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Scaling variational approach to non-confining potentials: The gaussian well
Volume 106A, number 5,6
PHYSICS LETTERS
10 December 1984
SCALING VARIATIONAL APPROACH TO NON-CONFINING POTENTIALS: THE GAUSSIAN WELL Christopher C. GERRY Department o f Physics, St. Bonaventure University, St. Bonaventure, N Y 14778, USA
and James B. TOGEAS Division o f Science and Mathematics, University of Minnesota, Morris, MN 56267, USA
Received 2 August 1984 Revised manuscript received 10 October 1984
The scaling variational method is applied to a model potential, the gaussian well, supporting only a finite member of bound states.
There has been some discussion in the recent literature [ 1] on the application of what is known as the scaling variational method [2] (SVM). The types of potentials that have been considered in ref. [1] are of two types, namely the isotropic anharmonic oscillator type with (1)
H = p 2 / 2 m + ½w2r 2 + Xr 2k
and the confining type H=p2/2m-Z/r+~k,
k = 0 , 1 , 2 ....
(2)
Both models (with X > 0) in fact are confining in that the spectra contains no continuum so that eq. (2) might be best referred to as the "quarkonium" model in light of its relevance to bound states of q?q [3]. In the cases when co = Z = 0, the application of the SVM analytically results in the correct functional dependence of the energy eigenvalues on the quantum numbers n and l and on ~, [1]. (These same scaring properties have also been derived as a simple application of the renormalization group [4].) For the pure quartic oscillator (k = 2, 60 = 0) the energy eigenvalues reported in ref. [ 1] are accurate to about 1% or less while for the linear potential (Z = 0, k = 1) they are considerably less accurate ( 5 - 1 0 % ) . In the case when ~ :/= 0, Z 4= 0 the accuracy can be greatly improved (especially for the latter) although the correct scaling behavior becomes ap0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (Noah-Holland Physics Publishing Division)
parent only in the limit of high quantum numbers. In this paper we wish to apply the SVM to a model potential with only a finite number of bound states, namely a gaussian well. In this work we shall employ the SO(2,1) group theoretical formulation of the SVM which has been presented elsewhere [5,6]. This formulation will be convenient for the problem at hand since the required matrix elements of the gaussian potential can be expressed rather neatly as an analytically continued SO(2,1) group transformation matrix element in the relevant unitary irreducible representation (UIR). What we show here is that a rather high degree of accuracy in the energy levels can be obtained even for the excited states. The hamiltonian under consideration then is H = p 2 / 2 m - g2 e x p ( _ a 2 r 2 ) .
(3)
This potential has been widely used in phenomenological theories of interactions, in particular as a model of the ground state of the deuteron [7]. Miiller [8] has given a perturbation expansion valid for large coupling constant g2. The realization of the SO(2,1) dynamical algebra relevant to this problem is given as [9] K0 = ~_(p2 + r 2 ) ,
(4a)
K1 = ~(p2 _ r 2 ) ,
(4b) 215
Volume 106A, number 5,6 1
K 2 = a ( r . p +p • r) ,
PttYSICS LETTERS (4c)
10 December 1984
,%,k(r0 = 2n(m + k) g2(m. klexp[ -2a2r/ I(K!)- K1) j I m , k ) .
satisfying the comnmtation relations
(t3)
[K o, t ( 1 ] = i K 2 ,
(Sa)
[K 2, KO] = i/( 1 ,
(5h)
Now the matrix elements in the last term of eq. (13) are in the form of matrix elements o f an SO(2,1 ) transformation of the parabolic type nanrely [ 10]:
[K 1 , K 2 ] ..... iK 0 .
(5c)
(; = exp[ i7(K 0
The Casimir operator is
O= o
(61
We shall employ the UIR o f SO(2,1) known as D+(k), the discrete principle series whose states ira, k ) diagonalize K o as K 0 Ira, k ) = (m + k)lm, k ) .
(71
where k > 0 such that the eigenvalues of Q are k(k - 1 ). k is the so-called Bargmann index. For a three-dimensional system one has k = 1 (l + ½) and thus m = n r the radical quantum number as can easily be seen by noting from eq. (4a) that 2K 0 is the hamiltonian o f an isotropic harmonic oscillator o f unit frequency. In one _3 dinaension one has k = a1 (even parity states) or k (odd parity states). The relation to the usual oscillator quantum number n is thus n = 2m (k = ¼) and n = 2m + 1 (k = ~) [6]. In our calculations we consider only the one-dimensional case. Now in terms of the operators o f e q . (4), eq. (3) reads H = 2(K 0 + K 1 ) - g2 exp [ - 2 a 2 ( K 0 - K 1)] ,
(8)
1
where we have set m = ~. Since K 2 is a generator of scale transformations, we define the scaled states
Ira, k)= exp(iOK2)lm, k ) ,
Emk(O ) = (m, klHIm, k ) = (m, klH(O)lm, k ) ,
i f ( 0 ) = e x p ( - iOK 2)H exp(i0K2)
= (c~*) 2(re+k) 2 F 1 (----m. 1 - m - 2k, 1;-/3*/3),
(15) where o<=1 - a 2 r / l
,
/3=a2r/--t .
j
(16a)
a2r/ 1 .
(16b)
Thus we get Emk(r/) = 2r/(m + k) - g2 Vk, m(O~, 1~) .
(17)
The gaussian well has few bound states. We shall consider only the lowest four. The energy functionals from eq. (17) are U0(r/) = lr/
El (r/) = ~n 3
1 I
where r/= e 0 is the scaling factor. Since K 1 is nondiagonal in the D+(k) representation eq. (10) becomes
o<* = l + a 2 r / /3* =
g(1 + a2/r/) 1/2
(m = 0, ~ =
g2 ( 1 + a 2/r/)3/2
(m = 0 , / ¢
E3(r/) =5- [7,7
( 1 1)
K1)]lm, k)= Vkm,m(~,fJ)
2a2r/ I(K 0
(10)
is the scaled hamiltonian operator. From the B a k e r Hausdorff Campbell formula one obtains (12) /~(0) = 2r/(K 0 + K 1 ) _ g 2 e x p [ _ 2 a 2 r / - l ( K 0 _ KI )] ,
216
(m, k l e x p [
1( E2(r/)= ~ 5r/
where
(14)
where T is analytically continued as 3' ~ - 2ia2r? 1. As has been shown elsewhere [ 11 ] the nratrix elements are expressed in terms o f Bargnrann [12] functions as
(9)
so that
ki"1 )] .
g2(2+a4/r/2)) (1 +a2/r/) 5 g2(2 + 3a4/r/2) ~
~-), (18a)
=}), (18b)
(m=l,k=~_) ' (18c)
(rn=l,k=]-). (18d) The SVM requires that we set dE/dr/= 0. The number o f bound states for a given a a n d g can be determined quite easily by setting E = 0 in the JWKB quantization rule and solving for the quantum number, its value being related to the upper bound on the number of bound states supported by the potential. The SVM calculations are in agreement with the JWKB results. For instance, for a = g = 1 the JWKB method predicts only one bound state. F r o m the SVM setting dE/dr/= 0 for eq. (18a) results in (1 + a2 /r/) 7/2 .!
Volume 106A, number 5,6
PHYSICS LETTERS
10 December 1984
Table 1 Energy eigenvaluesfor various g values, a
g
1 1
1 3
1
5
1
15
Energy
Exact
SVM
MOiler
JWKB
E0 E0 E1 Eo E1 E2 E3 E0 E1 E2 E3
-0.3540 -6.3775 -2.0406 -20.377 -11.965 -5.3608 -1.0131 -210.37 -181.90 -155.00 -129.74
-0.3347 -6.3637 -1.9470 -20.368 -11.912 -5.4030 -2.0996 -210.37 -181.89 -155.00 -129.80
-0.3770 -6.3787 -2.0345 -20.378 -11.964 -5.3202 -0.6910 -210.38 -181.90 -155.00 -129.74
-0.2071 -6.1927 -1.8639 -20.191 -11.779 -5.1774 -0.8518 -210.19 -181.71 -154.82 -129.56
r/-2 - (1 + r/-1) 5/2 = 0 ,
(19)
which in fact has no positive real roots. By contrast MOilers formula predicts spurious energy levels at these values. In table 1 we present our results for the energy eigenvalues along with exact results obtained by numerical integration of the Schr6dinger equation [13], the values obtained from Miiller's large g2 perturbation theory formula, and JWKB calculations. Table 2 contains error percentages for the various methods. For f'Lxeda and with increasing g we find that the SVM eigenvalues tend to approach the exact values even for some of the excited states. The method does seem to break down however near the top of the well as is evident for the third excited state of the case w h e n g = 5. In fact only in this instance does the JWKB calculation have higher accuracy then either the SVM or the MiJller formula. An unexpected find is that for highg, the SVM results are more accurate then Mtiller's formula at least for the ground states. Now the realization of the SO(2,1) algebra used in this paper is in fact applicable to models where the potential depends on even powers o f r such as those of eq. (1). On the other hand, there is another realization of SO(2,1) relevant to Coulomb type potentials as in eq. (2) [14]. In fact it is possible to formulate the SVM for a screened Coulomb problem in a similar fashion as presented here for the gaussian well. In ref. [11] we have used this method to calculate the energy levels, normalization factors, and wavefunctions (essentially scaled-wave hydrogenic-wave wavefunctions) for
Table 2 Error percentages for the different methods. a
g
1 1
1 3
1
5
1
15
Level
SVM
MtiUer
JWKB
0 0 1 0 1 2 3 0 1 2 3
5.45 0.22 4.6 0.044 0.44 -0.787 -107.25 0.0 0.0055 0.0 -0.05
-6.50 0.019 0.30 -0.005 0.008 0.757 30.63 0.0048 0.0 0.0 0.0
41.5 2.90 8.66 0.913 1.55 3.42 15.92 0.086 0.10 0.12 0.14
various screened Coulomb potentials. The VIRs required in that case are not the same as those of the oscillator problem.
Note added. Bozzolo and Plastino [ 15] have considered a similar variational method for generalized anharmonic oscillators including the gaussian potential V(x) = exp(kx2). We thank the referee for pointing out to us this paper. The authors are grateful to the Dean's Office at UMM for computing grants.
References [1] F.M. Fernandez and E.A. Castro, J. Chem. Phys. 79 (1983) 321; Phys. Rev. A27 (1983) 663. 217
Volume 106A, number 5,6 [2] [3] [4] 15[ [6[ [7] [8] [9]
218
PHYSICS LETTERS
P.-O. L6wdin, J. Mol. Spectr. 3 (1959) 46. C. Quigg and J.L. Rosner, Phys. Rep. 56 (1979) 167. R. Seki, Phys. Rev. D24 (1981) 1684. C.C. Gerry and S. Silverman, Phys. Lett. 95A (1983) 481. C.C. Gerry and S. Silverman. Phys. Rev. A29 (1984) 1574. H.A. Bethe and R.F. Bacher, Rev. Mod. Phys. 8 (1936) 82. H.J.W. Miiller, J. Math. Phys. 11 (1970)355. E. Chacon, D. Levi and Moshinsky, J. Math. Phys. 17 (1976) 1919.
10 December 1984
[ 10] B.G. Wybourne, Classical groups for physicists (Wiley, New York, 1974) p. 190. [11 ] C.C. Gerry and J. Laub, Phys. Rev. A30 (1984) 1229. [12] V. Bargmann, Ann. Math. 48 (1947) 568. [13] G. Henderson, W. Read and C.S. Ko, Program 434, Quantum Chemistry Program exchange, Indiana University (1982). [ 141 A.O. Barut, Dynamical groups and generalized symmetrics in quantum theory (University of Canterbury Press, Christchurch, 1971 ). [ 151 G. Bozzolo and A. Plastino, Phys. Rev. D24 (1981) 3113.
PHYSICS LETTERS
10 December 1984
SCALING VARIATIONAL APPROACH TO NON-CONFINING POTENTIALS: THE GAUSSIAN WELL Christopher C. GERRY Department o f Physics, St. Bonaventure University, St. Bonaventure, N Y 14778, USA
and James B. TOGEAS Division o f Science and Mathematics, University of Minnesota, Morris, MN 56267, USA
Received 2 August 1984 Revised manuscript received 10 October 1984
The scaling variational method is applied to a model potential, the gaussian well, supporting only a finite member of bound states.
There has been some discussion in the recent literature [ 1] on the application of what is known as the scaling variational method [2] (SVM). The types of potentials that have been considered in ref. [1] are of two types, namely the isotropic anharmonic oscillator type with (1)
H = p 2 / 2 m + ½w2r 2 + Xr 2k
and the confining type H=p2/2m-Z/r+~k,
k = 0 , 1 , 2 ....
(2)
Both models (with X > 0) in fact are confining in that the spectra contains no continuum so that eq. (2) might be best referred to as the "quarkonium" model in light of its relevance to bound states of q?q [3]. In the cases when co = Z = 0, the application of the SVM analytically results in the correct functional dependence of the energy eigenvalues on the quantum numbers n and l and on ~, [1]. (These same scaring properties have also been derived as a simple application of the renormalization group [4].) For the pure quartic oscillator (k = 2, 60 = 0) the energy eigenvalues reported in ref. [ 1] are accurate to about 1% or less while for the linear potential (Z = 0, k = 1) they are considerably less accurate ( 5 - 1 0 % ) . In the case when ~ :/= 0, Z 4= 0 the accuracy can be greatly improved (especially for the latter) although the correct scaling behavior becomes ap0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (Noah-Holland Physics Publishing Division)
parent only in the limit of high quantum numbers. In this paper we wish to apply the SVM to a model potential with only a finite number of bound states, namely a gaussian well. In this work we shall employ the SO(2,1) group theoretical formulation of the SVM which has been presented elsewhere [5,6]. This formulation will be convenient for the problem at hand since the required matrix elements of the gaussian potential can be expressed rather neatly as an analytically continued SO(2,1) group transformation matrix element in the relevant unitary irreducible representation (UIR). What we show here is that a rather high degree of accuracy in the energy levels can be obtained even for the excited states. The hamiltonian under consideration then is H = p 2 / 2 m - g2 e x p ( _ a 2 r 2 ) .
(3)
This potential has been widely used in phenomenological theories of interactions, in particular as a model of the ground state of the deuteron [7]. Miiller [8] has given a perturbation expansion valid for large coupling constant g2. The realization of the SO(2,1) dynamical algebra relevant to this problem is given as [9] K0 = ~_(p2 + r 2 ) ,
(4a)
K1 = ~(p2 _ r 2 ) ,
(4b) 215
Volume 106A, number 5,6 1
K 2 = a ( r . p +p • r) ,
PttYSICS LETTERS (4c)
10 December 1984
,%,k(r0 = 2n(m + k) g2(m. klexp[ -2a2r/ I(K!)- K1) j I m , k ) .
satisfying the comnmtation relations
(t3)
[K o, t ( 1 ] = i K 2 ,
(Sa)
[K 2, KO] = i/( 1 ,
(5h)
Now the matrix elements in the last term of eq. (13) are in the form of matrix elements o f an SO(2,1 ) transformation of the parabolic type nanrely [ 10]:
[K 1 , K 2 ] ..... iK 0 .
(5c)
(; = exp[ i7(K 0
The Casimir operator is
O= o
(61
We shall employ the UIR o f SO(2,1) known as D+(k), the discrete principle series whose states ira, k ) diagonalize K o as K 0 Ira, k ) = (m + k)lm, k ) .
(71
where k > 0 such that the eigenvalues of Q are k(k - 1 ). k is the so-called Bargmann index. For a three-dimensional system one has k = 1 (l + ½) and thus m = n r the radical quantum number as can easily be seen by noting from eq. (4a) that 2K 0 is the hamiltonian o f an isotropic harmonic oscillator o f unit frequency. In one _3 dinaension one has k = a1 (even parity states) or k (odd parity states). The relation to the usual oscillator quantum number n is thus n = 2m (k = ¼) and n = 2m + 1 (k = ~) [6]. In our calculations we consider only the one-dimensional case. Now in terms of the operators o f e q . (4), eq. (3) reads H = 2(K 0 + K 1 ) - g2 exp [ - 2 a 2 ( K 0 - K 1)] ,
(8)
1
where we have set m = ~. Since K 2 is a generator of scale transformations, we define the scaled states
Ira, k)= exp(iOK2)lm, k ) ,
Emk(O ) = (m, klHIm, k ) = (m, klH(O)lm, k ) ,
i f ( 0 ) = e x p ( - iOK 2)H exp(i0K2)
= (c~*) 2(re+k) 2 F 1 (----m. 1 - m - 2k, 1;-/3*/3),
(15) where o<=1 - a 2 r / l
,
/3=a2r/--t .
j
(16a)
a2r/ 1 .
(16b)
Thus we get Emk(r/) = 2r/(m + k) - g2 Vk, m(O~, 1~) .
(17)
The gaussian well has few bound states. We shall consider only the lowest four. The energy functionals from eq. (17) are U0(r/) = lr/
El (r/) = ~n 3
1 I
where r/= e 0 is the scaling factor. Since K 1 is nondiagonal in the D+(k) representation eq. (10) becomes
o<* = l + a 2 r / /3* =
g(1 + a2/r/) 1/2
(m = 0, ~ =
g2 ( 1 + a 2/r/)3/2
(m = 0 , / ¢
E3(r/) =5- [7,7
( 1 1)
K1)]lm, k)= Vkm,m(~,fJ)
2a2r/ I(K 0
(10)
is the scaled hamiltonian operator. From the B a k e r Hausdorff Campbell formula one obtains (12) /~(0) = 2r/(K 0 + K 1 ) _ g 2 e x p [ _ 2 a 2 r / - l ( K 0 _ KI )] ,
216
(m, k l e x p [
1( E2(r/)= ~ 5r/
where
(14)
where T is analytically continued as 3' ~ - 2ia2r? 1. As has been shown elsewhere [ 11 ] the nratrix elements are expressed in terms o f Bargnrann [12] functions as
(9)
so that
ki"1 )] .
g2(2+a4/r/2)) (1 +a2/r/) 5 g2(2 + 3a4/r/2) ~
~-), (18a)
=}), (18b)
(m=l,k=~_) ' (18c)
(rn=l,k=]-). (18d) The SVM requires that we set dE/dr/= 0. The number o f bound states for a given a a n d g can be determined quite easily by setting E = 0 in the JWKB quantization rule and solving for the quantum number, its value being related to the upper bound on the number of bound states supported by the potential. The SVM calculations are in agreement with the JWKB results. For instance, for a = g = 1 the JWKB method predicts only one bound state. F r o m the SVM setting dE/dr/= 0 for eq. (18a) results in (1 + a2 /r/) 7/2 .!
Volume 106A, number 5,6
PHYSICS LETTERS
10 December 1984
Table 1 Energy eigenvaluesfor various g values, a
g
1 1
1 3
1
5
1
15
Energy
Exact
SVM
MOiler
JWKB
E0 E0 E1 Eo E1 E2 E3 E0 E1 E2 E3
-0.3540 -6.3775 -2.0406 -20.377 -11.965 -5.3608 -1.0131 -210.37 -181.90 -155.00 -129.74
-0.3347 -6.3637 -1.9470 -20.368 -11.912 -5.4030 -2.0996 -210.37 -181.89 -155.00 -129.80
-0.3770 -6.3787 -2.0345 -20.378 -11.964 -5.3202 -0.6910 -210.38 -181.90 -155.00 -129.74
-0.2071 -6.1927 -1.8639 -20.191 -11.779 -5.1774 -0.8518 -210.19 -181.71 -154.82 -129.56
r/-2 - (1 + r/-1) 5/2 = 0 ,
(19)
which in fact has no positive real roots. By contrast MOilers formula predicts spurious energy levels at these values. In table 1 we present our results for the energy eigenvalues along with exact results obtained by numerical integration of the Schr6dinger equation [13], the values obtained from Miiller's large g2 perturbation theory formula, and JWKB calculations. Table 2 contains error percentages for the various methods. For f'Lxeda and with increasing g we find that the SVM eigenvalues tend to approach the exact values even for some of the excited states. The method does seem to break down however near the top of the well as is evident for the third excited state of the case w h e n g = 5. In fact only in this instance does the JWKB calculation have higher accuracy then either the SVM or the MiJller formula. An unexpected find is that for highg, the SVM results are more accurate then Mtiller's formula at least for the ground states. Now the realization of the SO(2,1) algebra used in this paper is in fact applicable to models where the potential depends on even powers o f r such as those of eq. (1). On the other hand, there is another realization of SO(2,1) relevant to Coulomb type potentials as in eq. (2) [14]. In fact it is possible to formulate the SVM for a screened Coulomb problem in a similar fashion as presented here for the gaussian well. In ref. [11] we have used this method to calculate the energy levels, normalization factors, and wavefunctions (essentially scaled-wave hydrogenic-wave wavefunctions) for
Table 2 Error percentages for the different methods. a
g
1 1
1 3
1
5
1
15
Level
SVM
MtiUer
JWKB
0 0 1 0 1 2 3 0 1 2 3
5.45 0.22 4.6 0.044 0.44 -0.787 -107.25 0.0 0.0055 0.0 -0.05
-6.50 0.019 0.30 -0.005 0.008 0.757 30.63 0.0048 0.0 0.0 0.0
41.5 2.90 8.66 0.913 1.55 3.42 15.92 0.086 0.10 0.12 0.14
various screened Coulomb potentials. The VIRs required in that case are not the same as those of the oscillator problem.
Note added. Bozzolo and Plastino [ 15] have considered a similar variational method for generalized anharmonic oscillators including the gaussian potential V(x) = exp(kx2). We thank the referee for pointing out to us this paper. The authors are grateful to the Dean's Office at UMM for computing grants.
References [1] F.M. Fernandez and E.A. Castro, J. Chem. Phys. 79 (1983) 321; Phys. Rev. A27 (1983) 663. 217
Volume 106A, number 5,6 [2] [3] [4] 15[ [6[ [7] [8] [9]
218
PHYSICS LETTERS
P.-O. L6wdin, J. Mol. Spectr. 3 (1959) 46. C. Quigg and J.L. Rosner, Phys. Rep. 56 (1979) 167. R. Seki, Phys. Rev. D24 (1981) 1684. C.C. Gerry and S. Silverman, Phys. Lett. 95A (1983) 481. C.C. Gerry and S. Silverman. Phys. Rev. A29 (1984) 1574. H.A. Bethe and R.F. Bacher, Rev. Mod. Phys. 8 (1936) 82. H.J.W. Miiller, J. Math. Phys. 11 (1970)355. E. Chacon, D. Levi and Moshinsky, J. Math. Phys. 17 (1976) 1919.
10 December 1984
[ 10] B.G. Wybourne, Classical groups for physicists (Wiley, New York, 1974) p. 190. [11 ] C.C. Gerry and J. Laub, Phys. Rev. A30 (1984) 1229. [12] V. Bargmann, Ann. Math. 48 (1947) 568. [13] G. Henderson, W. Read and C.S. Ko, Program 434, Quantum Chemistry Program exchange, Indiana University (1982). [ 141 A.O. Barut, Dynamical groups and generalized symmetrics in quantum theory (University of Canterbury Press, Christchurch, 1971 ). [ 151 G. Bozzolo and A. Plastino, Phys. Rev. D24 (1981) 3113.