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85-94  SQU  Journal For  Science, 14 (2009) 
 © 2009 Sultan Qaboos University 

85 

General Screened Coulomb Potential : 
Nonperturbative Solutions  

*Lalit Kumar Sharma, Phillip Monowe and Samuel Chimidza 

Department of Physics, University of Botswana, Gaborone, Botswana, *Email: 
sharmak@mopipi.ub.bw. 

  تقريبيةحلول دقيقة غير:  العام بالمسحجهد كولوم 

 اللت كومار شارما ، فيليب موني وصامويل شيميدزا 

. باستخدام حلول دقيقة غير تقريبية بالمسحأوجدت طاقات الربط لعدة ذرات متفاعلة عن طريق جهد كولومب  :خالصة
 .ات سابقةاسة توافقا تاما مع حسابوقد توافقت نتائج الدر ) r(ة مع تغير المسافة جيكما تم دراسة تغير الدالة المو

 
ABSTRACT: Using nonperturbative solutions, the binding energies for different atoms have 
been evaluated for the screened Coulomb potentials. The variation of the wavefunction with 
distance r  has also been studied. The results obtained are in excellent agreement with earlier 
calculations. 
 
KEYWORDS: Nonperturbative solutions, screened Coulomb potentials and binding energies.  

1.   Introduction 

t is well- known that there are some processes which although occurring outside the nucleus are dominated by 
distances which are small on the atomic scale (Compton wavelength distances). In atomic photo-effect for 

example, Pratt and Tseng (1972) have argued that for a wide range of photo energies, electron Compton 
wavelength distances are of primary importance. In such a case, the knowledge of screened wavefunction is 
desirable for including the effects of screening, at least for small distances r . Furthermore, internal conversion 
(see e.g. Singh and Varshni, 1984), threshold pair production (see e.g. Tseng and Pratt, 1971) and single quantum 
annihilation (see e.g. Tseng and Pratt, 1973) phenomena are also characterized by small distances on the atomic 
scale. The normalization screening theory has been applied successfully to explain the anomalously large photo-
defect cross-section in molecular hydrogen (see e.g. Cooper, 1974). There are many atomic and nuclear 
processes which are characterized by the behaviour of an electron wavefunction at the origin. One such 
phenomenon is the orbital electron capture, because in this process only the region of overlap between electron 
and nuclear wavefunction is involved.  

The screened Coulomb potential also finds importance in atomic phenomena involving electronic 
transitions. It has been treated numerically and analytically by various workers using different methods, such as 
WKBJ method (see e.g. Schiff, 1968), the quantum defect method (see e.g. Faridfathi and Sever, 2007) and 
different types of perturbation methods. Realizing the utility of the screened Coulomb potential, several 

I 



LALIT KUMAR SHARMA, PHILIP MONOWE and SAMUEL CHIMIDZA  

 86

variational calculations (see e.g. Greene and Aldrich, 1976, Lam and Varshni, 1971, Roussel and O’ Connel, 
1974, Harris, 1962) have been done and tables of energy as a function of screening parameters have been 
compiled. McEnnan et al. (1976) used analytic perturbation theory for non-relativistic cases, while Green and 
Aldrich (1976) applied a non-perturbative approach to the problem. Later Mehta and Patil (1978a), using an 

approximate and non-perturbative approach, considered a potential of the form: ( )
( )β+
−

=
r

Ze
rV

2

and found that 

S-wave energy level ( )βE , as a function of parameter β , has a singularity at the origin. These authors have 
studied the behaviour of the bound-state energy for this potential (a modified Coulomb potential), as a function 
of the parameter β . It may be of interest that this potential may also serve as an approximation to the potential 
due to a smeared charge rather than a point charge, and may be pertinent potential for the description of mesonic 
atoms. 

When a classical charged two-particle system is influenced by a plasma sea, the Coulomb potential is 
replaced by a static screened Coulomb potential, the so-called Debye potential (see e.g. Lam and Varshni, 1976), 
to explain the interaction. Ray and Ray (1980) obtained s -matrix, discrete energies and wavefunctions for the 
s - states of an exponential cosine screened Coulomb (ECSC) potential in Ecker-Weizel (EW) approximation. 
Dutt (1979) and Dutt et al. (1981) obtained bound s -state energies of an electron in ECSC potential by 
analytical method, using EW approximation. They further proposed an extension of EW approximation to treat 
the non-zero angular momentum bound-states of a class of screened Coulomb potentials and obtained discrete 
energies for the Yukawa potential. It may be of interest to note (see e.g. Alhaidri et al, 2008) that screened 
Coulomb potential also describes the shielding effect of ions embedded in plasmas.       
Motivated by the growing importance of the screened Coulomb potential, we have considered in this paper a 
general potential of the form 

( ) ( )nrr
Ze

rV
0

2

1 δ+
−=  ,                                                                  (1) 

where 0δ  is a screening parameter. 
The energy eigenvalues and wavefunctions for different values of n  for the potential (1) have been obtained by 
an approximate and a non-perturbative approach. 

2. Calculation of Eigenenergies 
The radial Schrödinger equation with potential (1) can be written as 

( )
( ) ( ) ( )rRErR

r
ll

rr
Z

dr
d

r
dr
d

r n
=⎥

⎦

⎤
⎢
⎣

⎡ +
+

+
−−

2
2

2 2
1

12
1

δ
.                                (2) 

In equation (2), we have used atomic units (unit of length 
2

2

0 me
=α  and unit of energy =

2

4me
− ) and  

( ) 00 δαδ
n= . 

Setting   ( ) ( )
r
rR

r =ψ , equation (2) is transformed to 

( ) ( ) ( )rErrV
rd

d eff ψψ =⎥
⎦

⎤
⎢
⎣

⎡
+− 12

2

2
1

 .                                                     (3) 

In equation (3), 



GENERAL SCREENED COULOMB POTENTIAL      

 87

( ) ( )
( )

21 2
1

1 r
ll

rr
Z

rV
n

eff ++
+
−

=
δ

 .                                                        (4) 

( )rV eff1   may also be written in the form 

( ) ( ) ( )
( )

( ) ( ) ( ) ⎥
⎦

⎤
⎢
⎣

⎡
+−++

+
+

++
−

=
−

−−

2
1

2
1

2221

2

111ln2

1
1ln1 nnn

n

nn

n
eff

rrr

rll
rr

rZ
rV

δδδ

δ
δδ

δ
 .                (5) 

As ( )rV eff2  tends to ( )rV
eff

1  for 1<
nrδ , i.e. ( ) 12 <<nrδ , and  hence we can neglect terms of order  

( )2nrδ  or higher in the expansions of expressions involving ( )1 nrδ+ . Thus, equation (3) with the help of 
equation (5) takes the form 
 

( ) ( )
( )

( ) ( ) ( )
( ) ( )rEr

rrr

rll
rr

rZ
rd

d

nnn

n

nn

n

ψψ

δδδ

δ

δδ
δ

=

⎥
⎥
⎥
⎥
⎥
⎥

⎦

⎤

⎢
⎢
⎢
⎢
⎢
⎢

⎣

⎡

⎥
⎦

⎤
⎢
⎣

⎡
+−++

+
+

++
−−

−

−

−

2
1

2
1

222

1

2

2

111ln2

1
1ln12

1

 .                     (6) 

Further, setting 

( ) ( )

2 21 ,
2

,
na r

E a

r r e δ

δ

ψ φ −

= −

=
                                                              (7) 

 
 equation  (6) becomes  

( )
( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

12
1

2

2 2 2

1 1
2 2

2 2 2
2 2 2

1
2 1 ln 1

1

2 ln 1 1 1

1 .
2 2

n
n

n n

n

n n n

n n

Z r rd d
an r

d r d r r r

l l r r

r r r

n a n
r r n a r r E r

δφφ φ
δ

δ δ

δ φ

δ δ δ

δ
φ δ φ φ

−
−

−

−

− −

− + −
+ +

+
+

⎡ ⎤
+ + − +⎢ ⎥

⎣ ⎦

− + − =

                                    (8) 

Now, a change of variable ( )nry δ+= 1ln  along with the approximations 

( ) ( )yyenr
n

n

y φ
δ

δφ ′⎟
⎠
⎞

⎜
⎝
⎛

=′

−

−

1

, 

 
 



LALIT KUMAR SHARMA, PHILIP MONOWE and SAMUEL CHIMIDZA  

 88

and 

( ) ( ) ( ) ( ) ( )[ ]yynnynyy
ny
ey

nr
yn

n

φφ
δ

δφ ′−−+′′−
⎥
⎥

⎦

⎤

⎢
⎢

⎣

⎡
⎟
⎠
⎞

⎜
⎝
⎛

=′′
−

−

11

21

, 

for 0,1 == ln (for the s states), transforms equation (8) to 

( ) ( ) ( ) ( ) ( ) ( ) 02211 2 =+′+′+−′′− yZyyyayyyy φ
δ

φφφ .                               (9) 

In equation (9), terms involving 2δ  (for the region 12 <<δ ) have been neglected. Further, to observe the 
usefulness of the results with 1≠n in potential (1), the study is restricted to those processes which occur in the 
region 1≅r . Thus from equation (8), 

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( )2

1
1 3 2 2

2 2

1 0,
2

Z
y y y y y y y a y y y

a
y y y

φ φ φ φ φ
δ

φ

′′ ′ ′ ′− − − + − +

− + + =
                       (10) 

 for   2=n , and  

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

2 21 3 2 2 2

2 1 0,

Z
y y y y y y a y y a y

ay y y

φ φ φ φ φ
δ

φ

⎛ ⎞′′ ′ ′ ′− + − + + + −⎜ ⎟
⎝ ⎠

− + =
                 (11) 

 for 1−=n  

2.1 Solutions for the case 1=n  

Assuming ( )yφ  to be of the form  

( ) σφ +
∞

=
∑= i
i

i yCy
0

 ,                                                                    (12) 

and substituting it in equation (9), one finds on equating the coefficients of like powers of y that 1=σ  for s -
states and the iC ’s satisfy the following recursion relation 

( )( )

( )( )21

2
121

1 ++

⎥⎦
⎤

⎢⎣
⎡

−+++
=+ ii

C
Z

aii
C

i

i
δ

.                                                        (13) 

If the series (13) is to terminate after a certain finite number of terms (i.e. 01 =+MC ), then   

( )( ) 02121 =−+++
δ
Z

aMM M .                                                          (14) 

This expression is identical with the similar expression obtained by Greene and Aldrich (1976) for =Z 1. 
Equation (14) now determines the quantity Ma and hence the energy. The wavefunction ( )rMlψ  may now be 
written as 



GENERAL SCREENED COULOMB POTENTIAL      

 89

( ) ( )[ ]∑
=

+− +=
M

M

M
M

ra
Ml rCer M

0

1ln σµ µψ ,                                          (15) 

where δ is replaced by a variational parameter µ  as trial function. Thus 
( ) raMl Mrekr µµψ −= . 

The condition of normalization finally yields, δ22 4 Mak = , where µ  has been set such that 

0=+
Ma

δ
µ . 

Now the expression (see e.g. Greene and Aldrich, 1976) 

( ) ( )
( )

⎥
⎦

⎤
⎢
⎣

⎡ +
+

+
−−= ∫

∞

22

2

0 2
1

12
1

r
ll

rr
Z

rd
d

rE
nlM δ

ψ ( )drrMlψ ,                         (16) 

yields the following for the binding energy 

⎥
⎦

⎤
⎢
⎣

⎡
−−== ∫

∞

42
][

2

1
0

1
δδ

ψψ
Z

drDE sls   ,                                              (17) 

where in the above relations the notations have the following meaning 

( )
( )

22

2

2
1

12
1

r
ll

rr
Z

rd
d

D
nl

+
+

+
−−=

δ
, 

( )reN ras s δψ δ += −− 1ln111   and ⎥⎦
⎤

⎢⎣
⎡

−=
2
1

1 δ
Z

a s . 

2.2  Solutions for the Case 2=n  

Again for 2=n  in potential (1), iC ’s satisfy the relation  

( )

( )1

1
1 2 2

2 2 2
,   with =0.

2 1

i

i

Z a
i i a C

C
i i

δ
σ+

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ + + − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦=
+

                                      (18)    

This yields for s -states and for  

0=M , 
δ
Z

a =0  , 

,1=M  ⎟
⎠
⎞

⎜
⎝
⎛

−=
5
3

51 δ
Z

a .                                                        (19) 

2.3 Solutions for the Case 1−=n   

Similarly for s -states and for 1−=n  and  0=σ , the recursion relation is 

( )

( )( )21

2222

1 ++

⎥
⎦

⎤
⎢
⎣

⎡
+⎟

⎠
⎞

⎜
⎝
⎛

−++
=+ ii

Ca
Z

aii
C

i

i
δ

 .             (20) 

 



LALIT KUMAR SHARMA, PHILIP MONOWE and SAMUEL CHIMIDZA  

 90

Equation (20) yields 

δ
Z

a =0 , 

and 

⎟
⎠
⎞

⎜
⎝
⎛

−=
4
3

21 δ
Z

a .                              (21) 

The above values of Ma finally give the expression for the wavefunctions  

( ) ( )[ ]
σ

δ δψ
+

=

− ∑ +=
MM

M

n
M

ra
Ml rCer

n
M

0
1ln .         (22) 

The wavefunctions given by equation (22) are only approximate and their resemblance to the exact 
wavefunctions depends on the extent to which ( )rV eff2 approximates ( )rV

eff
1  in the approximation 

( ) 1<<nrδ . 
3. Results and Discussions 

(A)  Using equation (17) in Table 1 the binding energy values (in keV) for 1=n  and for different atoms with 
Z  ranging between 34 and 84 have been given. The values of the screening parameters chosen and the 
corresponding experimental values have also been depicted in the same table, while Table 2 shows the 
binding energy values calculated for 1−=n . 

 
Table 1. Bound state energies E (keV) with 1=n  as a function of Z and δ  
 

Z  δ  E  
  Present study with 1n =  

in potential (1) 
Experimental (Reference 
Mehta and Patil, 1978b) 

34 0.09 -1.52 -1.27 
39 0.095 -1.84 -1.70 
44 0.100 -2.19 -2.21 
49 0.120 -2.93 -2.79 
54 0.130 -3.50 -3.46 
59 0.135 -4.12 -4.20 
64 0.16 -5.11 -5.02 
69 0.18 -6.20 -5.94 
74 0.20 -7.39 -6.95 
84 0.22 -9.22 -9.31 

 
 
 
 
 
 
 
 



GENERAL SCREENED COULOMB POTENTIAL      

 91

Table 2. Bound state energies E (keV) with 1−=n as a function of Z and δ  
 

Z  δ   E   
  Present study with 

1n =  in potential 
(1) 

By dispersion 
relation (Reference 
Mehta & Patil, 
1978b) 

By perturbative 
approach (Reference 

lyer, 1980) 

34 0.09 -1.52 -1.30 -1.36 
39 0.095 -1.84 -1.74 -1.55 
44 0.100 -2.19 -2.25 -2.12 
49 0.120 -2.92 -2.82 -2.80 
54 0.130 -3.49 -3.45 -3.53 
59 0.135 -4.11 -4.15 -4.44 
64 0.160 -5.10 -4.92 -5.22 
69 0.180 -5.81 -5.76 ----- 
74 0.200 -7.37 -6.66 ----- 
84 0.220 -9.20 -8.66 ----- 

 
 
(B) It should be noted here that the method of investigation used by us involves a minimum number of 
parameters and is quite straightforward as compared to the methods used by previous authors  ( see e.g. Greene 
and Aldrich, 1976, Rogers et al., 1970 and Roussel and O’ Connel, 1974). 

4.  Conclusions 

(A) From Table 1 it is seen that that the agreement between the calculated and the observed values is poor for 
lower values of Z but it improves for higher values of Z .  

(B) From Table 2, the results on the binding energy values for 1−=n  show a reasonably good agreement 
with the energy values obtained by Iyer (1980). It may also be mentioned here that the screened Coulomb 
potential (1) changes to a form considered by Mehta and Patil (1978a) for this value of n . The energy 
values obtained by us are comparable to those obtained in their study.  

(C) The potential considered in this paper is of general form, and the comparison of shapes obtained for the 
wavefunction ( )rMlψ  for 1=n  (Figures 1 and 2) reveals that within the interior of the atom, i.e. 1<rδ , 
shapes of all the screened Coulomb potentials are similar and correspond primarily to the Coulombic form. 
This enhances the chances of possible use of ( )rMlψ as the trial wavefunctions. 

 



LALIT KUMAR SHARMA, PHILIP MONOWE and SAMUEL CHIMIDZA  

 92

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8

ψ
 M

l (
r)

r
 

Figure 1. Unnormalised radial wave function ψMl (r) for 1s state and for n = 1 of Aluminium (Z = 13). Distances 
are in electron Compton wavelengths. 
 

0

0.5

1

1.5

0 0.1 0.2 0.3 0.4 0.5

ψ
 M

l (
r)

r
 

Figure 2. Unnormalised radial wave function ψMl (r) for 1s state and for n = 1 of Gold (Z = 79). Distances are in 
electron Compton wavelengths. 



GENERAL SCREENED COULOMB POTENTIAL      

 93

(D) Further, it is observed that there is a shift of maxima for 2=n  towards origin for increasing values of 
Z (Figure 3). For the same value of n , however, magnitude of the wavefunction is found to fall-off rapidly 
with increase in r  for higher values of Z . 

 

0

0.5

1

1.5

0.2 0.4 0.6 0.8 1 1.2 1.4

Z = 13

Z = 47

Z = 79

r

ψ
 M

l (
r)

 
Figure 3. Unnormalised radial wave function ψM l (r) for 2s state and for n = 2 of Aluminium (Z = 13), Silver 
(Z = 47) and Gold (Z = 79). Distances are in electron Compton wavelengths. 

 
(E) Finally, we conclude with the remark that fairly satisfactory results of binding energies for the potential (1) 

can be obtained only for 1=n  and that the utility of the results for the case 1≠n is restricted to the 
processes occurring in the region 1≅r . 

 

5.  References 

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COOPER, J.W. 1974. High Energy dependence of the molecular-hydrogen photo ionization cross-section. Phys. 
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DUTT, R. 1979. Bound s-state energies for the exponential cosine screened Coulomb potential in Ecker-Weizel 
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DUTT, R., RAY, A. and RAY, P.P. 1981. Bound states of screened Coulomb potentials. Phys. Lett. A. 83: 65-
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FARIDFATHI, G. and SEVER, R. 2007. Supersymmetric Solutions of PT-/non- PT-symmetric and non-
Hermitian screened Coulomb potential via Hamiltonian hierarchy inspired variation Method. J. Math 
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LALIT KUMAR SHARMA, PHILIP MONOWE and SAMUEL CHIMIDZA  

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HARRIS, G.M. 1962. Attractive Two-Body Interactions in Partially Ionised Plasmas. Phys. Rev. 125: 1131-
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β+

−=
r

Z
rV . Phys. Rev. A. 17: 43-

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Received   11 February 2009                      
Accepted   24 May 2009