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A variational method for excited states
PhysicsI~ettersA162 (1992) 227—231 North-Holland
PHYSICS LETTERS A
A variational method for excited states K.L. Chan, S.K. Lee, K.L. Liu, K. Young Department ofPhysics, The Chinese University ofHong Kong, Hong Kong
and C.K. Au Department ofPhysics, University of South Carolina, Columbia, SC 29208, USA and Department ofApplied Science, CityPolytechnic ofHong Kong, Hong Kong Received 4 November 1991; accepted for publication 9 December 1991 Communicatedby J.P. Vigier
A variational method is proposed for excited states in quantum mechanics, based on minimizing the deviation ofa comparison 2+ ~gx4. For Hamiltonian from the true Hamiltonian. Very good accuracy is found for the anharmonic oscillator with V= 1x n—.x, the method agrees with the WKB result in form (viz. E~=Cg’ ‘3n413), with the coefficient Cgood to 0.5%.
1. Formulation
~t~~n(x)~Wn(a*(n),x).
In this paper we formulate a variational method for excited states in quantum mechanics. Consider an eigenvalue problem
The method is formally justified by noticing that if the parameters a were sufficiently numerous that the family h(a) covers a large part of the space of op-
H ~P“—E / ~P “
erators h(a) were on the to Hilbert coincidespace, with in H,particular, then (1.4)ifwould one such be
fl
—
fl
l 1
fl/
‘
where n may generically denote a set of quantum numbers. In order to approximate ~I-’,~ and E~,we introduce a family of comparison Hamiltonians h (a) depending on parameter(s) a, assumed to be exactly soluble: (1.2) where the eigenstates I w~ (a)> are assumed to be normalized. The comparison Hamiltonian h would provide a good approximation to H if H— h were small; therefore we are led to minimize D~(a)=<~~(a)[H—h(a)]
2
I~~(a)>
(1.3)
in order to determine the optimal value of a, to be denoted as a * ( n). The variational approximation is then E
e (a * ( n))
(1.4)
exact. In this sense, the justification is analogous to that for the conventional variational method for the general state, in which case one looks for comparison wavefunctions (rather than Hamiltonians) that hopefully come close to the true wavefunction. Lee et al. [1] have used a similar method of a comparison Hamiltonian, but only on the ground state, and instead of minimizing D~(a), they used the condition that the first order perturbative correction due to H—h (a) should vanish. This amounts to a single condition, and works if there is only one parameter a. We shall see that this condition is recovered in our scheme, but that we always have enough conditions to determine all the parameters a, simply because the conditions come from minimization. .
Arias de Saavedra and Buendia [21 have proposed a similar method, but they did a perturbation
0375-960l/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
227
Volume 162, number 3
PHYSICS LETTERS A
to Nth order in H—h(a). The value of a is then determined by minimizing the Nth order estimate. Although the results in selected examples appear to be accurate, minimization of the eigenvalue, rather than of the deviation, would seem to lack a sound theoretical basis in the case of excited states. Moreover, the need for doing a high-order perturbative calculation in the first instance complicates the variational calculation. We therefore regard the present work as complementary to these variational techniques [1,21 forthe excited state. One of the most interesting results is a surprising connection with the WKB method, at least in some examples. Although the method has been formally defined, the practical problem is that h (a) has to be simple enough to be exactly soluble for all a, and yet general enough to be able to come close to H. The question is then: how well can one do in the face of these conflicting requirements? This can only be investigated through examples; it turns out that one can do extremely well, and the basic reason is the freedom to choose one h(a) for eachstate n. 2. Example
10 February 1992
One of these equations is general whenever the comparison Hamiltonian contains an overall multiplicative parameter: h = wE, so that minimizing ~i.
(H_wE)21n>
+w2 with respect to w gives =
KnIHIn>=w=
where we adopt the simplified notation
I n> = i yl,, (a)> This condition, in ensuring that the comparison Hamiltonian yields the correct expectation value, is equivalent to demanding that the first order correction due to H—h should vanish [1]. In the present case, (2.5) allows w to be determined in terms of 1u:
(~
l
1)
-
+
1 g A2(n) 4 n+~ /~ ___
where we use the shorthand 2kIn>Ak(n)/~k
,
(2.6)
(2.7)
for the coefficients Ak ( n), which are explicitly A
We illustrate the method with the anharmonic oscillator
(2.5)
,
2+6n+3)
A
2(n)=~(6n
3+3On2+4On+l5)
3(n)=~(20n A
4+l4On3+344n2+28On+lO5) (2.8) The other condition, obtained from ÔD,,/ä!L=0, upon eliminating w using (2.6), leads to a sixth order polynomial for ~t, 4(n)=j~(7On
—
1 d2 ~ +~x2+~gx~,
(2.1)
and attempt to approximate it by a family of harmonic oscillators +~px2),
~
(2.2)
where the parameters are a = (a), ~t). By writing the parameters in this form, the eigenvalues of h depend only on w and the eigenfunctions depend only on u:
e,,(w,u)=w(n+1)
,
(2.3) (2.4)
where ço,, are the normalized eigenfunctions of the “standard” harmonic oscillator described by h (1, 1). Now D,, (w, ~u) is calculable in closed form, and the minimization yields algebraic equations for w and 228
a 6+a 5+...+ao=O, (2.9) 6,i 5# where a 2], a 6=—a2=~[A2(n)—(n+-~) 5=a4=O, a3 = ~g[A3(n)
—
(n+ ~)A2(n)]
a1=~g[(n+~)A2(n)—A3(n)]
2[A
a0=~g
2—A
2(n)
4(n)]
.
(2.10)
This is solved numerically to find ~ ( n), and put into (2.6) to give w~( n), which then determines the approximate energies via (2.3). The numerical computational effort is minimal. The result for the energies are surprisingly good
Volume
162, number
PHYSICS
3
10 February
LETTERS A
1992
Table 1 The fractional error e (in %) of the approximate eigenvalues: t = (approximation - exact ) /exact; (a) for variational method, (b ) with second order correction as in (3.5), (c) with 9X 9 truncation of (3.1). In column (c), numbers in brackets mean there is agreement with the exact value to the number of significant figures shown. Since H for small g is close to a harmonic oscillator, the comparison is most significant for moderate and large g. n
g=400
g=40 (a)
(c) 0.6x (7) (8)
1O-4
2.39 I .40 -0.18
(7) (6) (6) 1.8x 1O-4 2.2x 10-4 2.8x 1O-3
-0.22 -0.08 0.06 0.16 0.24 0.30
0 1 2
-0.15
0.030 0.027 -0.030
3 4 5 6 7 8
-0.19 -0.06 -0.06 0.17 0.24 0.30
-0.044 -0.110 -0.161 - 0.200 -0.230 - 0.252
2.07
1.26
(a)
(b)
when compared with the known values [ 31. The maximum percentage error over all states n up to n=lO is 0.05%, 0.25%, 1.25%, 2.07Oh, 2.33% and 2.39% respectively for g=O.O4, 0.4, 4, 40, 400 and 4000. If the lowest states n=O, 1 are excluded, the maximum error is under 0.3% for g= 400 and 4000. Table 1 gives the fractional error for each state in selected cases of moderate and large g. In the large n limit, (2.9) can be solved explicitly, and we find E,, = Cg”3n4’3 ,
(2.11)
where C=O.871 823. The WKl3 approximation, which is exact as n-+m, also gives the same form as (2.11), but with C replaced by C’ =0.867 125, a difference of about 0.5%. We have verified the same agreement in form and good approximation for the coefficient for all xZk potentials; the coefficients agree to about 2% for k up to 10. In a practical sense then, the present method has already incorporated the WKB approximation; in contrast, in the perturbation-variational method [ 2 1, because the variation is performed on a relatively complex Nth order result, the large n asymptotics would be rather more difficult to extract. In short, despite its simplicity, the method proposed here is uniformly accurate over the entire gn plane. In this context, one may note that of the conventional approximation methods, perturbation is valid at small g, the usual variational scheme works
(b)
(c) (7) (6) 2.1 x10-4
0.035 0.030 0.000
(6) (6) (6) 3.0x 1o-4 3.4x 1o-4 2.7x 1O-3
-0.045 -0.115 -0.169 -0.210 -0.241 -0.263
only for the lowest n’s, while the WKB method is valid only for n+co. Fairly good results are also obtained for the wavefunctions and these will be reported elsewhere. Giachetti and Tognetti [ 41, and Feynman and Kleinert [ 51 have developed a method for path integrals, e.g.
Z(P)= sD[x(r)l
expi-Stx(~)l),
by seeking the best quadratic approximation each /I and each average position
(2.12) to S for
B
The present work is in fact motivated by these path integral techniques, and the use of a harmonic oscillator h amounts to seeking the best quadratic approximation to H for each n. The two approaches are broadly (but not exactly) equivalent, and the amount of labour and the accuracy are comparable, though it is in practice easier to obtain the free energy F (8) from our estimate of E,, than to do the reverse starting from the path integral estimate of F(P).
3. Corrections In the path integral
approach,
the variational
re229
Volume 162, number 3
PHYSICS LETTERS A
sult [4,5] may be regarded as zeroth order estimates, around which perturbative corrections can be formulated systematically [6]. In much the same way, one can seek corrections to the results of the previous sections. By expanding in the basis *1 ~ I yi,,~,a ~n1 ~ *
— =
assumed to be complete, (1.1) can be written exactly as the infinite matrix equation ~ (Hm,,
—
E
,nn)
C~= 0,
(3.1)
,
(3.2)
where H,nn
(3.3) Because I yl~> are eigenfunctions of different Hamiltonians (i.e. h with different values of a), they are not orthogonal, although we shall normalize them to unit length. Nevertheless, because the basis is in a sense optimized, the matrix elements I,,,,, decrease rapidly off the diagonal; for example for m = 0 and n2, 4, 6, 8, I,,,,,=0.081, 0.019, 0.006, 0.003, in the case of g= 4000 (large g being the “worst” situation). In order to solve (3.1), we write H
=Ht0~+,~.H”~, I =it°~+~W’~, (3.4)
inn
,nn
inn
inn
inn
10 February 1992
small n, even for large g, as shown in table 1. While this perturbative series is formally similar to generating a series in g for H in (2.1), the fact that the basis has been optimized makes a significant difference. By comparison, second order perturbation in g would give, for g=4000, E,,=—2.6x 106, —2.1 x l0~, —7.7 x l0~ versus the true values 6.69422, 23.9722, 47.0173 for n=0, 1,2 [3]. Yet another, and indeed simpler, method is to truncate (3.1) to a finite matrix and find the eigenvalues E by standard mathematical packages. A 9 x 9 truncation already gives excellent results, as shown in table 1. Again, a similar truncation using the basis of 2/dx2+x2) H0=~(—d would be highly inaccurate, e.g. for g= 4000 the values obtained forE,,, n=0, 1, 2 would be 22.4, 147.3, 618.2, in contrast to the exact values quoted above. The truncated matrix also yields, in a straightforward manner, the eigenfunctions as a superposition of I An overall comparison may be presented via the particle density at finite temperatures T= 1/fl: ~
p(/3,x)=Z’ ~,exp(—/3E,,)Iy,~(x)I2,
(3.6)
Z=
(3.7)
mn
where Ht0~,~ are the diagonal parts of H and j, and H~,j~) are the off-diagonal parts, with )~ being a formal small parameter. It is then straightforward to develop a perturbation expansion in and the first and second order corrections for the energy are given
~
exp(—flE,,)
.
Fig. 1 shows p versus x for g= 40, 400 and T= 1, ob-
,~.,
~
=
~ (H~—E~°~I~)(H~ —E~°~I~) (3.5)
The sum in the second order correction converges rapidly (to four figures with ten nontrivial terms ~, for the “worst” case of g=4000 studied), and the second order correction improves E,,, especially for ~‘
Since the even and odd sectors decouple, both the number of terms retained in (3.5) and the size ofthe truncation refer to the number of states in that given sector; all truncations are centred around the state in question, except in the case ofthe lowest states.
230
0
~‘
X
Fig. 1. The particle densities p(fl, x) versus x for $= I and (a) g= 40 (crosses and associated curve), (b) g=400 (triangles and associated curve). The lines are our results, using a 9 x 9 truncation; the points are the exact results obtained by integrating the Schrodinger equation with the known values of E,.
Volume 162, number 3
PHYSICS LETTERS A
tamed using the values of E,, and ~,, found from the truncated matrix, compared with the exact p, the latter obtained by integrating the Schrodinger equation for each n using the known value ofE,, [6] and employing the Runge—Kutta algorithm. The comparison is seen to be very good. The possibility of correcting the variational result in a simple way makes the present method especially attractive,
4. Discussion The formulation of the method is straightforward; for a sufficiently large range ofh (a), the variational estimate would be in principle exact, and even for a relatively poor h (a), a sufficiently large truncation of (3.1) would again be in principle exact. The pleasant surprise is that at least in the example dealt with a very simple choice of h (a) gives such good results (—~2% globally) and a small truncation of (3.1) yields such dramatic improvements (<3 x l0~% globally). It is clear that an anharmonic oscillator, especially for large g, cannot be approximated by a single harmonic oscillator, but the present method allows a different comparison h (a) for each state n, and this is the reason for the success. This feature is very similar to the freedom to choose a different effectiveaction for each temperature and each average displacement in the path integral formulation [4,5 1~ Although there are similarities with variational perturbation methods [1,21, there are two signifi-
10 February 1992
cant differences. At a conceptual level, the minimization of the deviation would appear to have a sounder basis; at a technical level, we allow the kinetic energy operator in the comparison Hamiltonian to have an arbitrary coefficient as well. The present method may also permit the incorporation of useful physical insight. For example, to discuss the excited state of atomic He, one could use the sum of two independent hydrogen-like Hamiltonians for h, with the apparent nuclear charges Z1, Z2 seen by the electrons as (two of) the variational parameters; for a state with one electron in the ground state and the other in a highly excited state, one would expect the minimization of the deviation D to give 2, Z1 1. Work in applying this method to excited states in such nontrivial atomic systems is in progress.
—
—
References
[1] Y.C. Lee, W.N. Mei and K.C. Liu, J. Phys. C 15 (1982) L469. [2] F. Arias de Saavedra and E. Buendia, J. Phys. A 22 (1989) 1933; Phys. Rev. A 42 (1990)5073; J. Killingbeck, J. Phys. A 14 (1981) 1005.
[31F.T. Hioe, D.
MacMillen and E.W. Montroll, Phys. Rep. 43
(1978) 305.
[41R. Giachetti and V. Tognetti, Phys. Rev. Lett.
55 (1985) 912; Phys. Rev. B 33 (1986) 7647. [5] R.P. Feynman and H. Kleinert, Phys. Rev. A 34 (1986) 5080. [6] S.K. Lee, K.L. Liu and K. Young, Systematic perturbative corrections to the variational approximation to the thermal properties ofa quantum system, Phys. Rev. A, lobe published.
231
PHYSICS LETTERS A
A variational method for excited states K.L. Chan, S.K. Lee, K.L. Liu, K. Young Department ofPhysics, The Chinese University ofHong Kong, Hong Kong
and C.K. Au Department ofPhysics, University of South Carolina, Columbia, SC 29208, USA and Department ofApplied Science, CityPolytechnic ofHong Kong, Hong Kong Received 4 November 1991; accepted for publication 9 December 1991 Communicatedby J.P. Vigier
A variational method is proposed for excited states in quantum mechanics, based on minimizing the deviation ofa comparison 2+ ~gx4. For Hamiltonian from the true Hamiltonian. Very good accuracy is found for the anharmonic oscillator with V= 1x n—.x, the method agrees with the WKB result in form (viz. E~=Cg’ ‘3n413), with the coefficient Cgood to 0.5%.
1. Formulation
~t~~n(x)~Wn(a*(n),x).
In this paper we formulate a variational method for excited states in quantum mechanics. Consider an eigenvalue problem
The method is formally justified by noticing that if the parameters a were sufficiently numerous that the family h(a) covers a large part of the space of op-
H ~P“—E / ~P “
erators h(a) were on the to Hilbert coincidespace, with in H,particular, then (1.4)ifwould one such be
fl
—
fl
l 1
fl/
‘
where n may generically denote a set of quantum numbers. In order to approximate ~I-’,~ and E~,we introduce a family of comparison Hamiltonians h (a) depending on parameter(s) a, assumed to be exactly soluble: (1.2) where the eigenstates I w~ (a)> are assumed to be normalized. The comparison Hamiltonian h would provide a good approximation to H if H— h were small; therefore we are led to minimize D~(a)=<~~(a)[H—h(a)]
2
I~~(a)>
(1.3)
in order to determine the optimal value of a, to be denoted as a * ( n). The variational approximation is then E
e (a * ( n))
(1.4)
exact. In this sense, the justification is analogous to that for the conventional variational method for the general state, in which case one looks for comparison wavefunctions (rather than Hamiltonians) that hopefully come close to the true wavefunction. Lee et al. [1] have used a similar method of a comparison Hamiltonian, but only on the ground state, and instead of minimizing D~(a), they used the condition that the first order perturbative correction due to H—h (a) should vanish. This amounts to a single condition, and works if there is only one parameter a. We shall see that this condition is recovered in our scheme, but that we always have enough conditions to determine all the parameters a, simply because the conditions come from minimization. .
Arias de Saavedra and Buendia [21 have proposed a similar method, but they did a perturbation
0375-960l/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
227
Volume 162, number 3
PHYSICS LETTERS A
to Nth order in H—h(a). The value of a is then determined by minimizing the Nth order estimate. Although the results in selected examples appear to be accurate, minimization of the eigenvalue, rather than of the deviation, would seem to lack a sound theoretical basis in the case of excited states. Moreover, the need for doing a high-order perturbative calculation in the first instance complicates the variational calculation. We therefore regard the present work as complementary to these variational techniques [1,21 forthe excited state. One of the most interesting results is a surprising connection with the WKB method, at least in some examples. Although the method has been formally defined, the practical problem is that h (a) has to be simple enough to be exactly soluble for all a, and yet general enough to be able to come close to H. The question is then: how well can one do in the face of these conflicting requirements? This can only be investigated through examples; it turns out that one can do extremely well, and the basic reason is the freedom to choose one h(a) for eachstate n. 2. Example
10 February 1992
One of these equations is general whenever the comparison Hamiltonian contains an overall multiplicative parameter: h = wE, so that minimizing ~i.
(H_wE)21n>
KnIHIn>=w
where we adopt the simplified notation
I n> = i yl,, (a)> This condition, in ensuring that the comparison Hamiltonian yields the correct expectation value, is equivalent to demanding that the first order correction due to H—h should vanish [1]. In the present case, (2.5) allows w to be determined in terms of 1u:
(~
l
1)
-
+
1 g A2(n) 4 n+~ /~ ___
where we use the shorthand 2kIn>Ak(n)/~k
,
(2.6)
(2.7)
for the coefficients Ak ( n), which are explicitly A
We illustrate the method with the anharmonic oscillator
(2.5)
,
2+6n+3)
A
2(n)=~(6n
3+3On2+4On+l5)
3(n)=~(20n A
4+l4On3+344n2+28On+lO5) (2.8) The other condition, obtained from ÔD,,/ä!L=0, upon eliminating w using (2.6), leads to a sixth order polynomial for ~t, 4(n)=j~(7On
—
1 d2 ~ +~x2+~gx~,
(2.1)
and attempt to approximate it by a family of harmonic oscillators +~px2),
~
(2.2)
where the parameters are a = (a), ~t). By writing the parameters in this form, the eigenvalues of h depend only on w and the eigenfunctions depend only on u:
e,,(w,u)=w(n+1)
,
(2.3) (2.4)
where ço,, are the normalized eigenfunctions of the “standard” harmonic oscillator described by h (1, 1). Now D,, (w, ~u) is calculable in closed form, and the minimization yields algebraic equations for w and 228
a 6+a 5+...+ao=O, (2.9) 6,i 5# where a 2], a 6=—a2=~[A2(n)—(n+-~) 5=a4=O, a3 = ~g[A3(n)
—
(n+ ~)A2(n)]
a1=~g[(n+~)A2(n)—A3(n)]
2[A
a0=~g
2—A
2(n)
4(n)]
.
(2.10)
This is solved numerically to find ~ ( n), and put into (2.6) to give w~( n), which then determines the approximate energies via (2.3). The numerical computational effort is minimal. The result for the energies are surprisingly good
Volume
162, number
PHYSICS
3
10 February
LETTERS A
1992
Table 1 The fractional error e (in %) of the approximate eigenvalues: t = (approximation - exact ) /exact; (a) for variational method, (b ) with second order correction as in (3.5), (c) with 9X 9 truncation of (3.1). In column (c), numbers in brackets mean there is agreement with the exact value to the number of significant figures shown. Since H for small g is close to a harmonic oscillator, the comparison is most significant for moderate and large g. n
g=400
g=40 (a)
(c) 0.6x (7) (8)
1O-4
2.39 I .40 -0.18
(7) (6) (6) 1.8x 1O-4 2.2x 10-4 2.8x 1O-3
-0.22 -0.08 0.06 0.16 0.24 0.30
0 1 2
-0.15
0.030 0.027 -0.030
3 4 5 6 7 8
-0.19 -0.06 -0.06 0.17 0.24 0.30
-0.044 -0.110 -0.161 - 0.200 -0.230 - 0.252
2.07
1.26
(a)
(b)
when compared with the known values [ 31. The maximum percentage error over all states n up to n=lO is 0.05%, 0.25%, 1.25%, 2.07Oh, 2.33% and 2.39% respectively for g=O.O4, 0.4, 4, 40, 400 and 4000. If the lowest states n=O, 1 are excluded, the maximum error is under 0.3% for g= 400 and 4000. Table 1 gives the fractional error for each state in selected cases of moderate and large g. In the large n limit, (2.9) can be solved explicitly, and we find E,, = Cg”3n4’3 ,
(2.11)
where C=O.871 823. The WKl3 approximation, which is exact as n-+m, also gives the same form as (2.11), but with C replaced by C’ =0.867 125, a difference of about 0.5%. We have verified the same agreement in form and good approximation for the coefficient for all xZk potentials; the coefficients agree to about 2% for k up to 10. In a practical sense then, the present method has already incorporated the WKB approximation; in contrast, in the perturbation-variational method [ 2 1, because the variation is performed on a relatively complex Nth order result, the large n asymptotics would be rather more difficult to extract. In short, despite its simplicity, the method proposed here is uniformly accurate over the entire gn plane. In this context, one may note that of the conventional approximation methods, perturbation is valid at small g, the usual variational scheme works
(b)
(c) (7) (6) 2.1 x10-4
0.035 0.030 0.000
(6) (6) (6) 3.0x 1o-4 3.4x 1o-4 2.7x 1O-3
-0.045 -0.115 -0.169 -0.210 -0.241 -0.263
only for the lowest n’s, while the WKB method is valid only for n+co. Fairly good results are also obtained for the wavefunctions and these will be reported elsewhere. Giachetti and Tognetti [ 41, and Feynman and Kleinert [ 51 have developed a method for path integrals, e.g.
Z(P)= sD[x(r)l
expi-Stx(~)l),
by seeking the best quadratic approximation each /I and each average position
(2.12) to S for
B
The present work is in fact motivated by these path integral techniques, and the use of a harmonic oscillator h amounts to seeking the best quadratic approximation to H for each n. The two approaches are broadly (but not exactly) equivalent, and the amount of labour and the accuracy are comparable, though it is in practice easier to obtain the free energy F (8) from our estimate of E,, than to do the reverse starting from the path integral estimate of F(P).
3. Corrections In the path integral
approach,
the variational
re229
Volume 162, number 3
PHYSICS LETTERS A
sult [4,5] may be regarded as zeroth order estimates, around which perturbative corrections can be formulated systematically [6]. In much the same way, one can seek corrections to the results of the previous sections. By expanding in the basis *1 ~ I yi,,~,a ~n1 ~ *
— =
assumed to be complete, (1.1) can be written exactly as the infinite matrix equation ~ (Hm,,
—
E
,nn)
C~= 0,
(3.1)
,
(3.2)
where H,nn
(3.3) Because I yl~> are eigenfunctions of different Hamiltonians (i.e. h with different values of a), they are not orthogonal, although we shall normalize them to unit length. Nevertheless, because the basis is in a sense optimized, the matrix elements I,,,,, decrease rapidly off the diagonal; for example for m = 0 and n2, 4, 6, 8, I,,,,,=0.081, 0.019, 0.006, 0.003, in the case of g= 4000 (large g being the “worst” situation). In order to solve (3.1), we write H
=Ht0~+,~.H”~, I =it°~+~W’~, (3.4)
inn
,nn
inn
inn
inn
10 February 1992
small n, even for large g, as shown in table 1. While this perturbative series is formally similar to generating a series in g for H in (2.1), the fact that the basis has been optimized makes a significant difference. By comparison, second order perturbation in g would give, for g=4000, E,,=—2.6x 106, —2.1 x l0~, —7.7 x l0~ versus the true values 6.69422, 23.9722, 47.0173 for n=0, 1,2 [3]. Yet another, and indeed simpler, method is to truncate (3.1) to a finite matrix and find the eigenvalues E by standard mathematical packages. A 9 x 9 truncation already gives excellent results, as shown in table 1. Again, a similar truncation using the basis of 2/dx2+x2) H0=~(—d would be highly inaccurate, e.g. for g= 4000 the values obtained forE,,, n=0, 1, 2 would be 22.4, 147.3, 618.2, in contrast to the exact values quoted above. The truncated matrix also yields, in a straightforward manner, the eigenfunctions as a superposition of I An overall comparison may be presented via the particle density at finite temperatures T= 1/fl: ~
p(/3,x)=Z’ ~,exp(—/3E,,)Iy,~(x)I2,
(3.6)
Z=
(3.7)
mn
where Ht0~,~ are the diagonal parts of H and j, and H~,j~) are the off-diagonal parts, with )~ being a formal small parameter. It is then straightforward to develop a perturbation expansion in and the first and second order corrections for the energy are given
~
exp(—flE,,)
.
Fig. 1 shows p versus x for g= 40, 400 and T= 1, ob-
,~.,
~
=
~ (H~—E~°~I~)(H~ —E~°~I~) (3.5)
The sum in the second order correction converges rapidly (to four figures with ten nontrivial terms ~, for the “worst” case of g=4000 studied), and the second order correction improves E,,, especially for ~‘
Since the even and odd sectors decouple, both the number of terms retained in (3.5) and the size ofthe truncation refer to the number of states in that given sector; all truncations are centred around the state in question, except in the case ofthe lowest states.
230
0
~‘
X
Fig. 1. The particle densities p(fl, x) versus x for $= I and (a) g= 40 (crosses and associated curve), (b) g=400 (triangles and associated curve). The lines are our results, using a 9 x 9 truncation; the points are the exact results obtained by integrating the Schrodinger equation with the known values of E,.
Volume 162, number 3
PHYSICS LETTERS A
tamed using the values of E,, and ~,, found from the truncated matrix, compared with the exact p, the latter obtained by integrating the Schrodinger equation for each n using the known value ofE,, [6] and employing the Runge—Kutta algorithm. The comparison is seen to be very good. The possibility of correcting the variational result in a simple way makes the present method especially attractive,
4. Discussion The formulation of the method is straightforward; for a sufficiently large range ofh (a), the variational estimate would be in principle exact, and even for a relatively poor h (a), a sufficiently large truncation of (3.1) would again be in principle exact. The pleasant surprise is that at least in the example dealt with a very simple choice of h (a) gives such good results (—~2% globally) and a small truncation of (3.1) yields such dramatic improvements (<3 x l0~% globally). It is clear that an anharmonic oscillator, especially for large g, cannot be approximated by a single harmonic oscillator, but the present method allows a different comparison h (a) for each state n, and this is the reason for the success. This feature is very similar to the freedom to choose a different effectiveaction for each temperature and each average displacement in the path integral formulation [4,5 1~ Although there are similarities with variational perturbation methods [1,21, there are two signifi-
10 February 1992
cant differences. At a conceptual level, the minimization of the deviation would appear to have a sounder basis; at a technical level, we allow the kinetic energy operator in the comparison Hamiltonian to have an arbitrary coefficient as well. The present method may also permit the incorporation of useful physical insight. For example, to discuss the excited state of atomic He, one could use the sum of two independent hydrogen-like Hamiltonians for h, with the apparent nuclear charges Z1, Z2 seen by the electrons as (two of) the variational parameters; for a state with one electron in the ground state and the other in a highly excited state, one would expect the minimization of the deviation D to give 2, Z1 1. Work in applying this method to excited states in such nontrivial atomic systems is in progress.
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References
[1] Y.C. Lee, W.N. Mei and K.C. Liu, J. Phys. C 15 (1982) L469. [2] F. Arias de Saavedra and E. Buendia, J. Phys. A 22 (1989) 1933; Phys. Rev. A 42 (1990)5073; J. Killingbeck, J. Phys. A 14 (1981) 1005.
[31F.T. Hioe, D.
MacMillen and E.W. Montroll, Phys. Rep. 43
(1978) 305.
[41R. Giachetti and V. Tognetti, Phys. Rev. Lett.
55 (1985) 912; Phys. Rev. B 33 (1986) 7647. [5] R.P. Feynman and H. Kleinert, Phys. Rev. A 34 (1986) 5080. [6] S.K. Lee, K.L. Liu and K. Young, Systematic perturbative corrections to the variational approximation to the thermal properties ofa quantum system, Phys. Rev. A, lobe published.
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