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Bound states of the exponential cosine screened Coulomb potential
Volume 83A, number 8
PHYSICS LETTERS
22 June 1981
BOUND STATES F ThE EXPONENTIAL COSINE SCREENED COULOMB POTENTIAL Aparna RAY and ~ritam P. RAY Department of Phys4~s,Visva-Bharati University, Santiniketan,
W.B. 731235, India
Received 27 Noyem r 1980
Revised manuscript r~ceived30 March 1981
A generalized yir 1 theorem and the Heilnian—Feynman theorem have been used to calculate perturbatively the energy levels, expectation v4ues of an arbitrary power of the position coordinates and normalization of the wave function. The method gives excell t results for low values of the screening parameter (X ~
The exponential cosi4e screened Coulomb (ECSC) potential with screening ~arameter A, V(r) = —r~e ~‘cosAr, has been treated by seve al approximation methods [1—4].In this note we a ply the generalized virial theorem (GVT) and the ellman—Feynman theorem (HFT) [5] to calculate p rturbatively the bound-state energy levels correct to y order of A without using perturbed wave function In this approach which is based on the behaviour the potential at small distances, we get an extra b nus in that the expectation values of an arbitrary po er of the position coordinate r are readily obtain correct to any order of A. To the best of our know dge expectation values of .
-
BR nl = EniRni, where H= d2/dr2 r~d/dr + Veff(r),
(1)
Veff(r) = V(r) + 1(1 + 1)/2r’. Following Kramers [12] we multiply eq. (1) by
(3)
1
rm+ dRni/d1 ~(m + l)rmR~~, and integrate over r from 0 to infinity. This leads after a few integrations by parts to the recurrence formula —
1(~51+1\2aT2z
~J ~nl1’m,—2l—i
~
<
dr) rm+l~v eff
(4
2
m—’2
i +
2(m
+
l)((Veff
E~
—
1)rm> ~m(m 1)(r )~ which holds for all m —21 1. In (4) the angular brackets ( ) represent the expectation value in the state Rni and the left-hand side arises from calculations of quantities at the lower limit of integration, where R~, N~1r’.N~1is the normalization of the wave function. Since for m = 0 eq. (4) reduces to the familiar virial theorem, we will call eq. (4) the generalizedvirial theorem (GVT). It may be noted that eq. (4) has also been derived along similar lines in ref. [13] - A similar relation for s states has also been derived very recently [14]. In order to develop the perturbation theory, we assume that the ECSC potential may be expanded as [15,16] —
physically important obs rvables other than energy have not been calculated rher for the ECSC potential. One could, however, estimate them by using the Ecker—Weizel wave func ions [4]. It is worth noting that such a perturbative pproach has been used earlier to analyse only the ener levels and expectation values of powers of radial c ordinates for other potentials [6—8].In this letter we show that the GVT and the HFT together can pr fitably be used to calculate the normalization of the ye function also. In particular, the famous Ferm —Segrè formula [9—11]for the probability density a the origin for s states follows very easily from the theorems. The radial Schrödinge equation for the ECSC p0tential may be written as in au) 0 031—9163/81/0000—0
(2)
—
—~
0/$ 02.50 © North-Holland Publishing Company
~
—
—
383
Volume 83A, number 8
22 June 1981
PHYSICS LETTERS
V(r)=—r~e~cosAr— ~
(5)
V~Akr~,
k=0
where
Vk = (_i/~2~~/2cos (7rk/4)/k!
(6)
Furthermore, we assume that the energy Eni and the expectation value (rm) can be expanded as
continuous variable and then apply the HFT to obtam 2). (14) aE~1/al=~V~ff/al)=!~(2l+ i)(r It is worth noting here that eq. (14) has been used very recently in a different context [14]. With the use of (7) and (8), eq. (14) yields C~”~’~—~-— —2 21+1 a7
En! =
,~ E~Ak,
(15)
(7) Thus using a computer, Eni and (rm) (m ~‘ —2! 2) may be calculated correct to any order of A with the —
~C(k)Xk, m
(rm> =
(8)
k=o
where ~ [5] gives
=
—n2/2 and
~k)
=
60k- Now the HFT
help of eqs. (11), (12), (13) and (15). In the following, however, we give an expression of the s-state energy for the ECSC potential correct to sixth order of A: Eno = + A ~n2 (Sn2 + l)A3 —
aEni/aA = (ö V/aA).
(9)
—
+ ~n4(.7n2 + 5)A4
—
~n4(2ln4+ 35n2
+
4)A5
With the use of (5), (7) and (8), eq. (9) leads to k
kE~~ =
—
~
s—i
(10)
-
Thus both Eni and (rm) would be known, once (which are the expectation values (rm)in the kth order) are determined. On substitution of (3), (5), (7) and (8) in eq. (4) we find after a little algebra, the recurrencerelations [8] withm>—21 —1:
—thn6(143n4+345n2+28)X6+.... (16) The expectation values of r and r2 in the ns state are given by (r)~ 2 + ~n6(l5n2 <2> 0 =~n i~n2(5n2+ 1)
+
21)A3
+
...
(17) (18)
+~n6(i43n4+345n2 +28)X3
+....
n2(A 1 C~.1+ A2C~°.~2),
(ii)
Normalization N~1for any state Rni(r) can also be calculated from eq. (4):
(12) and fork>2, =
+
]
~2 [A1c~’±1 + A2C~2
k
__________
~
+
2m + s + 1 V m+l 5
2N~
~(2l +
= dV/dr> 2~> 4l(r2~ V). (19) + 41 ~1(r Restricting ourselves to the s-state normalization in this letter, we have l)
1
~,—2i
—
(13)
m+s—i
R~
with
0(0)= N~0= 2(dV/dr>. A1
=
(2rn
+
1)/(m +
With expansion (5) for V, eq. (20) yields
1),
A2m(rn— 1)/4—ml(l+1)/(m+1). Using the recurrence relatioi~s(11), (12) and (13) it is possible to compute (Tm> for all m > —1 [8]. For all 2>, which obtained from thetoabove recurpowers —2 > ismnot—2! 2, we need determine (r rence relations. Following ref. [12] we treat 1 as a ~
384
(20)
—
k
E s0 ~~kV8s
R~0(0)= —2 k=0
—
1)C~~.
(21)
Explicit calculation to order six of A of the right-hand side of eq. (21) shows that
Volume 83A, number 8
PHYSICS LETTERS
R20(o) 4 dEn0/dfl, (22) which is the famous Feri~ni—Segrèformula [9—11,14]. It may be noted here th~tprevious work [101 described eq. (22) as exact only for the Coulomb and Hulthen potentials. Our ¶terivation of eq. (22) 6) for themdiECSC cates that it is also exact (at least to A potential. It is easy to see that t e effective expansion param2, in agreeeterinwith eqs.the (16), (17),(up (1 )o and (22) is An perturbament result A3) of analytic tion theory [15], and wi h that of high-order perturbation theory (up to A4) [16]. It is worth noting, however, that although ,~,(r~>and R~ 0can, in principle, be calculated corre t to any order of A, the applicability of the present method is limited to only small values of A. In this ote, instead of presenting detailed results, we emp size the salient features of our calculations. For inst nce, calculating the energy E15 to order as high as 5 we find that the energy series converges for A ~ c/a’ where A~ 0.71 [41 iS the critical screening par meter for the is state. For E2~,we also find that co vergence is obtained for A ~ A~/4,where A~for the p state is ~0.1 8. The convergence of the quantitie (rm> and R~1follows the same pattern. It is intere ing to note that for the Yukawa potential (r~e ~) also, the convergence of various energy series has ~een obtained [8] for values of A ~ Xj4. It is import+t, however, to remember that the above discussion has been based on direct convergence, without use~ofPadé approximants. We also observe that the ener~iesobtained for such low values of A by our methot more or less coincide with those obtained by first-o~ierperturbation theory [2] with the Coulomb Potenri as the unperturbedpotential. In this connectioi~we should point out that in the usual perturbation th~orythe expansion is in powers of the perturbation fr~mthe known potential, ,
22 June 1981
while our approach based on the expansion of the potential in the form (5) is in powers of the screening parameter A. Thus E~1,(rm> and N~1calculated by our method approach the true values more closely for small A when expansion (5) gives a good approximation of to our the ECSC potential. distinct perturbaadvantage method over the Another usual first-order tion theory is that the latter gives only the energy; it cannot give (rm> and N~ 1,which are so easily obtained in our approach. One of the authors (A.R.) is indebted to the CSIR for a Senior Research Fellowship. References [1] V.L. Bonch-Bruevich and V.B. Glasko, Soy. Phys. Doki. 4 (1959) 147; V.L. Bonch-Bruevich and Sh.M. Kogan, Soy. Phys. Solid State 1(1960)1118. [2] C.S. Lam and Y.P.Varshni, Phys. Rev. 6A (1972) 1391. [3] R. Dutt, Phys. Lett. 73A (1979) 310. [4] P.P. Ray and A. Ray, Phys. Lett. 78A (1980) 443. [5] A. Dalgarno, in: Quantum theory I. Elements, ed. D.R. Bates (Academic Press, New York, 1961) p. 190. [6] R.J. Swenson and S.H. Danforth, J. Chem. Phys. 57 (1972) 1734. [7] ~ ~fflngbeck,Phys. Lett. 65A (1978) 87. [8] M. Grant and C.S. Lai, Phys. Rev. A20 (1979) 718. [9] L.L. Foldy, Phys. Rev. 111 (1958) 1093. [10] N. Fröman and Främan, Phys. Rev. A6 (1972) 2064. [11] Z.R. Iwinski, (1979) 1924. Y.S. Kim and R.H. Pratt, Phys. Rev. C19 [12] H.A. Kramers, Quantum mechanics (North-Holland, Amsterthm, 1951) p. 251. P.O.
[13] C. Quigg and J.L. Rosner, Phys. Rep. 56 (1979) 167.
[14] J. Kilhingbeck and S. Galicia, J. Phys. A13 (1980) 3419. [15] J. McEnnan, L. Kissel and R.H. Pratt, Phys. Rev. Al 3 (1976) 532. [16] GJ. tafrate and L.B. Mendelsohn, Phys. Rev. A2 (1970) 561.
385
PHYSICS LETTERS
22 June 1981
BOUND STATES F ThE EXPONENTIAL COSINE SCREENED COULOMB POTENTIAL Aparna RAY and ~ritam P. RAY Department of Phys4~s,Visva-Bharati University, Santiniketan,
W.B. 731235, India
Received 27 Noyem r 1980
Revised manuscript r~ceived30 March 1981
A generalized yir 1 theorem and the Heilnian—Feynman theorem have been used to calculate perturbatively the energy levels, expectation v4ues of an arbitrary power of the position coordinates and normalization of the wave function. The method gives excell t results for low values of the screening parameter (X ~
The exponential cosi4e screened Coulomb (ECSC) potential with screening ~arameter A, V(r) = —r~e ~‘cosAr, has been treated by seve al approximation methods [1—4].In this note we a ply the generalized virial theorem (GVT) and the ellman—Feynman theorem (HFT) [5] to calculate p rturbatively the bound-state energy levels correct to y order of A without using perturbed wave function In this approach which is based on the behaviour the potential at small distances, we get an extra b nus in that the expectation values of an arbitrary po er of the position coordinate r are readily obtain correct to any order of A. To the best of our know dge expectation values of .
-
BR nl = EniRni, where H= d2/dr2 r~d/dr + Veff(r),
(1)
Veff(r) = V(r) + 1(1 + 1)/2r’. Following Kramers [12] we multiply eq. (1) by
(3)
1
rm+ dRni/d1 ~(m + l)rmR~~, and integrate over r from 0 to infinity. This leads after a few integrations by parts to the recurrence formula —
1(~51+1\2aT2z
~J ~nl1’m,—2l—i
~
<
dr) rm+l~v eff
(4
2
m—’2
i +
2(m
+
l)((Veff
E~
—
1)rm> ~m(m 1)(r )~ which holds for all m —21 1. In (4) the angular brackets ( ) represent the expectation value in the state Rni and the left-hand side arises from calculations of quantities at the lower limit of integration, where R~, N~1r’.N~1is the normalization of the wave function. Since for m = 0 eq. (4) reduces to the familiar virial theorem, we will call eq. (4) the generalizedvirial theorem (GVT). It may be noted that eq. (4) has also been derived along similar lines in ref. [13] - A similar relation for s states has also been derived very recently [14]. In order to develop the perturbation theory, we assume that the ECSC potential may be expanded as [15,16] —
physically important obs rvables other than energy have not been calculated rher for the ECSC potential. One could, however, estimate them by using the Ecker—Weizel wave func ions [4]. It is worth noting that such a perturbative pproach has been used earlier to analyse only the ener levels and expectation values of powers of radial c ordinates for other potentials [6—8].In this letter we show that the GVT and the HFT together can pr fitably be used to calculate the normalization of the ye function also. In particular, the famous Ferm —Segrè formula [9—11]for the probability density a the origin for s states follows very easily from the theorems. The radial Schrödinge equation for the ECSC p0tential may be written as in au) 0 031—9163/81/0000—0
(2)
—
—~
0/$ 02.50 © North-Holland Publishing Company
~
—
—
383
Volume 83A, number 8
22 June 1981
PHYSICS LETTERS
V(r)=—r~e~cosAr— ~
(5)
V~Akr~,
k=0
where
Vk = (_i/~2~~/2cos (7rk/4)/k!
(6)
Furthermore, we assume that the energy Eni and the expectation value (rm) can be expanded as
continuous variable and then apply the HFT to obtam 2). (14) aE~1/al=~V~ff/al)=!~(2l+ i)(r It is worth noting here that eq. (14) has been used very recently in a different context [14]. With the use of (7) and (8), eq. (14) yields C~”~’~—~-— —2 21+1 a7
En! =
,~ E~Ak,
(15)
(7) Thus using a computer, Eni and (rm) (m ~‘ —2! 2) may be calculated correct to any order of A with the —
~C(k)Xk, m
(rm> =
(8)
k=o
where ~ [5] gives
=
—n2/2 and
~k)
=
60k- Now the HFT
help of eqs. (11), (12), (13) and (15). In the following, however, we give an expression of the s-state energy for the ECSC potential correct to sixth order of A: Eno = + A ~n2 (Sn2 + l)A3 —
aEni/aA = (ö V/aA).
(9)
—
+ ~n4(.7n2 + 5)A4
—
~n4(2ln4+ 35n2
+
4)A5
With the use of (5), (7) and (8), eq. (9) leads to k
kE~~ =
—
~
s—i
(10)
-
Thus both Eni and (rm) would be known, once (which are the expectation values (rm)in the kth order) are determined. On substitution of (3), (5), (7) and (8) in eq. (4) we find after a little algebra, the recurrencerelations [8] withm>—21 —1:
—thn6(143n4+345n2+28)X6+.... (16) The expectation values of r and r2 in the ns state are given by (r)~ 2 + ~n6(l5n2 <2> 0 =~n i~n2(5n2+ 1)
+
21)A3
+
...
(17) (18)
+~n6(i43n4+345n2 +28)X3
+....
n2(A 1 C~.1+ A2C~°.~2),
(ii)
Normalization N~1for any state Rni(r) can also be calculated from eq. (4):
(12) and fork>2, =
+
]
~2 [A1c~’±1 + A2C~2
k
__________
~
+
2m + s + 1 V m+l 5
2N~
~(2l +
= dV/dr> 2~> 4l(r2~ V). (19) + 41 ~1(r Restricting ourselves to the s-state normalization in this letter, we have l)
1
~,—2i
—
(13)
m+s—i
R~
with
0(0)= N~0= 2(dV/dr>. A1
=
(2rn
+
1)/(m +
With expansion (5) for V, eq. (20) yields
1),
A2m(rn— 1)/4—ml(l+1)/(m+1). Using the recurrence relatioi~s(11), (12) and (13) it is possible to compute (Tm> for all m > —1 [8]. For all 2>, which obtained from thetoabove recurpowers —2 > ismnot—2! 2, we need determine (r rence relations. Following ref. [12] we treat 1 as a ~
384
(20)
—
k
E s0 ~~kV8s
R~0(0)= —2 k=0
—
1)C~~.
(21)
Explicit calculation to order six of A of the right-hand side of eq. (21) shows that
Volume 83A, number 8
PHYSICS LETTERS
R20(o) 4 dEn0/dfl, (22) which is the famous Feri~ni—Segrèformula [9—11,14]. It may be noted here th~tprevious work [101 described eq. (22) as exact only for the Coulomb and Hulthen potentials. Our ¶terivation of eq. (22) 6) for themdiECSC cates that it is also exact (at least to A potential. It is easy to see that t e effective expansion param2, in agreeeterinwith eqs.the (16), (17),(up (1 )o and (22) is An perturbament result A3) of analytic tion theory [15], and wi h that of high-order perturbation theory (up to A4) [16]. It is worth noting, however, that although ,~,(r~>and R~ 0can, in principle, be calculated corre t to any order of A, the applicability of the present method is limited to only small values of A. In this ote, instead of presenting detailed results, we emp size the salient features of our calculations. For inst nce, calculating the energy E15 to order as high as 5 we find that the energy series converges for A ~ c/a’ where A~ 0.71 [41 iS the critical screening par meter for the is state. For E2~,we also find that co vergence is obtained for A ~ A~/4,where A~for the p state is ~0.1 8. The convergence of the quantitie (rm> and R~1follows the same pattern. It is intere ing to note that for the Yukawa potential (r~e ~) also, the convergence of various energy series has ~een obtained [8] for values of A ~ Xj4. It is import+t, however, to remember that the above discussion has been based on direct convergence, without use~ofPadé approximants. We also observe that the ener~iesobtained for such low values of A by our methot more or less coincide with those obtained by first-o~ierperturbation theory [2] with the Coulomb Potenri as the unperturbedpotential. In this connectioi~we should point out that in the usual perturbation th~orythe expansion is in powers of the perturbation fr~mthe known potential, ,
22 June 1981
while our approach based on the expansion of the potential in the form (5) is in powers of the screening parameter A. Thus E~1,(rm> and N~1calculated by our method approach the true values more closely for small A when expansion (5) gives a good approximation of to our the ECSC potential. distinct perturbaadvantage method over the Another usual first-order tion theory is that the latter gives only the energy; it cannot give (rm> and N~ 1,which are so easily obtained in our approach. One of the authors (A.R.) is indebted to the CSIR for a Senior Research Fellowship. References [1] V.L. Bonch-Bruevich and V.B. Glasko, Soy. Phys. Doki. 4 (1959) 147; V.L. Bonch-Bruevich and Sh.M. Kogan, Soy. Phys. Solid State 1(1960)1118. [2] C.S. Lam and Y.P.Varshni, Phys. Rev. 6A (1972) 1391. [3] R. Dutt, Phys. Lett. 73A (1979) 310. [4] P.P. Ray and A. Ray, Phys. Lett. 78A (1980) 443. [5] A. Dalgarno, in: Quantum theory I. Elements, ed. D.R. Bates (Academic Press, New York, 1961) p. 190. [6] R.J. Swenson and S.H. Danforth, J. Chem. Phys. 57 (1972) 1734. [7] ~ ~fflngbeck,Phys. Lett. 65A (1978) 87. [8] M. Grant and C.S. Lai, Phys. Rev. A20 (1979) 718. [9] L.L. Foldy, Phys. Rev. 111 (1958) 1093. [10] N. Fröman and Främan, Phys. Rev. A6 (1972) 2064. [11] Z.R. Iwinski, (1979) 1924. Y.S. Kim and R.H. Pratt, Phys. Rev. C19 [12] H.A. Kramers, Quantum mechanics (North-Holland, Amsterthm, 1951) p. 251. P.O.
[13] C. Quigg and J.L. Rosner, Phys. Rep. 56 (1979) 167.
[14] J. Kilhingbeck and S. Galicia, J. Phys. A13 (1980) 3419. [15] J. McEnnan, L. Kissel and R.H. Pratt, Phys. Rev. Al 3 (1976) 532. [16] GJ. tafrate and L.B. Mendelsohn, Phys. Rev. A2 (1970) 561.
385