intégrale de parcours, nous solutionnons approximativement un potentiel de type hyperbolique avec termecentrifuge. La fonction de Green radiale est exprimée sous forme analytique et nous en tirons le spectre en énergie et les

fonctions d'onde normalisées pour les états liés et de diffusion correspondant a (1/2) − �(1/4) � (�2�2/2�D)l(l � 1) < � < (1/2) +

�(1/4) � (�2�2/2�D)l(l � 1). De plus, nous en déduisons le déphasage et la fonction de diffusion Sl pour chaque valeur dumomentcinétique l. Nous analysons aussi les cas particuliers correspondant aux ondes s (l = 0) et la barrière de potentiel (� = 1). [Traduitpar la Rédaction]

1. IntroductionThe hyperbolical potential

V(r) � D[1 � �coth (�r)]2 (1)

was proposed by Schiöberg [1] in 1986 in view of its applicationin the description of interatomic interactions emphasizing thatit gives a better fit to the experimental Rydberg–Klein–Rees(RKR) curves than the Morse potential for some diatomicmolecules. Here, D,�, and � are three adjustable positive realparameters.

The radial Schrödinger equation with this potential was ap-proximately solved [2] for any angular momentum l when theeffective potential is taken of the form

Veff(r) � V(r) ��2�2

2�

l(l � 1)

sinh2(�r)(2)

We note that this effective potential has a minimum at the point

r0 �1

2�ln �1 �

2�2 � (�2�2/�D)l(l � 1)

[� � (1/2)]2 � (�2�2/2�D)l(l � 1) � (1/4)� (3)

with value

Veff(r0) � D�1 �1

1 � (�2�2/2�D�2)l(l � 1)�2

��2�2

2�l(l � 1)

×[1 � � � (�2�2/2�D�)l(l � 1)][1 � � � (�2�2/2�D�)l(l � 1)]

[� � (�2�2/2�D�)l(l � 1)]2(4)

and as r0 > 0, the condition on the parameter � necessary for theexistence of bound states is given by

1

2� � 1

4�

�2�2

2�Dl(l � 1) � � �

1

2� � 1

4�

�2�2

2�Dl(l � 1) (5)

with D > (2�2�2/�)l(l + 1).For the special case corresponding to l = 0, this condition can

thus be simplified to

0 � � � 1 (6)

In this case, it should be pointed out that the multiparameterexponential-type potentials studied in ref. 3 with C = 0, q = 1, A =−4�D, B = 4�2D, and k = 1/2� are identical to V(r) − D(1 − �)2. Thecondition for existence of bound states A + B < 0 is equivalent tothe one defined in (6). We also note that the solution of continu-ous states for the l-wave Schrödinger equation with the sameeffective potential has been done recently [4]. The aim of thispaper is to clarify some points concerning the bound states (en-ergy levels and wave functions) obtained in refs. 2 and 4 for adiatomicmolecule in this potential. More precisely, we shall showthat the wave functions for such a system are continuous when �does not belong to the range defined by (5). Another reason forfurther study is the method used here. Treating (2) with pathintegration, we can build Green's function in closed form fromwhich we can easily extract all the eigenstates. We shall discuss insome detail the energy spectrum and thewave functions of boundand continuous states together with the phase shifts.

2. Green’s functionThe propagator for a particle of mass � in the spherically sym-

metric effective potential,

Received 9 July 2012. Accepted 28 September 2012.

A. Ghoumaid, F. Benamira, and L. Guechi. Laboratoire de Physique Théorique, Département de Physique, Faculté des Sciences Exactes, Université Mentouri,Route d'Ain El Bey, Constantine, Algeria.

Corresponding author: Larbi Guechi (e-mail: guechilarbi@yahoo.fr).

120

Can. J. Phys. 91: 120–125 (2013) dx.doi.org/10.1139/cjp-2012-0295 Published at www.nrcresearchpress.com/cjp on 1 February 2013.

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Ueff(r) � V(r) ��2

2�

l(l � 1)

r2

can be developed into a sum of partial waves of the form [5, 6]

K(r ′′, r ′;T) �1

r ′′r ′ �l�0

∞2l � 1

4Kl(r ′′, r ′;T)Pl(r

′′r ′/r ′′r ′) (7)

where Pl(r′′r ′/r ′′r ′) is a Legendre polynomial of degree l in r ′′r ′/r ′′r ′′ �

cos ′′ cos ′ � sin ′′ sin ′ cos(� ′′ � � ′) and

Kl(r ′′, r ′;T) � ��r(t)exp � i� �

0

T

��

2r2 � Ueff(r)�dt�

� limN¡∞ �

2i��(N�1)/2� �n�1

N

drnexp � i� �n�1

N�1

� �

2�( rn)2 � �Ueff(rn)�� (8)

represents the radial propagator in terms of a radial path inte-gral with the usual notation rn = r(tn), rn = (rn + rn−1)/2, rn = rn −rn−1, � = tn − tn−1 = T/(N + 1) and T is fixed. The potential Ueff(r) haseffectively a singularity at the origin (r = 0) that one can easilyeliminate by introducing for example an appropriate regulat-ing function [7] in the expression of the path integral defining thepropagator. As this singularity does not affect the final result, we canignore it to confine ourselves to the study of solutions of l-states,adopting a suitable approximation for the centrifugal potential. Fora short range potential, it turns out that the formula

1

r2≈ �2 13 �

1

sinh2(�r) for �r �� 1 (9)

is an adequate approximation to 1/r2.Taking into account (9), propagator (8) can be rewritten as

follows:

Kl(r ′′, r ′;T) � exp ��

i

��D(1 � �2) ��2�2

6�l(l � 1)�T�

× PMR(r ′′, r ′;T) (10)

where

PMR(r ′′, r ′;T) � ��r(t)exp � i� �

0

T

��

2r2 � 2�Dcoth (�r)

��2D � (�2�2/2�)l(l � 1)

sinh2(�r) �dt� (11)

is a propagator similar to that associated with the Manning–Rosen potential [8], which can be obtained from the general Mo-bius form of the Eckart potential by an appropriate choice ofparameters [9]. Recently, it has been analyzed by different authorsin the framework of the path integral [10–12] and therefore, fol-lowing a similar procedure, the partial Green's function for thehyperbolical potential can be expressed in closed form as

Gl(r ′′, r ′;E) �

i

��0

∞

dTexp i�ETKl(r ′′, r ′;T) �

i

��0

∞

dTexp i�� TPMR(r ′′, r ′;T) �

�

�2�

�(M1 � LE)�(LE � M1 � 1)

�(M1 � M2 � 1)�(M1 � M2 � 1)

× 2

1 � cosh (�r ′)

2

1 � cosh (�r ′′)(M1�M2�1)/2cosh (�r ′) � 1

cosh (�r ′) � 1

cosh (�r ′′) � 1

cosh (�r ′′) � 1(M1�M2)/2

2F1�M1 � LE, LE � M1 � 1,M1 � M2 � 1,cosh (�r�) � 1

cosh (�r�) � 1�× 2F1�M1 � LE, LE � M1 � 1,M1 � M2 � 1,

2

cosh (�r�) � 1� (12)

where we have used the following abbreviations

�LE � �1

2�

1

2��2�2�D

�2� �

M1,2 � �2��2D

�2�2� l �

1

22 ±1

2���2�2�D

�2� �

� �1

�2�E � D(1 � �2) ��2�2

6�l(l � 1)�

(13)

2F1(a,b,c;z) is the hypergeometric function and the symbols r> andr< denote max(r ′′,r ′) and min(r ′′,r ′) respectively.

3. Bound statesThe bound state energy spectrum can be obtained from the

poles of the radial Green's function (12). These poles are those of

the gamma function �(M1 – LE) found when its argument is anegative integer or zero, that is, when

M1 � LE � �nr nr � 0, 1, 2, 3,… (14)

Taking into account (13), the energy eigenvalues are then given by

Enr,l � �2�2�2

� �[(nr � �l)2 � 4k�]2

4(nr � �l)2� k(1 � �)2� �

�2�2

6�l(l � 1) (15)

where

�l �1

2(1 � �16k�2 � (2l � 1)2) k �

�D

2�2�2(16)

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To find the normalized wave functions corresponding to the lev-els Enr,l, we approximate the Gamma function �(M1 – LE) near thepoles M1 – LE ≈ –nr as follows

�(M1 � LE) ≈(�1)nr

nr!

1

M1 � LE � nr�

(�1)nr�1

nr!

4��nr,l(nr � �nr,l� �l)

(nr � �l)(E � Enr,l)(17)

and take into consideration the relation (see formula (9.131.2),p. 1043, in ref. 13)

2F1(a, b, c;z) ��(c)�(c � a � b)

�(c � a)�(c � b)2F1(a, b, a � b � c � 1;1 � z)

� (1 � z)c�a�b�(c)�(a � b � c)

�(a)�(b)× 2F1(c � a, c � b, c � a � b � 1;1 � z) (18)

We notice that the second term is null because the gammafunction �(a) is infinite (a = M1 – LE = −nr ≤ 0). This enables us towrite the contribution of bound states to the radial Green'sfunction as

Glb(r′′, r ′;E) � �

nr�0

nrmax Rnr,l� (r ′)Rnr,l

(r′′)

Enr,l � E(19)

where the radial wave functions, properly normalized, are

Rnr,l(r) � Nnr,l

exp (�2��nr,lr)(1 � e�2�r)�l

× 2F1(�nr, nr � 2�nr,l� 2�l, 2�nr,l

� 1;e�2�r) (20)

with

�nr,l�

4k� � (nr � �l)2

2(nr � �l)(21)

and Nnr,lis the normalization factor given by

Nnr,l�

2

�(2�nr,l� 1)���nr,l(nr � �nr,l

� �l)nr � �l

�(nr � 2�nr,l

� 1)�(nr � 2�nr,l� 2�l)

nr!�(nr � 2�l) �1/2 (22)

In terms of the Jacobi polynomials (see formula (8.962.1), p. 1036,in ref. 13), the wave functions can also be written as

Rnr,l(r)

� 2���nr,l(nr � �nr,l� �l)

nr � �l

nr!�(nr � 2�nr,l� 2�l)

�(nr � 2�nr,l� 1)�(nr � 2�l)�

1/2

× exp (�2��nr,lr)(1 � e�2�r)�lPnr

(2�nr,l,2�l�1)(1 � 2e�2�r) (23)

where the Jacobi polynomials Pn(�,�)(x) are defined for −1 < x < 1,

� > −1, and � > −1. As is to be expected, from the condition�nr,l

� � (1/2), it follows that bound states exist only in the rangedefined by (5).

On the other hand, it is clear that the radial wave functions (20)fulfill the boundary condition

limr¡∞

Rnr,l(r) � 0 (24)

when

�nr,l� 0 (25)

Therefore, it can be seen from (20) and (25) that the number ofdiscrete levels is equal to the largest integer satisfying theinequality

nrmax � 2�k� � �l (26)

These results coincide with those obtained by solving Schrödinger'sequation [14].

4. Scattering statesFor the values of nr superior to nr max, it is obvious that the

energy spectrum becomes continuous and contributes to the ex-pression of Green's function. To evaluate this contribution, westart by expressing (12) in the form

Glc(r ′′, r ′;E) �

i

2� �C

zdz

E � i� � (�2z2/2�)

�(M1 � LE)�(LE � M1 � 1)

�(M1 � M2 � 1)�(M1 � M2 � 1)� 2

1 � cosh (�r ′)

2

1 � cosh (�r ′′)�(M1�M2�1)/2

× �cosh (�r ′) � 1

cosh (�r ′) � 1

cosh (�r ′′) � 1

cosh (�r ′′) � 1�(M1�M2)/2

2F1�M1 � LE, LE � M1 � 1, M1 � M2 � 1;cosh (�r�) � 1

cosh (�r�) � 1�× 2F1�M1 � LE, LE � M1 � 1, M1 � M2 � 1;

2

cosh (�r�) � 1� (27)

We take for C the closed contour

C: �z � p p � [�R, R]z � Rei � (0,) (28)

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and consider the limit R ¡ ∞. By taking into account the asymptotic behavior of the hypergeometric functions, we can show that theintegral over the semicircle vanishes and (27) becomes

Glc(r ′′, r ′;E) �

i

2���∞

�∞

pdp

E�i� � (�2p2/2�)

�(M1 � LE)�(LE � M1 � 1)

�(M1 � M2 � 1)�(M1 � M2 � 1)� 2

1 � cosh (�r ′)

2

1 � cosh (�r ′′)�(M1�M2�1)/2

× �cosh (�r ′) � 1

cosh (�r ′) � 1

cosh (�r ′′) � 1

cosh (�r ′′) � 1�(M1�M2)/2

2F1�M1 � LE, LE � M1 � 1, M1 � M2 � 1;cosh (�r�) � 1

cosh (�r�) � 1�× 2F1�M1 � LE, LE � M1 � 1, M1 � M2 � 1;

2

cosh (�r�) � 1� (29)

where

�E � E �2�2�2

�(1 � �)2 �

�2�2

6�l(l � 1)

M1 � �1

2� �l � i

p

2�M2 � �

1

2� �l � i

p

2�

LE � �1

2� �4k� �

p2

4�2� �

1

2� �

(30)

With the abbreviations (30), we can rewrite (29) as well

Glc(r ′′, r ′;E) �

i

2�

1

�(2�l)�0

�∞

pdp

E�i� � (�2p2/2�)(z ′z ′′)�l��[�l � � � i(p/2�)]�[�l � � � i(p/2�)]

�[1 � i(p/�)][(1 � z ′)(1 � z ′′)i(p/2�)]

× 2F1�l � � � ip

2�, �l � � � i

p

2�, ip

�� 1;1 � z� 2F1�l � � � i

p

2�, �l � � � i

p

2�, 2�l;z� �

�[�l � � � i(p/2�)]�[�l � � � i(p/2�)]

�[1 � i(p/�)]

× [(1 � z ′)(1 � z ′′)�i(p/2�)]2F1�l � � � ip

2�, �l � � � i

p

2�,�i

p

�� 1;1 � z� 2F1�l � � � i

p

2�, �l � � � i

p

2�, 2�l;z�� (31)

with z = 2/(cosh(�r) + 1)Taking into account the identity [10]

z�(1/2)��l(1 � z)�i(p/2�)2F1�l � � � i

p

2�, �l � � � i

p

2�, 2�l;z � z�(1/2)��l(1 � z)�i(p/2�)

2F1�l � � � ip

2�, �l � � � i

p

2�, 2�l;z (32)

we can factorize the second function in (31) to obtain

Glc(r ′′, r ′;E) �

1

2�(2�l)�0

�∞

dp

E�i� � (�2p2/2�)(z ′z ′′)�l(1 � z ′)�i(p/2�)(1 � z ′′)i(p/2�)

2F1�l � � � ip

2�, �l � � � i

p

2�, 2�l;z�

× ��[�l � � � i(p/2�)]�[�l � � � i(p/2�)]

�[i(p/�)] 2F1�l � � � ip

2�, �l � � � i

p

2�, ip

�� 1;1 � z�

��[�l � � � i(p/2�)]�[�l � � � i(p/2�)]

�[�i(p/�)](1 � z�)�i(p/�)

2F1�l � � � ip

2�, �l � � � i

p

2�,�i

p

�� 1;1 � z�� (33)

Using (18) and letting

a � �l � � � ip

2�b � �l � � � i

p

2�c � 2�l (34)

we can simplify the expression in square brackets contained in(33) to arrive finally at

Glc(r′′, r ′;E) � �

0

�∞

dp

E�i� � (�2p2/2�)Rp,l

� (r ′)Rp,l(r′′) (35)

where the suitably normalized wave functions are

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Rp,l(r) �1

�2��[�l � � � i(p/2�)]�[�l � � � i(p/2�)]

�[i(p/�)] � 1

�(2�l)e�ipr

× (1 � e�2�r)�l2F1�l � � � ip

2�, �l � � � i

p

2�, 2�l;1 � e�2�r (36)

with

Ep,l ��2p2

2��

2�2�2k

�(1 � �)2 �

�2�2

6�l(l � 1) (37)

The scattering function Sl for each partial wave l is obtained fromthe asymptotic behavior of the wave function Rp,l(r) With the helpof the formula (18), it is easy to show that

Rp,l(r)¡r¡∞

1

�2��[�l � � � i(p/2�)]�[�l � � � i(p/2�)]

�[i(p/�)] �× � �[�i(p/�)]

�[�l � � � i(p/2�)]�[�l � � � i(p/2�)]e�ipr

��[i(p/�)]

�[�l � � � i(p/2�)]�[�l � � � i(p/2�)]eipr�

� �2

cos �pr � arg

�[i(p/�)]

�[�l � � � i(p/2�)]�[�l � � � i(p/2�)]� (38)

which is written yet

Rk,l(r)¡r¡∞�2

sin pr �

l

2� �l (39)

from which we can deduce the phase shift �l as

�l � (l � 1)

2� arg

�(ip/�)

�[�l � � � i(p/2�)]�[�l � � � i(p/2�)](40)

and Sl is found analytically [15] to be

Sl � e2i�l � (�1)l�1�(ip/�)�[�l � � � i(p/2�)]�[�l � � � i(p/2�)]

�(�ip/�)�[�l � � � i(p/2�)]�[�l � � � i(p/2�)](41)

The preceding expression of the phase shifts coincides with thatobtained by solving Schrödinger's equation [4] because the pres-ence of scattering states is not subject to any condition on theparameter � unlike the existence of bound states.

5. Special cases

5.1. First case: s statesSetting l = 0 in (2), the effective potential reduces to hyperbolical

potential (1). The energy spectrum and the wave functions of thebound states can be deduced from (15) and (20)

Enr � �2�2�2

� [(nr � �0)2 � 4k�]2

4(nr � �0)2� k(1 � �)2

nr � 0, 1, 2, 3,…�2�k� � �0 (42)

Rnr(r) � 2��

�nr(nr � �nr� �0)

nr � �0

�(nr � 2�nr� 1)�(nr � 2�nr

� 2�0)nr!�(nr � 2�0) �1/2 1

�(2�nr� 1)

exp (�2��nrr)(1 � e�2�r)�0

× 2F1(�nr, nr � 2�nr� 2�0, 2�nr

� 1;e�2�r) (43)

with

�nr�

4k� � (nr � �0)2

2(nr � �0)�0 �

1

2(1 � �16k�2 � 1) (44)

and

0 � � � 1 (45)

Likewise, the wave functions relative to the continuous spectrum are deduced from (36).

Rp(r) �1

�2��[�0 � � � i(p/2�)]�[�0 � � � i(p/2�)]

�[i(p/�)] � 1

�(2�0)e�ipr(1 � e�2�r)�0

2F1�0 � � � ip

2�, �0 � � � i

p

2�, 2�0;1 � e�2�r (46)

with

Ep ��2p2

2��

2�2�2k

�(1 � �)2 (47)

In this case, the phase shift (40) reduces to

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�0 �

2� arg

�(ip/�)

�[�0 � � � i(p/2�)]�[�0 � � � i(p/2�)](48)

and the scattering function (41) becomes

S0 � ��(ip/�)�[�0 � � � i(p/2�)]�[�0 � � � i(p/2�)]

�(�ip/�)�[�0 � � � i(p/2�)]�[�0 � � � i(p/2�)](49)

5.2. Second case: potential barrierBy setting � = 1, the orbital angular momentum l is null and potential (2) reduces to a potential barrier given by

V(r) �4De�4�r

(1 � e�2�r)2(50)

In this case, condition (26), which defines the number of discrete levels becomes

nr max � 2�k � 2�k �1

16�

1

2(51)

This is absurd because nr max is a positive integer. Consequently, potential (50) is characterized by the absence of bound states. Thecontinuum states have the energies Ep = �2p2/2�. The corresponding wave functions can be deduced from (36)

Rp(r) �1

�2��[(1/2) � �� � i(p/2�)]�[(1/2) � �� � i(p/2�)]

�(i(p/�)) � 1

�(1 � �� � ��)e�ipr(1 � e�2�r)(1������)/2

× 2F1 12 � �� � ip

2�,1

2� �� � i

p

2�, 1 � �� � ��;1 � e�2�r (52)

with

�± � �4k �1

4± �4k �

p2

4�2

Thus, from (40) and (41), it follows that

�0 �

2� arg

�(ip/�)

�[(1/2) � �� � i(p/2�)]�[(1/2) � �� � i(p/2�)](53)

S0 � ��(ip/�)�[(1/2) � �� � i(p/2�)]�[(1/2) � �� � i(p/2�)]

�(�ip/�)�[(1/2) � �� � i(p/2�)]�[(1/2) � �� � i(p/2�)](54)

6. ConclusionIn this paper, we have presented a complete path integral treat-

ment for a central hyperbolic potential. By using an appropriateapproximation for the centrifugal potential, we were able to de-rive the radial Green's function in closed form for an orbital an-gular momentum l. From this, we were able to obtain the energyspectrum and the normalized wave functions of bound and scat-tering states in the range of parameter �. The phase shifts and the

scattering function have been deduced from the scattering states.When � = 1, we have shown that there are no bound states.

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