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1

Solution of Two-Body Bound State Problems with Confining

Potentials

M. R. Hadizadeh1∗ and Lauro Tomio1,2

1Instituto de F́ısica Teórica (IFT), Universidade Estadual Paulista (UNESP),

Barra Funda, 01140-070, São Paulo, Brazil, and

2Instituto de F́ısica, Universidade Federal Fluminense, 24210-346, Niterói, RJ, Brazil,

(Dated: December 2, 2013)

Abstract

The homogeneous Lippmann-Schwinger integral equation is solved in momentum space by using

confining potentials. Since the confining potentials are unbounded at large distances, they lead to

a singularity at small momentum. In order to remove the singularity of the kernel of the integral

equation, a regularized form of the potentials is used. As an application of the method, the mass

spectra of heavy quarkonia, mesons consisting from heavy quark and antiquark (Υ(bb̄), ψ(cc̄)), are

calculated for linear and quadratic confining potentials. The results are in good agreement with

configuration space and experimental results.

PACS numbers: 2.39.Jh, 12.39.Pn, 14.40.Pq, 14.65.-q

∗ hadizade@ift.unesp.br

1

http://arxiv.org/abs/1104.3891v1
mailto:hadizade@ift.unesp.br


I. NUMERICAL SOLUTION OF NON-RELATIVISTIC EQUATION

The solution of non-relativistic and semi-relativistic Schrödinger equation with confining

potentials is interesting in various phenomena in physics, from particle to atomic physics.

Many numerical methods have been developed to study such systems in configuration and

momentum spaces, such as recent asymptotic exact solution of two- and three-body problems

[1, 2]. In this work we use a regularization method to study the two-body systems, which

interact by confining potentials. The bound state of two equal mass particles in momentum

space and in a PW representation is described as:

ψl(p) =
1

E − p2

m

∫ ∞

0

dp′ p′2 Vl(p, p
′)ψl(p

′), (1)

Vl(p, p
′) = 2π

∫ +1

−1

dxPl(x) V (p, p
′, x). (2)

Since the integral equation (1) is singular for confining potentials, consequently the calcu-

lated energy eigenvalues would not be in agreement with the exact analytic binding energies.

To overcome this problem one can use the regularized form of confining potentials to remove

the singularity of the kernel. To this aim one can keep the divergent part of the potential

fixed after exceeding a certain distance, which creates an artificial barrier. The influence

of tunneling barrier is manifested by significant changes in the energy eigenvalues at small

distances. For numerical solution of integral equations (1) and (2) and for discretization of

continuous momentum and angle variables we have used Gauss-Legendre quadrature grids

with hyperbolic plus linear (200 mesh points) and linear (100 mesh points) mapping cor-

respondingly. The momentum integration interval [0,∞) is covered by a combination of

hyperbolic and linear mappings of Gauss-Legendre points from the interval [−1,+1] to the

intervals [0, p1] + [p1, p2]
︸ ︷︷ ︸

hyperbolic

+ [p2, p3]
︸ ︷︷ ︸

linear

as:

Phyperbolic =
1 + x

1
p1

+ ( 2
p2

− 1
p1
) x
, Plinear =

p3 − p2
2

x+
p3 + p2

2
. (3)

The used values for p1, p2 and p3 in our calculations are 1.0, 3.0, 10.0. In the following

we present the calculated energy eigenvalues for linear and quadratic harmonic oscillator

potentials and we investigate the agreement to energy eigenvalues obtained from analyti-

cal solution of Schrödinger equation. Fourier transformation of regularized form of these

2



potentials to momentum space is given by:

V (r) = a1r : V (p, p′, x) = a1rcδ
3(q)

+
a1

2 π2 q4

(

2 cos(q rc)− 2 + q rc sin(q rc)

)

(4)

V (r) = a2r
2 : V (p, p′, x) = a2r

2
cδ

3(q)

+
a2
π2 q5

(

3q rc cos(q rc) + (q2r2c − 3) sin(q rc)

)

, (5)

where the potentials are kept fixed at cut-off rc and q = |q| = |p−p′| =
√

p2 + p′2 − 2pp′x.

The PW projection of these potentials Vl(p, p
′) can be obtained by solution of Eq. (2).

In Tables (I) and (I) we have listed our numerical results for energy eigenvalues. The

S-wave energy levels for linear potential are compared with corresponding configuration

space results and are in excellent agreement with exact energies. As indicated in Table (I)

our numerical results are in excellent agreement with corresponding exact energies En,l =

(2n + l + 3
2
)~ω. In the following we test also the accuracy of our numerical calculations

TABLE I. Energy eigenvalues for a linear potential. The S-wave (P and D-waves) energy levels

are calculated for m = 1.0 (1.84), a1 = 1.0 (0.18263) and rc = 20.0.

state 1S 2S 3S 4S 5S 6S

E 2.3381 4.0879 5.5205 6.7867 7.9441 9.0226

E [3] 2.3373 4.0865 5.5190 6.7814 7.9514 9.0119

E = − a1Z0

(ma1)
1

3

† 2.3381 4.0879 5.5205 6.7867 7.9441 9.0226

state 1P 2P 3P 4P 5P 6P

E 0.8830 1.2831 1.6280 1.9454 2.2369 2.5107

state 1D 2D 3D 4D 5D 6D

E 1.1160 1.4789 1.8044 2.1041 2.3844 2.6496

† Z0 are zeros of the Airy function.

for coulomb potential. PW representation of Fourier transformation of coulomb potential

V (r) = −a−1/r to momentum space, i.e. V (p, p′, x) = −a−1

2π2 q2
, can be obtained analytically

as Vl(p, p
′) = −a−1

π p p′
Ql(

p2+p′2

2p p′
), where Ql is the Legendre polynomial of second kind. Clearly

in the calculation of Vl(p, p
′) one should overcome the moving singularity which appears in

Ql at p = p′. To avoid it, one can calculate Vl(p, p
′) by solution of integral equation (2)

3



and by using the Gauss-Legendre quadrature integration. In Table (I) we have compared

our numerical results for coulomb energy levels with corresponding exact energies. Our

numerical results confirm the degeneracy of energy levels for different values of l.

TABLE II. Energy eigenvalues of a quadratic potential for m = 1.0, a2 = 0.25 and rc = 10.0.

state 1S 2S 3S 4S 5S 6S 7S 8S 9S

E 1.500 3.500 5.500 7.500 9.500 11.500 13.500 15.500 17.500

state 1P 2P 3P 4P 5P 6P 7P 8P 9P

E 2.500 4.500 6.500 8.500 10.500 12.500 14.500 16.500 18.499

state 1D 2D 3D 4D 5D 6D 7D 8D 9D

E 3.500 5.500 7.500 9.500 11.500 13.500 15.500 17.500 19.499

TABLE III. Coulomb energy levels for m = 1 and a−1 = 1.

state n = 1 n = 2 n = 3 n = 4 n = 5

E −0.2467 −0.0619 −0.0278 −0.1568 −0.0101

En = −
ma2

−1

4n2 −0.2500 −0.0625 −0.0278 −0.1562 −0.0100

II. HEAVY QUARKONIA MASS SPECTRUM

In this section we solve the integral equation (1) to calculate the mass spectra of heavy

quarkonia, mesons consisting from heavy quark and antiquark. We consider both linear and

quadratic confinements. The momentum space representation of the regularized form of

these potentials can be obtained as:

V (r) = −
a−1

r
+ a1r + a0 :

V (p, p′, x) =

(

−
a−1

rc
+ a1rc + a0

)

δ3(q)

+
−a−1

2 π2 q2

(

1−
sin(q rc)

q rc

)

+
a1

2 π2 q4

(

2 cos(q rc)− 2 + q rc sin(q rc)

)

, (6)

V (r) = −
a−1

r
+ a2r

2 + a0 :

V (p, p′, x) =

(

−
a−1

rc
+ a2r

2
c + a0

)

δ3(q)

+
−a−1

2 π2 q2

(

1−
sin(q rc)

q rc

)

+
a2
π2 q5

(

3q rc cos(q rc) + (q2r2c − 3) sin(q rc)

)

. (7)

4



In tables (II) and (II) the calculated Bottomonium and Charmonium mass spectra are

compared with the results obtained by Faustov et al. [3] and also experimental data [4].

Our numerical results show that the regularized form of the confining potentials leads to

energy eigenvalues which are in good agreement with configuration space calculations and

also experimental data. The study of two-body bound states with other confining potentials

and also in a relativistic frame is in progress.

TABLE IV. Charmonium ψ(cc̄) mass spectrum calculated for the sum of linear and quadratic

confining potentials with the coulomb potential. The parameters of calculation for linear plus

coulomb (quadratic plus coulomb) potentials are as: a0 = −0.29 (−0.05)GeV , a−1 = 4
3αs; αs =

0.47 (0.345), a1 = 0.18GeV 2 (a2 = 0.174GeV 3), rc = 10.0 (3.0) fm and mc = 1.56 (1.55)GeV .

State linear+coulomb quadratic+coulomb Exp. [4]

Present Faustov et al. [3] Present Faustov et al. [3]

1S 3.062 3.068 3.076 3.070 3.0675

2S 3.696 3.697 3.720 3.730 3.663

3S 4.144 4.144 4.331 4.331 4.159†

1P 3.529 3.526 3.492 3.508 3.525

2P 3.997 3.993 4.108 4.095

3P 4.384 4.383 4.652 4.670

1D 3.832 3.829 3.811 3.841 3.770§

2D 4.237 4.234 4.396 4.415

† 3S1 state

§ 3D1 state

ACKNOWLEDGMENTS

We would like to thank the Brazilian agencies FAPESP and CNPq for partial support.

[1] Joseph P. Day, Joseph E. McEwen and Zoltán Papp, Few Body Syst. 47, 17 (2010).

[2] J. McEwen, J. Day, A. Gonzalez, Z. Papp and W. Plessas, Few Body Syst. 47, 227 (2010).

5



TABLE V. Same as table (II) but for Bottomonium Υ(bb̄) mass spectrum. mb = 4.93 (4.95)GeV

and αs = 0.39 for linear plus coulomb potentials. Other potential parameters are the same as

previous table.

State linear+coulomb quadratic+coulomb Exp. [4]

Present Faustov et al. [3] Present Faustov et al. [3]

1S 9.425 9.447 9.730 9.447 9.4604†

2S 10.006 10.012 10.014 10.007 10.023†

3S 10.350 10.353 10.379 10.389 10.355†

4S 10.628 10.629 10.724 10.742 10.580†

1P 9.909 9.900 9.892 9.898 9.900

2P 10.263 10.260 10.265 10.259 10.260

3P 10.546 10.544 10.594 10.593

1D 10.158 10.155 10.135 10.147

2D 10.450 10.448 10.488 10.486

† 3S1 state

[3] R. N. Faustov, V. O. Galkin, A. V. Tatarintsev, A. S. Vshivtsev, Int. J. Mod. Phys. A 15, 209

(2000).

[4] Particle Data Group (R. M. Barnett et al.), Phys. Rev. D 54, 1 (1996)

6


	Solution of Two-Body Bound State Problems with Confining Potentials
	Abstract
	I Numerical solution of non-relativistic equation 
	II Heavy quarkonia Mass spectrum
	 acknowledgments
	 References