IL NUOVO CIMENTO VOL. 112 B, N. 4 Aprile 1997 NOTE BREVI
Exact bound states of the relativistic Schrödinger equation
for the central potential V(r) 42aOr1bOr
1 O2S. K. BOSE
Department of Physics, University of Gorakhpur - Gorakhpur, 273009 India (ricevuto il 9 Febbraio 1994; approvato il 15 Gennaio 1997)
Summary. — Using a suitable ansatz, we obtain here a set of infinite number of bound states consisting of eigenvalues and the corresponding eigenfunctionals of the relativistic Schrödinger equation for the central potential V(r) 42aOr1bOr1 O2. For each solution, a separate relation interrelating the parameters of the potential and the orbital angular-momentum quantum number l is given. The eigenfunction(s) obtained here is (are) normalizable.
PACS. 03.65 – Quantum mechanics.
One of the important issues of quantum mechanics is to solve the Schrödinger equation (the relativistic and the nonrelativistic one) for the potentials of physical importance. Unfortunately, however, only for a few potentials the Schrödinger equation is found to be exactly solvable. In recent years, a lot of attention has been paid towards obtaining exact solution of the Schrodinger equation for the potentials of physical interest [1-6]. With the development of fast computers it is, although, possible to solve the Schrödinger equation for any potential to a desired degree of accuracy. The exact solutions, however, have an aesthetic appeal. Further, the exact solutions can serve as benchmark to test the accuracy of various nonperturbative methods.
In the present paper, we obtain a set (consisting of infinite number of eigenvalues and the corresponding eigenfunctions) of exact solutions of the Schrödinger equation for the potential
V(R) 42a r 1
b r1 O2 ,
by using an appropriate ansatz in the form of a series in r . The solutions hold when for each solution, a separate relation, interrelating the parameters of the potential and the orbital angular-momentum quantum number l is satisfied. The eigenfunctions(s) thus obtained are normalizable. The above potentials can be useful for application in particle, atomic and molecular physics.
S.K.BOSE 636
1. – Exact solution of the relativistic Schrödinger equation for the potential V(r) 42aOr1bOr1 O2
The radial part of the Schrödinger equation for the potential (a) (ˇ 4c41) is
y
m
E 2g
2a r 1 b r1 O2hn
2 2 m021 1 r2 d drg
r 2 d drh
2 l(l 11) r2z
R(r) 40 . (1)To solve the second-order differential equation (1), we use the ansatz [4]
R(r) 4exp [ar12br1 O2]
!
n 40
anrn O21n,
(2)
where a , b and n are constants to be determined later. On substitution of the ansatz (2) in the relation (1), we obtain the following three-term recurrence relation by setting the coefficient of rn O21n21 equal to zero:
An nan1 Bn 11n an 111 Cn 12n an 124 0 , (3) where An n4 b21 2 Ea 1 (n 1 2 n 1 2 ) a 1 b2, (4a) Bn n4 22 ab 1
g
n 12n1 3 2h
b (4b) and Cn n4g
n 2 1 nhg
n 2 1 n 1 1h
2 l(l 1 1 ) 1 a 2, (4c)and we have set a2
4 m022 E2, ab 4eb. We should choose
a 42
k
m2 02 E2, (5)so that R(r) is finite for r KQ.
Now if a0 is the first nonvanishing coefficient in the series (2), we should have
Cn 04 0 or n 4n64 2 1 2 6
m
g
l 1 1 2h
2 2 a2n
1 O2 ,gg
l 1 1 2h
2 2 a2F 0h
, (6)in which only n 4n1 is physically acceptable. Again, if ap is the last nonvanishing
coefficient in the series (2), we should have ap 11, ap 124 0 and then the recurrence relation (3) yields
Apn4 0 or [(b21 2 aEpl)]m022 (Epl)2( 1 b2(epl)2]24 (p 1 n11 2 ) 2
]m022 (Epl)2(3.
(7)
Also, for a nontrivial solution, the coefficients An
EXACT BOUND STATES OF THE RELATIVISTIC SCHRO¨DINGER EQUATION ETC. 637
satisfy the determinantal relation
det
N
Bn 0 A0n0
Cn 1 B1n C2n Q Q Q0
An p 21BpnN
4 0 . (8)Now the different exact solutions can be generated by setting p 40, 1 R, etc. Thus
a) for p 40, the relation (7) yields
[ (El
0)2b21 ( 2 E0la 1b2) ]m022 (E0l)2(2]24 4(n11 1 ) 2
]m022 (E0l)2(3. (9)
The condition (9), in this case, yields the energy eigenvalue
E0l4 m0
{
1 1 a2 (n11 1 )2}
21 O 2 . (10)The relations (9) and (10) imply the interrelation
{
b2 1 a 2b2 (n111 )21 2 a3m 0 (n111 )k
a21(n111 ) 2}
2 4 4 (n111 )2{
m2 0a6 (n111 )21m2}
, (11)between the parameters a , b of the potential and the orbital angular-momentum quantum number l . The eigenfunction, from the ansatz (2), can be written as
c0 , l , m(r , u , W) 4exp
y
2 am0rk
a2 1 (n11 1 ) 2 2 2 b a (n11 1 ) r 1 O2z
a 0rn1Ylm(u , W) . (12)b) For p 41, the relation (7) implies
[ (El
1)2b21 ( 2 E1la 1b2) ]m022 (E1l)2(] 4 ( 2 n11 3 ) 2
]m022 (E1l)2(3. (13)
The condition, on the other hand, now means
b2
{
g
n 11 3 4h
El 1k
m022 (E1l)2 1 a}{
g
n11 5 4h
El 1k
m022 (E1l)2 1 a}
4 (14) 4g
n11 3 4h
k
m 2 02 (E1l)2, where we have used the relations (4a)-(4c) and (5). The relations (13) and (14) can be used to find the energy eigenvalue and an interrelation between the parameters a , b of the potential and the orbital angular-momentum quantum number l . Theeigen-S.K.BOSE 638
function, from the ansatz (2), can be written as
c1 , l , m(r , u , W) 4 (15) 4 exp
y
2k
m022 (E1l)2r 2 2 El 1bk
m2 02 (E1l)2 r1 O2z
rn1(a c1 a , r1 O2) Y m l (u , W) ,where n1is given by the relation (6), and a0and a1, using the relations (3) and (4a), (4b) are related as ]m22 (E1l)2( a02 2
m
abk
m022 (E1l)21g
n11 5 4h
(E l 1) bn
a14 0 . (16)Continuing this way, we can generate a set of exact solutions although the calculations are more involved for the higher values of p .
R E F E R E N C E S
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[4] BOSES. K. and VARMAN., Phys. Lett. A., 141 (1989) 141; 147 (1990) 85; BOSES. K., J. Math. Phys. Soc., 26 (1992) 129.
[5] CHHAJLANYS. C. and MALNEVV. N., J. Phys. A, 23 (1990) 3711; SINGHL. A., SINGHS. P. and SINGH K. D., Phys. Lett. A, 148 (1990) 389.
