A variation principle for arbitrary operators

6.B

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Nuclear Physics 41 (1963) 497--503; (~) North-Holland Publishing Co., Amsterdam

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Not to be reproduced by photoprint or microfilm without written permission from the publisher

A VARIATION PRINCIPLE FOR ARBITRARY OPERATORS L. M. D E L V E S

Applied Mathematics Department, University of New South Wales, Kensington, N.S. IV., Australia Received 28 September 1962 A variation principle, linear in the Hamiltonian, is given for the expectation value o f a n arbitrary operator W over bound states. A particular case o f the results is a variation principle for the wave function itself. In general, the principle does n o t give a bound. A n example is given to illustrate the method.

Abstract:

1. Introduction For most complex systems of interest in nuclear physics the equations of motion have not been integrated directly in any realistic approximation. Instead, approximations @T to the wave function are found by inserting an assumed form for @T(di) containing variable parameters d~ into a variation principle for the energy and mlnlmlslng:

ET= minfO*(a,)(7"+V)g,~(a,)d~, flq,T(a,)led~=1.

(1.1)

This defines a "best" ~0T(E) for calculating the energy; however, other properties of the wave function, such as the quadrupole moment, Coulomb radius, etc., are usually also calculated directly from ~0T(E), that is, we approximate the expectation value of other operators W by

w =f

f o*(E)WOT(E)d = ,.

(1.2)

Such estimates of W are not variational, and are therefore inherently less reliable than the estimate (1.1) for E; if we write

~bT

=

~+~,

then ET = E + O(~2), while in general (w>~- = + o(~). It would therefore be desirable to have a variation principle for ( W) analogous to (1.1) from which estimates of error 0(8 2) could be made. We give such a principle here, for arbitrary Hermitian operators W. In particular, the operator W = 5(X-Xo) leads to a variation principle for the wave function itmlf. 497

498

L.M.

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2. Variation Principles for Arbitrary Operators We consider first for simplicity the one-dimensional SchrSdinger equation Ida2 - k 2 +

V] Go(r)= 0,

flPo2dr =

1,

(2.1)

for a bound state Go of energy - k o2. We wish to find a variation principle for the expectation value of an operator W over d/o, that is, for ( W ) = f~/'o W¢odr.

(2.2)

Consider the associated equation: [d~2 - k 2 ( 2 ) + V + 2W 1 ¢(r, 2) = 0,

(2.3)

considered for a given 2 as an eigenvalue equation for k(2). Differentiating (2.3) with respect to ;t and setting 2 = 0 we have = (k~ - W ) $ o ( r ),

(2.4)

where ¢'#) =

4:0'

dk2 I k2t = - ~ 4=0'

and ¢1 satisfies the same boundary conditions as Go: I#t(rc) = 0 at the hard core radius r o ¢1(r) ~ 0 exponentially as r --} co. Equation (2.4) is an inhomogeneous equation for $1; it has solutions only for one value of k 2. Multiplying (2.4) on the left by ~ko and integrating we find

f ¢o [-~ -k2 + V] ~t dr = O = f ~ko(k2- W)d/odr, since the operator

(2.5)

(d2[dr2)-k 2+ V is self-adjoint. Thus we have k 2 = (W>.

(2.6)

This result is well known from perturbation theory; however, our results do not depend on the existence of a perturbation expansion, but only on the existence of d¢ and dk2 at 2 = 0. d2 d2

VARIATION pRINCIPLE

~t99

We can rewrite (2.5) as a functional equation for ktZ:

k~ = a f ~,o I-~ -k~)+Vl ~,dr+ f C,oWi,odr,

(2.7)

with arbitrary constant a. This equation is stationary for variations of ~bt since

,~,[k:] = a f ,~¢~ [~-~rz-k2 + VJ ~odr = O.

(2.8)

However, variation of ~o gives

,5oEk:]= a f r~'o [~--~2r2-k~)+ Vl ~'xdr+2f 'SOoW~'odr = f 6~koEkZt+ (2-- a)W2¢ o dr, according to (2.4). Now S 6¢/o ¢/o = 0 + O ( ~ o 2) since $o satisfies j" ~,o2 dr = 1; so that 6Ek2] = 6o[k2] +3lE k2] -

(2-a)fr¢o W¢odr.

(2.9)

Hence (2.7) is a variational principle for < IV> = k 2 if we take a = 2:

(W) = 2f ¢o [ ~ -kg+V? Oldr+ f OoW¢odr.

(2.10)

Conversely, setting the variation of (2.10) equal to zero, subject to the normalization (2.1) for ~ko, leads to equations (2.1) and (2.4) for fro and ~l. 3. Method of Use

Equation (2.10) can be used in either of the usual ways. If approximate solutions ffor and @xr of (2.1) and (2.4) are known, of accuracy eo and el, then they can be inserted into (2.10) to give a value for < W) of accuracy O(eo2, eoet). Alternatively, and more usually, assumed forms for ¢o and ~k1 can be used containing a number of parameters, and the parameters varied until (2.10) is stationary. It is, however, necessary to have some way of fixing the normalization of a trial ~klx. (~kor is normalized by (2.1)). The true solution ~kI is normalized by the inhomogeneous equation (2.4), and we must have recourse to (2.4) to normalize ¢1r; this can be done in several ways. The simplest is to ensure that ¢1r satisfies (2.4) at a point; the point may be chosen for convenience or used as a variational parameter. Possibly a better way, for operators W that are not very sharply peaked, is to use the integral condition \

(2.11,

500

L.M.

DELVES

satisfied by the true solutions $1, ~o- Imposition of this condition does not alter the variational character of (2.10). It is then necessary to use some trial value for kt2; the obvious choice is

k2T = f$o W$odr" In general, k 2 is not known exactly either; however, we can use for k 2 a variationally determined k2x, thus retaining the variational character of (2.10). Further, ko2 and kt2. can be removed from (2.10) and (2.1 l), since $1 can always be chosen orthogonal to $o. This follows since we can choose the normalization of $(2) in equation (2.3) to be independent of 2:

f~b2(r, 2)dr

= 1,

df

2-( r

= o,

hence

f ~bl~odr = O. In practice, the effort involved in choosing orthogonal functions may make this not worth while. 4. Variation Principle for the Wave Function

Perhaps the most interesting special case of our result is a variation principle for $2(Xo) at the point r = Xo, which follows by choosing W = 6(r-xo). Equation (2.10) then gives the variation principle

q/2(Xo)= q/2x(Xo)+2 f ~oT ( 5 --k° + V) d,qTdr +O(e~, eo~),

(4.1)

where ~vv is a trial function for qq. The true function ~1 satisfies (4.2) it has a discontinuity of slope at Xo; dq/x I d~l dr [xo+~ dr

= -~o(Xo).

(4.3)

xo- - g

This relation gives a convenient way of normalizing the trial q/iv in terms Of~ox(Xo). A variation principle for the wave function has been given previously by Biedenharn and Blatt 1). This principle required trial functions for a complete set of eigcnfunctions, while (4.1) requires only trial functions for ~Oo and ~1, and is in this respect much easier to use. Neither principle gives a bound.

VARIATION PRINCIPLE

501

5. Extension to Complex Systems The variation principle (2.10) extends trivially to many dimensions, since the selfadjointness of the Hamiltonian was the only property used. We find, for an Hermitian operator IV, the variation principle (w> -

f c,WC,od,

= f~,:dT+V--E)¢,Td~+ fOoT(T+V-E)O*,d*

(5.1)

+ ~¢,*~ WC,o~d~+O(d, ~o~), 3 where ~bOTand ~klx are trial functions (of accuracy ~o, el) for ~ko and ~1; and ~bo, ~ satisfy I T + V-Eo]

= 1,

o = O,

[~+ V - E o ] ¢ l = [ ( w ) - W]~o.

(5.2)

A variation principle for ~b*(Xo)~bo(Xo) at a point x o in the space involved follows by writing W = ,5(X-Xo)

in the necessary number of dimensions; however, a quantity of more interest may often be S ff~ ~ko dS over some subspace S' for given values of the remaining coordinates; principles for such quantities follow by using for W the appropriate lowerdimensional delta function. 6. A Simple Example We illustrate the use of (2.10) by considering the expectation values of the operator x ~ for the one-sided simple harmonic oscillator. Consider the system ~bo satisfying

dV'~o--X2

~/0 = O,

~/0(0)= ~ko(O0) = O,

fo

~ko2(x)dx = 1. (6.1)

The lowest state of this system has 2o = 3,

~ko = 2rc-~xe -*x2,

( x " ) ---

~b2(x)x,,dx _ ( n + l ) ! !

(6.2)

2½"

We choose the following simple trial function: Oor = 2~'t xe-~x.

(6.3)

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L.M.

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Variation of y in the principle for the energy

[2o] = min f ~or (X2-- S~) 'PoTdX gives the best values AT for 2 and 7a for y (best from the point of view of determining 2): AT = 2~/3 = 3.464,

7~. = 3~ = 1.316.

(6.4)

We can get a non-variational approximation o to x" with the wave function ~'or by calculating directly the integral ~ ~,o2rx"dx. We find

r

(.+2)! 2.**y.

_

(6.5)

We shall refer to the value of this integral with y = Ya as the zeroth approximation denoted o. We now choose a trial function ~qr which is orthogonal to ~'o (as we may always do, see sect. 4) and which we normalize by satisfying (2.4) at x = 0:

~qr = x'[Ae-~'X + Bxe-"q •

(6.6)

This is orthogonal to ~ko, and satisfies (2.4) at x = 0 with the approximate value ~ for k~, if m = 3,

a = -}y'o,

B =

(a+~')6A 5(27)s

(6.7)

A variational expression for is then

, = Stat 2 ~o(7) ~x~x z +AT -x2

~/i(~,y)dx+

~,~(?)xndx ,

(6.8)

and thisexpression is stationaryfor the unique positivevalues of ~ and ?: (210 I '

o% = \ Ys : -?s'

?'

= [ 75(n+2)7 L2(2~2)J

t

'

(6.9)

for which values

, - (n+2)! 112 ~

3(210)I"]

2n+ v

(6.10)

_I"

The firstfactor in (6.10) is seen to be of the form ~ for y ---Ys- Since y, > Ya for all n, thisfactor is always lessthan o. Moreover, the second factor is always less than unity, and approaches zero as n -o oo. Thus l < o for all n, and 1 -* O,

o

n ---~ o o .

VARIATION PRINCIPLE

503

Since (x"> o grossly overestimates the true result for large n, this implies that the variational result is very much better than the non-variational result for large n. The approximations (X~>o, l are plotted against n in table 1, together with the exact result (6.2). It is seen that for n > 2, the variational result is better than the zeroth approximation, while for n = 2, it is worse. This is understandable as a special feature of the harmonic oscillator, for which = ½2.

(6.11)

Thus, if we pick a trial function for 2 from a class o f functions satisfying (6.11) exactly, a variational calculation of 2 is also a variational calculation of (x2>. The function ~/or given by (6.3) is of this class; it gives 2 = 2x/3, o = x/3 and therefore a variational result for , although for no other . It is also interesting to note that (2.10) does not guarantee the result (x°>l = 1; in fact; we find for our trial functions

(x°>l=[fd/2dxI

1

=0.9467.

It is encouraging that the deviation from unity is as small as this, for such a poor trial function. TABLE 1 Diagonal matrix elements o f x ~ for the simple harmonic oscillator

n 0 2 4 6 8 10 12

Exact I 1.5 3.75 14.375 63.69 350.3 2.277× 10s

Zeroth approx.

Variational approx.

~'s

0.9467 0.9204 2.564 16.55 167.0 2.522x 10~ 5.685 x lif t

1.995 1.647 1.538 1.485 1.452 1.431 1.416

1 1.732 7.5 60.63 787.5 1.376x 104 3.613 x 105

Also tabulated in table 1 are the values of ?, for which (6.10) is stationary. The deviation of these values from Ta given by (6.4) is a measure o f the degree to which a wave function optimized for calculating the energy differs from one of the same form optimized for calculating (x'>. I am grateful to Dr. G. H. Derrick and Professor J. M. Blatt for helpful discussions of this work. Reference 1) L. C. Biedenharn and J. M. Blatt, Phys. Rev. 93 (1954) 230