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Energy approximants and perturbation theory
Volume 74A, number 6
PHYSICS LETTERS
10 December 1979
ENERGY APPROXIMANTS AND PERTURBATION THEORY I.K. DMLTRIEVA Heat and Mass Transfer Institute, Minsk, USSR
and G.I. PLINDOV Nuclear Power Engineering Institute, Minsk, USSR Received 2 August 1979
It is shown that the use of the quantum-mechanical virial and Heliman—Feynman theorems provides a natural way to construct the approximants for the eigenvalues of the Schrodinger equation with the correct asymptotic behaviour.
In many physical problems, from molecular physics to quantum field theory, the necessity appears to study how the energy of a system depends on a param eter, A, which enters the problem linearly. For this purpose perturbation theory is frequently used to find the energy values for hamiltonians which do not allow an exact solution, and the function E(A) may be expressed as the series E(X) =
E EkX”. k=O
(0
However, since in the majority of physical problems the perturbation series is semi-convergent or even divergent, regularization methods [1] should be employed. We will show that the application of the virial theorem and the Hellmann—Feynmantheorem makes it possible to propose a procedure which considerably speeds up the convergence of the generally accepted regularization methods. Consider a system making a finite motion defined by the hamiltonian H(A), H(A)
=
T + V1 + Ay2.
(2)
Here T is the kinetic energy operator, V1 and V2 are the potential parts being explicit functions of the coordinates. Suppose that the eigenvalues of the hamil-
tonian E(X) are expressed as a series of type (1). It is well known that the mean values of the hainiltonian (2) obey the general virial and Heilman—Feynman theorems (these are often used conformaMy to pertur. bation theory, cf. ref. [2]): 2
VV1)+X(rVV2),
and dE/dX = (~‘(X)Ia~iax I~’(X)>, respectively, where < ) denotes the mean value of an operator. Combining these theorems gives the first-order differential equation which relates the energy to the mean values of the operators V1 and r V(V1 + + A dEfdA (3) If V1 (r) and V2(r) are homogeneous functions of the coordinates of order of n and m, respectively, eq. (3) assumes a surprisingly simple form: E(X) = ~ + ~A(r• VV2) +
-
E(A) = ~(n +2) W1) + ~(m +2) A dE/dA (4) Integrating this equation for E(O) = E0, gives the -
relation between the energy and the mean value of the operator W1):
f
2) du,
(5)
E(X) = E0 + X2/(m+2) 0 f(u) u_121(m+ 387
Volume 74A, number 6
PHYSICS LETTERS
where 2 f(X)
=
n+2 E0
~—~--~
— ~—~--~
10 December 1979
it directly follows that the lower bounds of (V1 (A)) give a sequence of upper bounds of E [N,N 1] and vice versa. We assume that we are dealing with Stieltjes series. We now discuss briefly the structure of the energy approximants. Taking into account that f(u) in (5) maybe defined as: —
(V~ (A))
.
Now the problem is to find a flA) such that it correctly describes this function both for small and large A. From (4) and (1) it follows that for A 0, (V1(X)) may be presented as a formal A-expansion: -~
2
N
f(u)=~a1(1+b1u)_1, i=1
it is easy to obtain 1(m+2)FN(A—21(m+2)), —~
VkAk.
(6)
Substitution of (6) into (5) naturally gives the mitial series expansion (1). The power series (6) should be regularized to obtain a new result. The expansion coefficients of series (6) behave similarly to those of series (1). Therefore, the problem of regularization ~
E[N,N+/] —E0 =A2
where FN(x) is expressed as a sum of logarithms and arc tangents. For A -÷0,E[N,N +j] is given as: 2N+-j E [N, N + f]
=
E 0+
a diverging series (6) is not more complex than that for If(6) series is(I). a Stieltjes series, then any sequence of Padé approximants [N, N + j] (f ~ —1) converges to an analytical function [1]. The Carleman condition ~
ly ~—1/(2k+1) = k=0’ ki
is enough in order that all these sequences tend to a common limit which gives the analytic continuation of function (V1(A)). Substitution of the sequence of the the Pade approximants [N,N + j] for series (6) into (5) and allowance for the fact that all their poles are beyond the integration domain (i.e., they lie on the negative real axis) yield the sequence of energy approximants converging to the exact value ofE(A) for any A. The convergence of this procedure, especially for large A, will be considerably better than that of the Padé approximants for series (1), since the asymptotic relation E(A) for A oo defined in eq. (5). If the Carleman condition is not satisfied, then the convergence of all sequences of Padé approximants [N,N + j] to a common limit is not provided; nevertheless, it is known that the sequences of the Padé approximants [N,NJ and [N,N 1] for series (6) give the best upper and lower bounds for the values of the function described by this series [1]. From (5) —~
—
388
~
EkAk + O(A2N+i+ 2/(m +2)).
= const., which ensures(7) Forcorrect A 0O~asymptotic FN(A_ the behaviour E(A -÷oo) —~A21(m+2). The correct asymptotic form of the energy is also preserved when other regularization methods for (V 1(A)) are used. For illustration of the specific features of the model
2/(m+2))
—~
proposed, let us consider the problem of the anharmonic oscillator which has been studied thoroughly during the last years (see refs. [3—5]and references therein)~ 2/~2 + + ~2m] ~4’~x) = E”P~x). (8) [d Here _0o 3 is of special interest because the convergence of the energy Padé approximants to the eigenvalues of (8) is not proved and rather strong arguments exist against this convergence [4]. It is therefore of great interest to apply the proposed method to calculations of the energies of a generalized oscillator with anharmonicity ~8• The coefficients of the asymptotic expansion (1) for the ground state of the octic oscillator are presented in ref. [41. The use of these coefficients and ...,
—
Volume 74A, number 6
PHYSICS LETTERS
10 December 1979
Table 1 Energy approximants for the ground state ofthe anharmonic oscillator.
x
10—2
10—1
1
10
102
Ep[18,181 — 1 E[1,1] — 1 E[3,3] — 1
3.58 X 10—2 3.07 X 10—2 3.56 .x 10—2
0.078 0.087 0.12
0.089 0.18 0.24
0.33 0.41
0.56 0.78
5.9 X 10~ 4.52 X 10~ 4.32 X 10—2 3.95 X 102
0.35 0.27 0.19 0.169
1.1 0.90 0.57 0.491
2.3 2.0 1.26 1.11
4.2
16
3.7
15
[4] this work this work
2.45 2.19
10.1 9.23
this work [5]
91 1 Ep[18,1 E[1,0] 1 E[3,2] 1 Eq[1,0I 1 Eexa~ 1 —
— —
—
—
4.37 X 10~
0.37
4 x io~
Ref. [4]
3.6
this work
2.0 2.6
this work
relation (4) yields the formal expansion (V 1 (A)>: 2(V1(X)> = I
—
105 607 835 2 —4—A + 64 A 125A~+
(9) 2810842 256 It is not difficult to show that the leading term in the —
asymptotic relation for Vk for k
-~
00
also equal to
(3k)!.
The analysis shows that series (9) is a Stieltjes series, best upper and[N,N] lowerand bounds are defined by the whose Padé approximants [N,N 1]. Substitution of the Padé approximant (V 1 (A)> into (5) gives an analytical expression for the energy approximants E [N,N + /] which was used in the calculation of table I. expression because of This its tedious form.is not presented here The data of table 1 show that the simplest approximants of this method, namely, E[ 1,11 and E [1,0] govern the ground-state energy better than the Padé approximants of series (1), E~[18,18] and E~[18,19] startingwithA=0.1. Similarestimatesmaybe obtained for an arbitrary excited state. Up to now we have considered the case when the potentials V 1(r) and V2(r) are homogeneous functions of the coordinates. The generalization to the case of more complex potentials is made immediately. Assume that the leading term V2(r) for r °°is proportional to rm. Then, using V2(r) = arm + ~i (r), it is easy to obtain from (3) the following relation: —
-~
~ + ~2m + 1\ ~2V’~ i)
—~ —
.1 t’1,1~ U~L~UI~,
where —
1
— —
A = q(I
—
0~ q ~
— 2ff(q),
E(A) = (I
—
i~q)1/
In this case, the perturbation V 8 1~y2.Ex. 2 the y above procedure panding E(q) in q and employing yields for the ground state: —
E(q)= 1 —~-~-~q —q’/5 16
—
297 098 025 ~9’s 512
+
?(‘~~ 185u~/~
g
\
64
~du
11
/
Taking into account that the integrand is a Stieltjes series, its simplest Padd approximant [1,01 is constructed. It gives an upper estimate ofR(q). One can see eighth line of(10) tablenoticeably 1 that the improves use of eq. (4a)from alongthe with transform the estimate of E(X) for large A. Note that if the singularity V2(r) for r is less than a power one, then eq. (4a) with m = 0 may be -+
~2(A) — ~(r• V(V1 + A~1(r))> + (V1) —
~ ~-~a1 (A
Integration of (4a) gives an expression for E(A) in (~ complete agreement with (5) if (V1> is replaced by 2(A)>.Further, the use of the Padé approximant (~2(X)> makes it possible toEN(A), construct a sequence of upper and lower bounds of whose convergence to the exact value depends on the form of the potential V 1 (r) and the function ~2(A). For ifiustration let us consider again the equation for the anharmonic oscillator (8) rearranged using the nonlinear transformation ofscaling the parameter A = q(l 2)/(n+2) and the coefficient~y = (1 Aq)—(m+ ~q)_hf2x [6]:
~Am(~1(r)).
00
used to estimate E(A). In this case the obtained esti389
Volume 74A, number 6
PHYSICS LETTERS
mates of the energy will be better than the Padé approximants of series (1). And finally it may be noted that the construction of the (V1 (A)) or (~2(X))Padé approximants is not the only possible method of regularization. If the Padé method leads to slowly convergent (or completely divergent) approximants of the energy, then it is possible to construct the analytic continuation of(V1(A)) or (~2(A)>using the more tedious but still more effective Padé—Borel method.
390
10 December 1979
References [1] G.A. Baker, Advances in Theoretical Physics, Vol. 1(1965).
[2] 1. Killingbeck, Phys. Lett. 65A (1978) 87. [3] B. Simon, Ann. Phys. (NY) 58 (1970) 76. [4] G. Turchetti, S. Graffi and V. Giecchi, Nuovo Cimento 4B (1971) 313. [5] K. Banerjee, Proc. R. Soc. A364 (1978) 265. [6] I.K. Dmitrieva and G.I. Plindov, J. de Phys. Lett., to be published.
PHYSICS LETTERS
10 December 1979
ENERGY APPROXIMANTS AND PERTURBATION THEORY I.K. DMLTRIEVA Heat and Mass Transfer Institute, Minsk, USSR
and G.I. PLINDOV Nuclear Power Engineering Institute, Minsk, USSR Received 2 August 1979
It is shown that the use of the quantum-mechanical virial and Heliman—Feynman theorems provides a natural way to construct the approximants for the eigenvalues of the Schrodinger equation with the correct asymptotic behaviour.
In many physical problems, from molecular physics to quantum field theory, the necessity appears to study how the energy of a system depends on a param eter, A, which enters the problem linearly. For this purpose perturbation theory is frequently used to find the energy values for hamiltonians which do not allow an exact solution, and the function E(A) may be expressed as the series E(X) =
E EkX”. k=O
(0
However, since in the majority of physical problems the perturbation series is semi-convergent or even divergent, regularization methods [1] should be employed. We will show that the application of the virial theorem and the Hellmann—Feynmantheorem makes it possible to propose a procedure which considerably speeds up the convergence of the generally accepted regularization methods. Consider a system making a finite motion defined by the hamiltonian H(A), H(A)
=
T + V1 + Ay2.
(2)
Here T is the kinetic energy operator, V1 and V2 are the potential parts being explicit functions of the coordinates. Suppose that the eigenvalues of the hamil-
tonian E(X) are expressed as a series of type (1). It is well known that the mean values of the hainiltonian (2) obey the general virial and Heilman—Feynman theorems (these are often used conformaMy to pertur. bation theory, cf. ref. [2]): 2
VV1)+X(rVV2),
and dE/dX = (~‘(X)Ia~iax I~’(X)>, respectively, where < ) denotes the mean value of an operator. Combining these theorems gives the first-order differential equation which relates the energy to the mean values of the operators V1 and r V(V1 + + A dEfdA (3) If V1 (r) and V2(r) are homogeneous functions of the coordinates of order of n and m, respectively, eq. (3) assumes a surprisingly simple form: E(X) = ~
-
E(A) = ~(n +2) W1) + ~(m +2) A dE/dA (4) Integrating this equation for E(O) = E0, gives the -
relation between the energy and the mean value of the operator W1):
f
2) du,
(5)
E(X) = E0 + X2/(m+2) 0 f(u) u_121(m+ 387
Volume 74A, number 6
PHYSICS LETTERS
where 2 f(X)
=
n+2 E0
~—~--~
— ~—~--~
10 December 1979
it directly follows that the lower bounds of (V1 (A)) give a sequence of upper bounds of E [N,N 1] and vice versa. We assume that we are dealing with Stieltjes series. We now discuss briefly the structure of the energy approximants. Taking into account that f(u) in (5) maybe defined as: —
(V~ (A))
.
Now the problem is to find a flA) such that it correctly describes this function both for small and large A. From (4) and (1) it follows that for A 0, (V1(X)) may be presented as a formal A-expansion: -~
2
N
f(u)=~a1(1+b1u)_1, i=1
it is easy to obtain 1(m+2)FN(A—21(m+2)), —~
VkAk.
(6)
Substitution of (6) into (5) naturally gives the mitial series expansion (1). The power series (6) should be regularized to obtain a new result. The expansion coefficients of series (6) behave similarly to those of series (1). Therefore, the problem of regularization ~
E[N,N+/] —E0 =A2
where FN(x) is expressed as a sum of logarithms and arc tangents. For A -÷0,E[N,N +j] is given as: 2N+-j E [N, N + f]
=
E 0+
a diverging series (6) is not more complex than that for If(6) series is(I). a Stieltjes series, then any sequence of Padé approximants [N, N + j] (f ~ —1) converges to an analytical function [1]. The Carleman condition ~
ly ~—1/(2k+1) = k=0’ ki
is enough in order that all these sequences tend to a common limit which gives the analytic continuation of function (V1(A)). Substitution of the sequence of the the Pade approximants [N,N + j] for series (6) into (5) and allowance for the fact that all their poles are beyond the integration domain (i.e., they lie on the negative real axis) yield the sequence of energy approximants converging to the exact value ofE(A) for any A. The convergence of this procedure, especially for large A, will be considerably better than that of the Padé approximants for series (1), since the asymptotic relation E(A) for A oo defined in eq. (5). If the Carleman condition is not satisfied, then the convergence of all sequences of Padé approximants [N,N + j] to a common limit is not provided; nevertheless, it is known that the sequences of the Padé approximants [N,NJ and [N,N 1] for series (6) give the best upper and lower bounds for the values of the function described by this series [1]. From (5) —~
—
388
~
EkAk + O(A2N+i+ 2/(m +2)).
= const., which ensures(7) Forcorrect A 0O~asymptotic FN(A_ the behaviour E(A -÷oo) —~A21(m+2). The correct asymptotic form of the energy is also preserved when other regularization methods for (V 1(A)) are used. For illustration of the specific features of the model
2/(m+2))
—~
proposed, let us consider the problem of the anharmonic oscillator which has been studied thoroughly during the last years (see refs. [3—5]and references therein)~ 2/~2 + + ~2m] ~4’~x) = E”P~x). (8) [d Here _0o
—
Volume 74A, number 6
PHYSICS LETTERS
10 December 1979
Table 1 Energy approximants for the ground state ofthe anharmonic oscillator.
x
10—2
10—1
1
10
102
Ep[18,181 — 1 E[1,1] — 1 E[3,3] — 1
3.58 X 10—2 3.07 X 10—2 3.56 .x 10—2
0.078 0.087 0.12
0.089 0.18 0.24
0.33 0.41
0.56 0.78
5.9 X 10~ 4.52 X 10~ 4.32 X 10—2 3.95 X 102
0.35 0.27 0.19 0.169
1.1 0.90 0.57 0.491
2.3 2.0 1.26 1.11
4.2
16
3.7
15
[4] this work this work
2.45 2.19
10.1 9.23
this work [5]
91 1 Ep[18,1 E[1,0] 1 E[3,2] 1 Eq[1,0I 1 Eexa~ 1 —
— —
—
—
4.37 X 10~
0.37
4 x io~
Ref. [4]
3.6
this work
2.0 2.6
this work
relation (4) yields the formal expansion (V 1 (A)>: 2(V1(X)> = I
—
105 607 835 2 —4—A + 64 A 125A~+
(9) 2810842 256 It is not difficult to show that the leading term in the —
asymptotic relation for Vk for k
-~
00
also equal to
(3k)!.
The analysis shows that series (9) is a Stieltjes series, best upper and[N,N] lowerand bounds are defined by the whose Padé approximants [N,N 1]. Substitution of the Padé approximant (V 1 (A)> into (5) gives an analytical expression for the energy approximants E [N,N + /] which was used in the calculation of table I. expression because of This its tedious form.is not presented here The data of table 1 show that the simplest approximants of this method, namely, E[ 1,11 and E [1,0] govern the ground-state energy better than the Padé approximants of series (1), E~[18,18] and E~[18,19] startingwithA=0.1. Similarestimatesmaybe obtained for an arbitrary excited state. Up to now we have considered the case when the potentials V 1(r) and V2(r) are homogeneous functions of the coordinates. The generalization to the case of more complex potentials is made immediately. Assume that the leading term V2(r) for r °°is proportional to rm. Then, using V2(r) = arm + ~i (r), it is easy to obtain from (3) the following relation: —
-~
~ + ~2m + 1\ ~2V’~ i)
—~ —
.1 t’1,1~ U~L~UI~,
where —
1
— —
A = q(I
—
0~ q ~
— 2ff(q),
E(A) = (I
—
i~q)1/
In this case, the perturbation V 8 1~y2.Ex. 2 the y above procedure panding E(q) in q and employing yields for the ground state: —
E(q)= 1 —~-~-~q —q’/5 16
—
297 098 025 ~9’s 512
+
?(‘~~ 185u~/~
g
\
64
~du
11
/
Taking into account that the integrand is a Stieltjes series, its simplest Padd approximant [1,01 is constructed. It gives an upper estimate ofR(q). One can see eighth line of(10) tablenoticeably 1 that the improves use of eq. (4a)from alongthe with transform the estimate of E(X) for large A. Note that if the singularity V2(r) for r is less than a power one, then eq. (4a) with m = 0 may be -+
~2(A) — ~(r• V(V1 + A~1(r))> + (V1) —
~ ~-~a1 (A
Integration of (4a) gives an expression for E(A) in (~ complete agreement with (5) if (V1> is replaced by 2(A)>.Further, the use of the Padé approximant (~2(X)> makes it possible toEN(A), construct a sequence of upper and lower bounds of whose convergence to the exact value depends on the form of the potential V 1 (r) and the function ~2(A). For ifiustration let us consider again the equation for the anharmonic oscillator (8) rearranged using the nonlinear transformation ofscaling the parameter A = q(l 2)/(n+2) and the coefficient~y = (1 Aq)—(m+ ~q)_hf2x [6]:
~Am(~1(r)).
00
used to estimate E(A). In this case the obtained esti389
Volume 74A, number 6
PHYSICS LETTERS
mates of the energy will be better than the Padé approximants of series (1). And finally it may be noted that the construction of the (V1 (A)) or (~2(X))Padé approximants is not the only possible method of regularization. If the Padé method leads to slowly convergent (or completely divergent) approximants of the energy, then it is possible to construct the analytic continuation of(V1(A)) or (~2(A)>using the more tedious but still more effective Padé—Borel method.
390
10 December 1979
References [1] G.A. Baker, Advances in Theoretical Physics, Vol. 1(1965).
[2] 1. Killingbeck, Phys. Lett. 65A (1978) 87. [3] B. Simon, Ann. Phys. (NY) 58 (1970) 76. [4] G. Turchetti, S. Graffi and V. Giecchi, Nuovo Cimento 4B (1971) 313. [5] K. Banerjee, Proc. R. Soc. A364 (1978) 265. [6] I.K. Dmitrieva and G.I. Plindov, J. de Phys. Lett., to be published.