- Email: [email protected]
Bound s-state energies for the exponential cosine screened Coulomb potential in Ecker-Weizel approximations
Volume 73A, number 4
PHYSICS LETTERS
1 October 1979
BOUND s-STATE ENERGIES FOR THE EXPONENTIAL COSINE SCREENED COULOMB POTENTIAL IN ECKER—WEIZEL APPROXIMATIONS Ranabir DUTT Department of Physics, Visva-Bharati University, Santiniketan-731235, West Bengal, India Received 27 June 1979
2 e~’~cos(ôr)/r, Bound s-state energies of an electron in the exponential cosine screened Coulomb potential, V(r) = —e are obtained analytically using the Ecker—Weizel approximation. For reasonable values of the screening parameter &, the predicted results are in excellent agreement with the variational calculation of Lam and Varshni. The effect of screening on the shift of the quantum numbers from their normal values is also studied for various s-states.
An approximate analytic solution of the problem of the static screened Coulomb potential (Yukawa potential) was first obtained by Ecker and Weizel [I]. It was observed, however, that when the screening parameter is large, the calculated energies for the bound s-states are quite small compared to the exact results obtained by a numerical method [2]. The reason for this discrepancy was pointed out by Lam and Varshni [31.They proposed that a quantity ~which appears in the expression for the bound s-state energies and represents some sort of mean distance of the electron in the considered quantum state, should be identified in a different way than was done in the Ecker—-Weizel calculation. Instead of choosing ~to be the expectation value of the radius, (r) = ~ n2 (using atomic units, m = e = h = 1), Lam and Varshni considered this to be the Bohr radius of the quantum state n, i.e., rn2. It is important to note that this improved choice of ~ corresponds to a radial distance for which the probability of finding the bound electron in a hydrogen like atom is at a maximum [4]. In the present note, we investigate the problem of the exponential cosine screened Coulomb (ECSC) p0tential, V(r) = —e2(e’~’7r)cos(~r)which exhibits a stronger screening effect than the Yukawa potential, due to the presence of the oscillatory (cosine) part. The ECSC potential which is used in the study of electron—positron interaction in a positronium atom in a solid [5] ,has only been studied so far numerically using 310
perturbative and variational techniques [6,7]. It is then worthwhile to obtain the approximate analytic solution for this problem and compare the results with the numerical calculations. The ECSC potential is of course central, which a!lows us to split the Schrodinger equation into separate angular and radial equations. For the bound s-states. the radial equation becomes r 2 e~” 1 1 d cos(~r) Ej ~(r) = 0. (1) dr Eq. (1) can be reduced to the form 1 d2v dv e~r ct,~ + cos(ör) v = 0 (2) dr by substituting
[~
—~
-~ —
+—
—
—-
=
—~—
.
r
—
~(r) e 0n v(r), in which = 2E \1/2 fl
~
(3) (4
fl)
It is convenient to change variables from r toy e~’~), in which case eq. (2) becomes
(1
=
—
2
d v dy 2-
dv
y(l —y)—-—
3nY~~ —
!
-
2ycos(log(l ölo (I —y)) g Y —
~
U
=
U, (5)
where ~
=
1 + 2c~n/6.
(6)
Volume
PHYSICS LETTERS
73A, number 4
1 October 1979
Following the Ecker—Weizel approximation, we assume that the last term in eq. (5) is a slowly varying function and hence it may be treated as constant given by
probability density function hPI2r2 is at a maximum for the Hulthén type potential. Since the s-state eigenfunctions for the Hulthén potential are obtained in closed (normalized) form [8], the quantity fls obtained
2 (1 —e5~)cos(br), 7=—
as ~n2+~+O(~2).
—
(7)
where rrepresents an average distance of the bound electron in the appropriate quantum state, Using (7) in eq. (5), we get the eigenvalue condition which when combined with (6~givesthe eigen-energy of the ns state E~ 5=~
r1
“1
[—
‘
e~1~ —
—
12
‘cos(~)
—
~
n5j
(9)
Using (9) in (8), we compute the energies for different values of the screening parameter ~ and for different s-states. The results for ls to 4s states are shown in table 1 along with the values obtained from the oneparameter variational calculation of Lam and Varshni [7]. The two calculations agree within 10% for ~ as
.
(8)
To compute the energies for various s-states from (8), it is very crucial to make the right choice of r~The major difference between our work and that of Lam and Varshni lies in the determination of Ffor the static screened Coulomb potential [3]. From the perturbative calculations [7],it is clear that the ECSC potential resembles the Hu!thén potential more closely than the Coulomb potential. We therefore identify Pwith the radial distance at which the
large as about 0.5 ~ where ~ is the critical screening parameter above which no bound state with negative energy exists. The parameter ~ is different for differ. ent quantum states and is obtained by maximizing expression (8), which gives ~
2—
—
)cos(~r)——n ~ r=0. (10) c c Solving eq. (10), ~ may be determined for different values of n. For the ground state, our results lead to = 0.7 173 which is in excellent agreement with the values, 0.72 obtained by Bouch-Bruevich and Glasko [6] (1 —e
C
2
Table 1 Energy eigenvalues as a function of the screening parameter for various ns states. Screening
is
2s
3s
4s
parameter our results
oneparameter
our results
variational (ref. [7])
oneparameter
our results
variational (ref. [7])
oneparameter
our results
variational (ref. [7])
oneparameter variational (ref. [71)
0.0002
0.499800 0.499500 0.499000 0.498001 0.495006 0.489995
0.499800 0.499500 0.499000 0.498000 0.495000 0.490001
0.124800 0.124500 0.124006 0.123003 0.120054 0.115069
0.124800 0.124500 0.124000 0.123000 0.120002 0.115013
0.055356 0.055056 0.054557 0.053562 0.050598 0.045745
0.055356
0.0005 0.0010 0.0020 0.0050 0.010
0.055056 0.054556 0.053556 0.050564 0.045619
0.031050 0.030751 0.030253 0.029262 0.026333 0.021646
0.031050 0.030750 0.030250 0.029252 0.026275 0.021437
0.020
0.479972
0.480008
0.105309
0.105104
0.036473
0.036025
0.013293
0.012572
0.040 0.060 0.080 0.10 0.20 0.30 0.40 0.50 0.60
0.459897 0.439804 0.419721 0.399673 0.301088 0.208596 0.127238 0.062311 0.018745
0.460061 0.440201 0.420464 0.400885 0.306332 0.219399 0.142375 0.077481 0.027708
0.086491 0.068902 0.052853 0.038608
0.085769 0.067421 0.050384 0.034935
0.020363
0.018822
_________________________________
_____________
311
Volume 73A, number 4
PHYSICS LETTERS
1 October 1979
Table 2 Quantum number shift for ns states as a function of the reduced screening parameter o /1 cis tc (
0.005 0.010 0.020 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500
=
3s (tc
0.71 73)
0.0036 0.0072 0.0147 0.0380 0.0808 0.1293 0.1845 0.2478 0.3210 0.4169 0.5070 0.6270 0.7723
=
Ss (tc
0.0959)
Lsnfn
~o
sn/n
0.0036 0.0072 0.0147 0.0380 0.0808 0.1293 0.1845 0.2478 0.3210 0.4169 0.5070 0.6270 0.7723
0.0130 0.0262 0.0531 0.1375 0.2920 0.4671 0.6658 0.8936 1.1556 1.4607 1.8182 2.2440 2.7564
0.0043 0.0087 0.0177 0.0458 0.0973 0.1557 0.2219 0.2979 0.3852 0.4869 0.6061 0.7480 0.9188
-
=
7s (~ = 0.01 77)
0.0347)
an
ao/o
0.0213 0.0442 0.0883 0.2307 0.4893 0.7834 1.1161 1.4991 1.9381 2.4516 3.0513 3.7687 4.6294
0.0043 0.0088 0.0177 0.0461 0.0979 0.1567 0.2232 0.2998 0.3876 0.4903 0.6103 0.7537 0.9259
an/n 0.0311 0.0624 0.1265 0.3240 0.6848 1.0977 1.5621 2.0995 2.7121 3.3426 4.2694 5.2756 6.4762
0.0044 0.0089 0.0181 0.0463 0.0978 0.1568 0.2232 0.2999 0.3874 0.4904 0.6099 0.7537 0.9252
—
using numerical methods and 0.7115 obtained by Lam
like ions where it is found to be nearly independent of
and Varshni [7]. It may not be unreasonable to assume that the bound electron in the ECSC potential behaves as if it
n.
is under the influence of a modified Coulomb poten-
tions with Dr. B. Talukdar.
I am happy to acknowledge several useful conversa-
tial. One may thus write E
2, (I!) 0 l!2n* where n’~is the effective quantum number. Comparing (8) with (11), we determine n’~and also the quantum number shift ~n = ~ n) for different values of the
References
reduced screening parameter ~ For the sake of comparison, we present the results for ~n and the relative quantum defect ~.n/n only for is, 3s, 5s and 7s
[1] G. Ecker and W. Weizel, Ann. Phys. (Leipzig) 17(1956) 126. 121 F.J. Rogers, Il.C. Graboske Jr. and D.J. Harwood, Phys. Rev. Al (1970) 1577. 131 C.S. Lam and Y.P. Varshni, Phys. Lett. 59A (1976) 363. 141 E. Merzbacher, Quantum mechanics (Wiley, New York, 1961) pp. 197.
—
states in table 2. It is interesting to note that although for a given quantum state ~n increases with ~/ö c the quantity &‘i/n remains approximately constant and independent of the principal quantum number n at any fixed value of ö/~c.This feature shows a similarity with the quantum defect in alkali atoms and alkali,
312
151 E.P. Prokop’ev, Soy. Phys. Solid State 9 (1967) 993. [61 V.L. Bouch-Bruevich and V.B. Glasko, Soy. Phys. Dokl. 171 4(1959)147. C.S. Lam and Y.P. Varshni, Phys. Rev. 6A (1972) 1391. [81 S. Flugge, Practical quantum mechanics (Springer, Berlin. 1974).
PHYSICS LETTERS
1 October 1979
BOUND s-STATE ENERGIES FOR THE EXPONENTIAL COSINE SCREENED COULOMB POTENTIAL IN ECKER—WEIZEL APPROXIMATIONS Ranabir DUTT Department of Physics, Visva-Bharati University, Santiniketan-731235, West Bengal, India Received 27 June 1979
2 e~’~cos(ôr)/r, Bound s-state energies of an electron in the exponential cosine screened Coulomb potential, V(r) = —e are obtained analytically using the Ecker—Weizel approximation. For reasonable values of the screening parameter &, the predicted results are in excellent agreement with the variational calculation of Lam and Varshni. The effect of screening on the shift of the quantum numbers from their normal values is also studied for various s-states.
An approximate analytic solution of the problem of the static screened Coulomb potential (Yukawa potential) was first obtained by Ecker and Weizel [I]. It was observed, however, that when the screening parameter is large, the calculated energies for the bound s-states are quite small compared to the exact results obtained by a numerical method [2]. The reason for this discrepancy was pointed out by Lam and Varshni [31.They proposed that a quantity ~which appears in the expression for the bound s-state energies and represents some sort of mean distance of the electron in the considered quantum state, should be identified in a different way than was done in the Ecker—-Weizel calculation. Instead of choosing ~to be the expectation value of the radius, (r) = ~ n2 (using atomic units, m = e = h = 1), Lam and Varshni considered this to be the Bohr radius of the quantum state n, i.e., rn2. It is important to note that this improved choice of ~ corresponds to a radial distance for which the probability of finding the bound electron in a hydrogen like atom is at a maximum [4]. In the present note, we investigate the problem of the exponential cosine screened Coulomb (ECSC) p0tential, V(r) = —e2(e’~’7r)cos(~r)which exhibits a stronger screening effect than the Yukawa potential, due to the presence of the oscillatory (cosine) part. The ECSC potential which is used in the study of electron—positron interaction in a positronium atom in a solid [5] ,has only been studied so far numerically using 310
perturbative and variational techniques [6,7]. It is then worthwhile to obtain the approximate analytic solution for this problem and compare the results with the numerical calculations. The ECSC potential is of course central, which a!lows us to split the Schrodinger equation into separate angular and radial equations. For the bound s-states. the radial equation becomes r 2 e~” 1 1 d cos(~r) Ej ~(r) = 0. (1) dr Eq. (1) can be reduced to the form 1 d2v dv e~r ct,~ + cos(ör) v = 0 (2) dr by substituting
[~
—~
-~ —
+—
—
—-
=
—~—
.
r
—
~(r) e 0n v(r), in which = 2E \1/2 fl
~
(3) (4
fl)
It is convenient to change variables from r toy e~’~), in which case eq. (2) becomes
(1
=
—
2
d v dy 2-
dv
y(l —y)—-—
3nY~~ —
!
-
2ycos(log(l ölo (I —y)) g Y —
~
U
=
U, (5)
where ~
=
1 + 2c~n/6.
(6)
Volume
PHYSICS LETTERS
73A, number 4
1 October 1979
Following the Ecker—Weizel approximation, we assume that the last term in eq. (5) is a slowly varying function and hence it may be treated as constant given by
probability density function hPI2r2 is at a maximum for the Hulthén type potential. Since the s-state eigenfunctions for the Hulthén potential are obtained in closed (normalized) form [8], the quantity fls obtained
2 (1 —e5~)cos(br), 7=—
as ~n2+~+O(~2).
—
(7)
where rrepresents an average distance of the bound electron in the appropriate quantum state, Using (7) in eq. (5), we get the eigenvalue condition which when combined with (6~givesthe eigen-energy of the ns state E~ 5=~
r1
“1
[—
‘
e~1~ —
—
12
‘cos(~)
—
~
n5j
(9)
Using (9) in (8), we compute the energies for different values of the screening parameter ~ and for different s-states. The results for ls to 4s states are shown in table 1 along with the values obtained from the oneparameter variational calculation of Lam and Varshni [7]. The two calculations agree within 10% for ~ as
.
(8)
To compute the energies for various s-states from (8), it is very crucial to make the right choice of r~The major difference between our work and that of Lam and Varshni lies in the determination of Ffor the static screened Coulomb potential [3]. From the perturbative calculations [7],it is clear that the ECSC potential resembles the Hu!thén potential more closely than the Coulomb potential. We therefore identify Pwith the radial distance at which the
large as about 0.5 ~ where ~ is the critical screening parameter above which no bound state with negative energy exists. The parameter ~ is different for differ. ent quantum states and is obtained by maximizing expression (8), which gives ~
2—
—
)cos(~r)——n ~ r=0. (10) c c Solving eq. (10), ~ may be determined for different values of n. For the ground state, our results lead to = 0.7 173 which is in excellent agreement with the values, 0.72 obtained by Bouch-Bruevich and Glasko [6] (1 —e
C
2
Table 1 Energy eigenvalues as a function of the screening parameter for various ns states. Screening
is
2s
3s
4s
parameter our results
oneparameter
our results
variational (ref. [7])
oneparameter
our results
variational (ref. [7])
oneparameter
our results
variational (ref. [7])
oneparameter variational (ref. [71)
0.0002
0.499800 0.499500 0.499000 0.498001 0.495006 0.489995
0.499800 0.499500 0.499000 0.498000 0.495000 0.490001
0.124800 0.124500 0.124006 0.123003 0.120054 0.115069
0.124800 0.124500 0.124000 0.123000 0.120002 0.115013
0.055356 0.055056 0.054557 0.053562 0.050598 0.045745
0.055356
0.0005 0.0010 0.0020 0.0050 0.010
0.055056 0.054556 0.053556 0.050564 0.045619
0.031050 0.030751 0.030253 0.029262 0.026333 0.021646
0.031050 0.030750 0.030250 0.029252 0.026275 0.021437
0.020
0.479972
0.480008
0.105309
0.105104
0.036473
0.036025
0.013293
0.012572
0.040 0.060 0.080 0.10 0.20 0.30 0.40 0.50 0.60
0.459897 0.439804 0.419721 0.399673 0.301088 0.208596 0.127238 0.062311 0.018745
0.460061 0.440201 0.420464 0.400885 0.306332 0.219399 0.142375 0.077481 0.027708
0.086491 0.068902 0.052853 0.038608
0.085769 0.067421 0.050384 0.034935
0.020363
0.018822
_________________________________
_____________
311
Volume 73A, number 4
PHYSICS LETTERS
1 October 1979
Table 2 Quantum number shift for ns states as a function of the reduced screening parameter o /1 cis tc (
0.005 0.010 0.020 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500
=
3s (tc
0.71 73)
0.0036 0.0072 0.0147 0.0380 0.0808 0.1293 0.1845 0.2478 0.3210 0.4169 0.5070 0.6270 0.7723
=
Ss (tc
0.0959)
Lsnfn
~o
sn/n
0.0036 0.0072 0.0147 0.0380 0.0808 0.1293 0.1845 0.2478 0.3210 0.4169 0.5070 0.6270 0.7723
0.0130 0.0262 0.0531 0.1375 0.2920 0.4671 0.6658 0.8936 1.1556 1.4607 1.8182 2.2440 2.7564
0.0043 0.0087 0.0177 0.0458 0.0973 0.1557 0.2219 0.2979 0.3852 0.4869 0.6061 0.7480 0.9188
-
=
7s (~ = 0.01 77)
0.0347)
an
ao/o
0.0213 0.0442 0.0883 0.2307 0.4893 0.7834 1.1161 1.4991 1.9381 2.4516 3.0513 3.7687 4.6294
0.0043 0.0088 0.0177 0.0461 0.0979 0.1567 0.2232 0.2998 0.3876 0.4903 0.6103 0.7537 0.9259
an/n 0.0311 0.0624 0.1265 0.3240 0.6848 1.0977 1.5621 2.0995 2.7121 3.3426 4.2694 5.2756 6.4762
0.0044 0.0089 0.0181 0.0463 0.0978 0.1568 0.2232 0.2999 0.3874 0.4904 0.6099 0.7537 0.9252
—
using numerical methods and 0.7115 obtained by Lam
like ions where it is found to be nearly independent of
and Varshni [7]. It may not be unreasonable to assume that the bound electron in the ECSC potential behaves as if it
n.
is under the influence of a modified Coulomb poten-
tions with Dr. B. Talukdar.
I am happy to acknowledge several useful conversa-
tial. One may thus write E
2, (I!) 0 l!2n* where n’~is the effective quantum number. Comparing (8) with (11), we determine n’~and also the quantum number shift ~n = ~ n) for different values of the
References
reduced screening parameter ~ For the sake of comparison, we present the results for ~n and the relative quantum defect ~.n/n only for is, 3s, 5s and 7s
[1] G. Ecker and W. Weizel, Ann. Phys. (Leipzig) 17(1956) 126. 121 F.J. Rogers, Il.C. Graboske Jr. and D.J. Harwood, Phys. Rev. Al (1970) 1577. 131 C.S. Lam and Y.P. Varshni, Phys. Lett. 59A (1976) 363. 141 E. Merzbacher, Quantum mechanics (Wiley, New York, 1961) pp. 197.
—
states in table 2. It is interesting to note that although for a given quantum state ~n increases with ~/ö c the quantity &‘i/n remains approximately constant and independent of the principal quantum number n at any fixed value of ö/~c.This feature shows a similarity with the quantum defect in alkali atoms and alkali,
312
151 E.P. Prokop’ev, Soy. Phys. Solid State 9 (1967) 993. [61 V.L. Bouch-Bruevich and V.B. Glasko, Soy. Phys. Dokl. 171 4(1959)147. C.S. Lam and Y.P. Varshni, Phys. Rev. 6A (1972) 1391. [81 S. Flugge, Practical quantum mechanics (Springer, Berlin. 1974).