P H Y S I C A L R E V I E W V O L U M E 152 , N U M B E R 4 23 D E C E M B E R 1966

Perturbation Method Based on the Principle of Moments* CHI-YU H U

Department ofPhysics and Astronomy, California State College, Long Beach, California (Received 25 July 1966; revised manuscript received 13 September 1966)

A perturbation method closely related to the classical problem of moments has been studied. The lowest order ground-state energy by this method is explicitly comparable to an ordinary third-order perturbation series. Application of this method to simple systems which possess exact solutions showed promising results.

I. INTRODUCTION

HOW can one do a high-order perturbation calcu­lation with the evaluation of only a few integrals?

This note will demonstrate such a possibility at least for a certain class of well-behaved interactions.

The method of moments has been studied extensively in applied mathematics.1 This method is applied to quantum mechanics in the following sections. Formulas for the lowest two orders of approximation are given in Sec. III. Comparison of the lowest order ground-state energy with an ordinary third-order perturbation series is given in Sec. IV. Finally, the method is applied to the simple-harmonic-oscillator Hamiltonian

P2

2m

and to the Hamiltonian

P2

H= hF0tanh2(#/a). 2m

The reason for choosing these two simple Hamilton-ians is that their exact solutions are known and can be conveniently compared with the results from the present method. Application of this method to systems in­volving only a few particles is straightforward and does not lose much in simplicity.

II. THE METHOD OF MOMENTS

Consider a system described by a Hamiltonian

H=H0+h. (1)

H has an energy spectrum given by

H\n)=En\n). (2)

The solutions of H0 are assumed to be known and are given by

Let2 Ho\n)=&n\n).

| 0 )=E<» |0 ) | «> . n=0

(3)

(4)

* This work was supported by a summer grant from the Cali­fornia State College at Long Beach.

1 Yu. V. Vorobyev, Method of Moments in Applied Mathematics (Gordon and Breach Science Publishers, Inc., New York, 1965).

2 Unless otherwise specified, the summation sign denotes a summation over the discrete set together with an integration over the continuous set of eigenfunctions.

152

From Eqs. (2) and (4), the successive moments of H with respect to the ground state 10) can be written

\i=(o|#|o)=x;cw£n,

where

\2=(0\H*\0)=XCnEn\

\m=(0\H«\Q)='£CnEnm,

c»=K»|o)|», E c = i .

(5)

In the theory of moments, the complete set of expansion coefficients {Cn} is called the distribution. An approximate distribution {Cn(N)} can be chosen such that the first 2N+1 moments of H are identical to those of {Cn}- This distribution is obtained by solving the following simultaneous equations for the unknowns C„(i\0and£w(i\T):

\l=i:Cn(N)En(N),

X2=ECn(^)Ew2(iV)3

71=0

(6)

N= 1,2,3,....

The rest of the moments of {Cn(N)} are given by

N Xm(iV)= E Cn(N)En™(N)^\m, m^2N+2.

According to the converse of the first and the second limits theorems in the theory of moments,3 if the upper limit of (\n)lln/n is finite, each distribution specified by the integer N, '

{Cn(N)} = Cn(N), »=0, 1,2, =0, n>N,

N:

8 Maurice George Kendall, The Advanced Theory of Statistics (Charles Griffin and Company, London, 1948), Vol. I, pp. 105-115.

1116

152 P E R T U R B A T I O N M E T H O D BASED ON P R I N C I P L E OF M O M E N T S 1117

provides an approximation to {Cn} in such a way that4 for the case of N= 1 is carried out in the Appendix. In Sec. IV, it is made plausible that even this lowest order approximation is comparable to a third-order perturbation expansion.

Hm{C»CY)> = {C»}, N-*co

lira {En(N)} = {En}.

An estimate of the error due to this approximation

m . SOLUTIONS FOR N= 1, 2 Rewrite Eq. (6) into the following form:

1 Xi X2

W + i j

1 Eo(N) Eo*(N)

1 Ex(N) Ef(N)

1 EN{N) ENKN)

0 0 0 0 0 0

E^{N) Ei"(N) E0

N+1(N) E^+^N) E0

N+2(N) E!N+2(N)

Eo^+^N) E!m+l(N)

ENN(N) 0 0

ENN+*(N) 0 1

E^'+KN) 6 6

C0(N) Ci(N) C2(N)

CN'(N) 0 0

6

The unknowns En(N) and Cn(N) can be separated Of particular interest is the ground-state energy by inverting the matrix. Consequently, the apparently Eo(N). Closed forms for E0(l) and E0(2) can readily be nonlinear equation (6) reduces to a simple eigenvalue obtained from Eqs. (7) and (8); they are problem.5 For N= 1 we have,

foriV=2

E„(1)+JE1(1) =

•E„(l)Ei(l) =

Co(l)=

d ( l ) =

we have, £„(2)+£i(2)+£2(2)

(x3--A1A2XA4—A1A3) —

A3—A1A2

— a , A2-A12

A1A3—A22

- — Q X2-Xi2

S i W - X i

E1(l)-E0(l)' \i-E0(l)

~ EiW-Eoil)'

(x5-x2x3)(x2-x12)

(7)

(Xsi-XA:,)2- (X2-Xi2) (X4-X22) Eo(2)£i(2)+£0(2)£2(2)+E1(2)£2(2)

(X4-X1X3)(X4-X22)-(X3-X1X2)(X5-X2X3)

—P,

= ?, (X3-XiX2)2- (X2-Xi2) (X4-X2

2) /A3A1—A22\ A4A1—A2A3

£0(2)E1(2)£2(2) = -p[ ) = \ X2-X12 / X 2 - X i 2

£2(2)£ l(2)-X1(£1(2)+£2(2))+X2

(8)

= — r,

Co(2) =

Ci(2) =

C2(2) =

(£i(2)-£0(2))(£2(2)-£o(2)) -JE2(2)JEo(2)+X1(£2(2)+£0(2))-X2

(JE1(2)-Eo(2))(£2(2)-E1(2)) ' £o(2)Eo(2)-X1(£0(2)+£i(2))+X2

(E2(2)-£x(2))(£2(2)-£o(2)) 4 A more rigorous proof of the convergence problem can be

found in Ref. 1. 6 A different approach to the solution of this problem can be

found in Ref. 1.

£o(l) = !a - [ ( f«) 2 - /3] 1 / 2 , (9)

£o(2)= -\p+2(-W'2 cos(!«M-120°), (10) 0=f(3<7-£2), b=(\/n)(2p*-9pq+21r),

c o s 0 = - W ( - * « ) 3 / 2 .

IV. COMPARISON OF £„(1) WITH A THIRD-ORDER PERTURBATION EXPANSION

Using the identity

E \n){n\=i, n=0

Eq. (9) can be put into the form

£o( l )=(0 |# |0)+§A

-C( |A)2+E (0\h\n)(n\h\0)J'\ (11)

E E (o|%)M#|<K^|o) A= — (0|Z7|0)

E (0\h\n)(n\h\0) n?#>

E E (0\h\n)(n\h\n')(n'\h\0)

E (0\k\n)(n\h\0) -E—EQ,

E—EQ-

E (0|A|w)[(w|ff |#)-(0|H|0)](»|^|0)

E {Q\h\n){n\h\0)

1118 C H I - Y U H U 152

Expanding Eq. (11) and keeping only terms up to third order, we have

E0(l) = -Eo+(0|ft|0)-

E (0\h\n)(n\h\0)

E—EQ

E E (0|A|n)(»|A|»')(n'|A|0)

(E-Eo)2 . (12)

Equation (12) is explicitly comparable to an ordinary third-order perturbation expansion.6

V. APPLICATIONS

A. The Simple Harmonic Oscillator H0=P2/2m+i^Jc2, h=ibx2

This problem is trivial. It is, however, illustrative, because the perturbation series can be written down explicitly to all order,7 as as well as E0(l) and E0(2).

The exact solution for the ground state is, of course, the well-known expression

• j y / 2

K)

=ii"(1+*r*QVAQ"-ix*(i)' +ix*(-)

(32)AKJ J (13)

The first seven terms correspond to a sixth-order perturbation expansion.7

The quantities a, (3, p, q, and r that appear in Eqs. (7) to (10) can be evaluated by the calculation of a limited number of integrals. For example

X2-Xi2=(0|^2 |0)-(0|^|0)2 ,

X3-X1X2=(0|^5rA|0)-(0|^|0)(0|^2|0)

+£o(0|A2 |0)-2£o(0|^|0)2,

XiX3-X22=[£o+(0|^|0)](0|A^|0) - [£o (0 | ^ | 0 )+ (0 | ^ | 0 ) ] 2 , (14)

so that a and fi that appear in Eq. (7) can be obtained by evaluating the three integrals (0|&|0), (0|&2|0), and (0\hHh[0). These integrals can often^be^calculated very easily.

Equations (9) and (10) give E0(l) and E0(2) in closed forms. They are expanded up to sixth order for the

6 The inequality Wj^wVO under the summation signs means that terms with w = 0, w' = 0, and n = nf are to be excluded from the summation.

7 L. I. Schiff, Quantum Mechanics (McGraw-Hill Book Com­pany, Inc., New York, 1949), p. 153,

convenience of comparison with Eq. (13). We have

+ & X & ( - ) - * X — - ( - ) + • • • ] . (15)

r b /b\2 / & \ 8 /b\A

Eo(2) = |feo0^l+|--|^-j +A(-J -* X ^(^ j

+ 8 X 3 2 U 16X(32)AK)+ 1 (16)

It is clear that Eqs. (15) and (16) are, respectively, better than the third- and the fifth-order perturbation expansion of Eq. (13).

B. A Soluble Potential Well

The Hamiltonian

H=p2/2m+V0 tanh2(#/a), (17)

has been solved exactly.8 The bound-state energies are given by

¥ rl/SmVoa2

En=Vo -2ma2L2\

«=0 , l , 2 , . - - , iV .

)l/2 -i2

-(»+*) J , (18)

The total number of bound states is N+l9 where

)l /2 /2mV0a2 x1'2

N<[ \ ¥

When N is large, Eq. (18) can be expanded for small n, giving

E«c~ft«o(»+i)-( V^+|)2+- • • , \2maV

where co0=(2Fo/W)1/2.

Thus, the lowest few states approximate those of a simple harmonic oscillator

f Ho= \~hkx2,

2m 2 To

k = mco02 = .

a2

To apply our method, Eq. (17) is rewritten as H=H0+h,

h= F0[tanh2(#/tf)-- (*/<*) *]• 8 P. M. Morse and Herman Feshbach, Methods of Theoretical

Physics (McGraw-Hill Book Company, Inc., New York, 1953), pp. 1651 and 1673.

152 P E R T U R B A T I O N M E T H O D BASED ON P R I N C I P L E OF M O M E N T S 1119

TABLE I. Exact and lowest order ground-state energy for different values of R.

R

No. of bound states Eo/Vo Eo(l)/V0

DE 0( l ) -£o]/£o LE0+(0\k\0)3/Vo [£o+(0 | /^ |0)-Eo]/Eo

1

1 0.61803 0.66633 7.8% 0.77368

25.1%

9

3 0.28238 0.28373 0.4% 0.29458 4.3%

25

5 0.18100 0.18124 0.1% 0.18421 1.7%

49

7 0.13302 0.13310 0.06% 0.13431 0.97%

The integrals (0|A|0), (0|A2|0), and (0\hHh\0) have been computed numerically for R—2mVoa2/^t2=l, 9, 25, and 49. They correspond to N=0, 2, 4, and 6 or 1, 3, 5, and 7 bound states, respectively. The results of -Eo(l) are listed in Table I together with the exact ground-state energy EQ and the results of a first-order perturbation theory.

VI. DISCUSSION

An inspection of Eq. (21) shows that EQ(1) is de­termined most accurately if C0 is the largest coefficient in the expansion of Eq. (5). The success of the method depends on the unperturbed wave function 10) as well as on the energy spectrum En of the system H. A rela­tively large energy gap between the ground state and the first excited state of H and a fast dropping off of Cn with increasing En are both essential in reducing the errors. Clearly, the higher moments Xt- will not converge unless the coefficient Cn drops off essentially to zero at some finite En, if the energy spectrum of H increases without bound.

It is noticed in Sec. V that the existence of the un­perturbed Hamiltonian EQ and a complete set of un­perturbed solutions is more of a luxury there than a necessity. One can have the freedom of choosing |0). In fact, it can be shown1 that Eo(N) is completely identical to a variational problem with the trial wave function

£ anH"\0), n=o

where an,n=0,1, • • •, N are the variational parameters. Of course, 10) must be so chosen that it is essentially orthogonal to eigenstates belonging to the unbounded part of the spectrum of H. If Cn drops off to zero at some finite En, the convergence condition mentioned in Sec. II is obviously satisfied. This condition does not seem to be hard to satisfy for well-behaved interactions. It is, however, not so easy for singular interactions.

Alternatively, if H is a bounded linear operator,1

i.e., its energy spectrum is bounded, the convergence condition is always satisfied for any |0). But such is never the case for a real system. One may try to con­struct a bounded linear operator f(H), such as the resolvent of H, to replace H in Eqs. (5) and (6). Any such attempt will complicate the evaluation of the moments.

The author wishes to thank Professor Eugen Merz-bacher, Professor S. A. Moszkowski, and Dr. Bruce L. Scott for many helpful discussions. She is also grateful to the California State College at Long Beach for a summer grant. Special thanks are due to Mrs. Judy Anderson for her kind help with the typing.

Let

APPENDIX: ERROR FOR N=l

E0(l) = Eo+AEo, Co(l) = Co+AC0,

E1(1) = E1+AE1) C1(1) = C1+AC1. (19)

Substitute the quantities defined by Eq. (19) into Eq. (6). Neglect terms of second order. We have approximately

A C 0 + A C i = £ C n ,

£oACo+EiACi+C0A£o+CiA£i= £ C A ,

(20)

£o2ACo+£i2ACi+2Co£oA£0+2Ci£iA£1= £ CnEn\

£o3AC0+£i3AC1+3C(>£o2A£o+3Ci£i2AEl= £ CnEnK n=2

Solving the linear equations (20) for the errors, we have

1 AE0=

A £ i =

(LC,(£ . -£ i )»(£ . -£o)) , C0(Ei-E0)

2 »H2

1 „ , - — — ( £ Cn(En-E1)(En-E0y),

(21)

AC0

ACi=-

(Ei-EoY »-*

- 1

( E C„(E n -E 1 ) 2 (2£ B +£ 1 -3£ 0 ) ) ,

(Ei-Eo)3 »-2 ( E Cn(En-Etf{2En+E,-3Ex)).