Largeω Asymptotic Expansions and Padé Approximants for the HarmonicOscillatorG. R. Osche and G. J. Iafrate Citation: The Journal of Chemical Physics 56, 1104 (1972); doi: 10.1063/1.1677329 View online: http://dx.doi.org/10.1063/1.1677329 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/56/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Sudden expansion or squeezing of a harmonic oscillator Am. J. Phys. 63, 1021 (1995); 10.1119/1.18049 Padé Approximants Comput. Phys. 6, 82 (1992); 10.1063/1.4823047 Padé Approximants to the Normal Dispersion Expansion of Dynamic Polarizabilities J. Chem. Phys. 52, 1435 (1970); 10.1063/1.1673148 Harmonic Oscillator Expansions of the Hydrogen Atom J. Chem. Phys. 51, 857 (1969); 10.1063/1.1672093 Uniformly Valid Asymptotic Approximation for the Quantized Anharmonic Oscillator J. Chem. Phys. 47, 4540 (1967); 10.1063/1.1701665

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THE JOURNAL OF CHEMICAL PHYSICS VOLUME 56. NUMBER 3 1 FEBRUARY 1972

Large-w Asymptotic Expansions and Pade Approximants for the Harmonic Oscillator

G. R. OSCHE*

Monmouth College, West Long Branch, New Jersey 07740

AND

G. J. IAFRATE

United States Army Electronics Technology and Devices Laboratory (ECOM), Fort Monmouth, New Jersey 07703

(Received 17 May 1971)

An approximate solution to the Schrodinger equation for a particle in various perturbed harmonic­oscillator potentials is obtained by utilizing large-w (w is the natural frequency of unperturbed oscillator) asymptotic expansion theory. Both the basic asymptotic and perturbation theory solutions are obtained and are shown to be related. Explicit high-order perturbation expansions for the energy and mean-square displacement of the symmetric anharmonic oscillator are obtained and are seen to be divergent but asymptotic. It is shown that by rearranging these expansions as Pade approximants, accurate numerical values are obtained for these observables, even for nonperturbative values of the coupling constant.

I. INTRODUCTION

The problem considered here is that of a one-dimen­sional harmonic oscillator perturbed by two general classes of potentials, namely

Vex) =~mw2X2+XXN,

where N is a positive integer and

Vex) =!mw2x2B(AX), where

with

co

B(Xx) = L: Bj(Xx)i j=-O

Bo=1.

(1)

(2)

(3)

( 4)

In both cases, numerous physical applications to these potentials can be found in various areas of physics1•2 and chemical physics.3 .4 The potential of Eq. (1) with N =3,4 represents, respectively, the well­known cubic and symmetric anharmonic oscillator, whereas the potential of Eq. (2) with

B(Xx) = exp( -XZx2) (5)

is the Gaussian well. Due to the simple denumerable spectrum of the un­

perturbed oscillator, specific cases of the above general potentials have been the subject of extensive approxi­mate methods in quantum mechanics. These include high-order Rayleigh-Schrodinger5 perturbation theory, Feynman diagrammatic techniques,! variational tech­nique,6 and WKB7 theory. In this paper, the large parameter asymptotic approach developed by Mendel­sohn8-l0 and one of the authors (G.J.I.)11·12 for use in

general form of the energy expansion is derived for any N and energy state n.

In Sec. III, a large-w solution is obtained in the neighborhood of the origin by solving the stretched differential equation (perturbation theory· equation). By demanding that the solutions be well behaved near the origin, the energy terms are immediately deter­mined and the "matching condition"ll is automatically satisfied. It is shown that by utilizing a finite expansion of the perturbation theory wavefunctions in terms of Hermite polynomials, perturbation theory results can be obtained for any N and energy state n to arbitrarily high order. Specific results are given to sixth order in perturbation theory for the cubic and symmetric oscil­lator as well as to eighth-order result for the screened harmonic oscillator [Eq. (2) ] with

B (Xx) = cosXx. ( 6)

General results are also given to second order for the screened harmonic oscillator with arbitrary B.

In Sec. IV, the perturbation expansions for the energy and mean-square displacement of the symmetric oscil­lator are analyzed and are found to be divergent but asymptotic. It is shown that by re-expressing these ex­pansions as a series of rational fractions (Pade approxi­mants), accurate eigenvalues are obtained, even for large values of the expansion parameter.

II. BASIC ASYMPTOTIC EXPANSION

The Schrodinger equation for a particle in the po­tential of Eq. (1) is

- (fiz/2m) (d2f/dx2)+~mw2x2"'+XxN"'=E"'. (7)

atomic problems is employed to obtain the wavefunc- In the large-w expansion theory, the assumed solu­tions and the perturbation theory energy expansions tion to Eq. (7) for the wavefunction is the formal to arbitrarily high order. For atoms, the largeness pa- asymptotic expansion rameter is the nuclear charge, Z, while for the per- co

turbed oscillator, it is chosen to be w, the natural fre- f= exp[ -ws(x) ] L ajCAx)w- i . (8) quency of the unperturbed oscillator. j=-O

In Sec. II of the paper, the basic asymptotic expan- In order to obtain the proper form of the energy ex­sion (the correct large-w asymptotic expansion away pansion, it is necessary to identify the natural expan­from the neighborhood of the origin) is obtained to sion parameter. In this regard, transforming Eq. (7) into second order in w- l for the potential of Eq. (1), and the the stretched dimensionless variables ~= (mw/fi)I/2X,

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A S Y M P TOT ICE X PAN S ION FOR H ARM 0 N leo SCI L L AT 0 R 1105

one obtains the equation

- t( d2'lt / dr) + !r'lt + f~N'lt = (E/ftvJ) 'It, (9)

where (10)

Inspection of Eq. (9) indicates that f is the natural expansion parameter. Therefore, in the large-w limit, the proper form for the energy expansi9n is

E 00

- = L fjEj, (11) ftvJ Hi

wherej=O, 1, 2, ... for N even,j=O, 2, 4, ... for N odd, and f is given in Eq. (10).

Thus, in looking for the solution to Eq. (7), we sub­stitute Eqs. (8) and (11) into Eq. (7) to get

f. [w2- j (!mx2- fI,2 (SI)2) a3+wl-j j-O 2m

( Ji2" n

2 1 I.l: ) . {N n2" X - s a·+ - s u· -nEoa' +w-3 AX a·- - a· 2m 3 m 3 3 J 2m J

-n EI [~(~r/IWI-(I/2)(N+2)kEkaj}]=0. (12)

Setting the coefficient of w2 to zero in Eq. (12), one obtains

[!mxL (li,2/2m) (s')2]ao=0. (13)

The solution to Eq. (13) is immediately found to be

s(x) =t(mx2/n) , (14)

where ao is arbitrary.

and

Equating the coefficient of w to zero in Eq. (12) while using s(x) of Eq. (14), one gets the first-order differen­tial equation for ao as

xao'+ (!-Eo)ao=O. (15)

The solution to Eq. (15) is

ao=xEo-(I/2). (16)

Since Eo is equal to n+! [see Eq. (11) with e .. ..,A~O], ao in Eq. (16) reduces to

(17)

Setting the coefficient of w-a to zero in Eq. (12), we obtain the recursive first-order inhomogeneous differ­ential equation

nxaa+I'-nnaa+l+Ax-"aa- (h2/2m)aa"

-n f. f. aj [~(!!)NI2Jk E k8a,[(N+2)k/21+j-l, 3=-0 k-l n m

(18)

where a=O, 1,2, .... Dividing by n and utilizing the integrating factor

exp[ - f(n/x)dx] = (xn)-I= (ao)-I, (19)

Eq. (18) can be immediately integrated to obtain

Ix {n aal/(y) AyN-1 aa+l(X) =ao(x) dy[ao(y)]-l - --- - --aa(Y)

2m y n

+ EEl ajy-l [~(~r/IEk8a'[(N+2)k'21+j-l}. (20)

For a=O, 1, the results of integrating Eq. (20) are

(21)

~= (A2/2n2) (rN+n/N2) + (A/2m) [!n(n-1) - (N+n) (N+n-1)/N][x·V+n- 2/(N-2) ]N+2

- (h2/32m2)n(n-1) (n- 2) (n-3)xn- 4+ (Aim) [El - (n+!) ]xn Inx8N,2. (22)

Notice that for N =2, a2 in Eq. (22) contains an lnx divergence as X goes to zero. In this regard, we shall show, by finding the correct asymptotic expansion in the neighborhood of the origin, that

El = (n+!)8N,2,

so that no logarithmic divergence occurs in the basic asymptotic solution when N = 2. In point of fact, it will be shown, through matching, that the basic asymptotic expansion contains no admissible lnx terms but is simply a reordering of the perturbation theory solution (solution near the origin). Thus, the energy corrections may be determined by setting the coefficients of lnx terms occurring in the basic asymptotic expansion to zero.

Therefore, for a=2, by integrating Eq. (20) while choosing the E's so that lnx terms vanish, we obtain

a3=_~x3N+n _~(n(n-1) _ (N+n)(N+n-1») [ r N+n- 2 ] 6li3 N3 4mli 2 N (N-l) (N-2) N;o'1.2

+ nA (n(n-1) _ (N+n)(N+n-1)) [(N+n-2) (N+n-3) XN+n- 4] 4m2 2 N (N-2)(N-4) N;o'2,4

nA [XN+n-4] A2 [(2N+n) (2N+n-1)] - - n(n-1)(n-2) (n-3) + - r N+n- 2 Nfl 32m2 (N-4) Nr'4 8mn (N-1)N2

+ (li3/384m3)n(n-1) (n-2) (n-3) (n-4) (n-5)xn- 6, (23) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

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1106 G. R. OSCHE AND G. J. IAFRATE

with

and (24)

(25)

If this procedure is carried to higher orders, it will be observed that corrections corresponding to higher values of N will enter in the asymptotic energy and wavefunction.

III. PERTURBATION THEORY: SOLUTION NEAR THE ORIGIN

In order to show that the basic asymptotic expan­sion is "matched" to a solution well behaved near the origin (and therefore contains no lnx terms), we look for the large-w solution to Eq. (7) which is valid for small x. In this regard, due to the large-w limiting pro­cedure, when ~ is finite and w is taken as large (with n=m=1), x=UwI/2~ and the solution to Eq. (9) becomes the solution to Schrodinger's equation [Eq. (7) ] near the origin.

The large-w solution to Eq. (9) is simply the per­turbation theory solution, namely,

expansion of Hermite polynomials in the form

bk=L Ak,jHj(~). j

(32)

Since Hermite polynomials obey a general recursive relation of the form (see Appendix for the explicit forms)

i+N ~NHi(~) = L r(i,j, N)Hj(~), (33)

j=.-N

the use of Eqs. (32) and (33) in Eq. (30) indicates that the truncation form for the solution is

n+Nk bk= t Ak.#i(~) (34)

.i=n-Nk

(26) with

and

(27)

where E is given in Eq. (10) (with n=m= 1). Note also that the energy expansion of Eq. (27) is identical to the expansion given in Eq. (11). We will now show that by selecting the E/s of Eq. (27) so that the expansion of Eq. (26) is a valid solution to Eq. (9) for a given N, the results of Eqs. (24) and (25) are obtained in­eating the removal of lnx terms from the basic asymp­totic expansion.

Substituting Eqs. (26) and (27) into Eq. (9) while setting equal powers of E to zero, we obtain the familiar hierarchy of differential equations

~

-i'l'/'+ir'l'k-Eo'l'~= L Ej'l'k_j_~N'l'k_I, (28) ;-1

with k= 1,2, •... Putting the additional simplifying substitution

\}fk= exp( -r/2)bkW (29)

into Eq. (28), while using the fact that Eo=n+i, we obtain the set of inhomogeneous Hermite differential equations

k

b/'-2~b/+2nbk=2~Nbk_I-2 L Ejbk- h (30) j-I

where bo=HnCt) , (31)

an nth-order Hermite polynomial. The well-behaved solution to Eq. (30) is simply an

Ao,j=c5j ,n' (35)

Combining Eqs. (30), (33), and (34), we obtain n+Nk n+N(k-I) i+N L (n-j)Ak,jHJ = :E :E Ak_I,jr(j, s, N)H.

j=n-Nk .i=n-N(k-Il 8='-N

k-l n+Nw - :E Ek- w :E Aw,jHj • (36)

1V:=z{) j=n-Nto

For k = 1, Eq. (36) reduces to n+N L {[(n-j)A1,j-r(n,j, N)JHj}+E1Hn=0. (37)

j=n-N

In setting the coefficients of HJ equal to each other, we see that for.i =n

EI = r(n, n, N) (38) with

AI,n=O,

while for j rf n, we obtain

AI,j= (n-j)-Ir(n,j, N).

(39)

(40)

Thus, the first-order correction to the wavefunction is n+N

bI = :E (n-j)-Ir(n,j, N)Hj. (41) i=n-N,jr!n

For k=2, Eq. (36) reduces to _2N n+N j+N :E (n-j)A 2,jHj= :E A1,j L r(s,j, N)Hj

j-n-2N j=an-2N s=n-N

n+2N n+N :E AI,j :E r(s,j, N)H;-E2Hn

j=n+I =j-N

n+N -EI 1: AI,jH;. (42)

j-n-N

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A S Y M P TOT ICE X PAN S ION FOR H ARM 0 N leo SCI L L AT 0 R 1107

Examination of Eq. (42) shows that for j=n, we obtain

n+N E2= L (n-j)-1r(j,n,N)r(n,j,N). (43)

j-n-N

In addition, wavefunction results are obtained for this case by considering Eq. (42) with jrfn. This process can be continued indefinitely by simply repeating the above recursive procedure to obtain higher-order cor­rections to the perturbation theory solutions.

In order to relate the stretched (perturbation) solu­tion to the basic asymptotic expansion, we invoke the "matching principle" often used in boundary layer problems.I3 For nomenclature purposes, let y,j(w, x) represent the truncated basic asymptotic expansion after w-j term and consider '!Fm (t, 1;) to be the truncated expansion after em. One re-expresses y,j(w, x) in terms of variables e, 1;, and '!Fm(t, 1;) in terms of w, x.

We then obtain

y,j(w, x)----t~i(e, 1;) (44)

(45) and

Truncating if;,. after mth order and ~m after jth order, we then define

and

[~m(W, X)]T=~j

(truncated expansion after Em term)

(46)

(truncated expansion after w- i term). (47)

The "matching condition" is then stated as

(48)

Substituting Eqs. (21) and (22) into Eq. (8), we obtain a large-w expansion correct to order w-2 as

f [ X xN+n fi ] [ fi2 y,(x) = exp( -wx2/2) xn+ - - - - - n(n-1)xn- 2 w-1+ - n(n-1) (n-2) (n-3)xn-4 fI]V 4m 32m2

+ ~x2N+n + ~(N(n-1)-2(n+N)(n+N-1)) ]!.-}. (49) 2fi2 N2 4m N(n-2) N;el w2

Letting I;=W1/2X, while collecting powers of E=XW-1/2(N+2), Eq. (49) (with fi=m=l) reduces to

if;2(E, 1;) =wn /2 exp( -1;2/2) {un- tn(n-1)t;n-2+~(n-1)(n-2)(n-3)t;n-4J

+ [_ f;N+n + !(n(n-1)N-2(n+N)(n+N-1)) I;N+n-2]t+ (f;2N+n)t2}. N 4 N(N-2) NY"2 2N2

(50)

A general comparison of if;2 with results obtained for the expansion of Eq. (26) would indicate that Eq. (50) is an incomplete form of the perturbation theory wave­function to order e2• This is clearly illustrated by noting that the coefficient of eO in Eq. (50) is simply 2" times the first three terms of the nth-order Hermite poly­nomial; whereas, in perturbation theory [Eq. (31)J, the full Hermite polynomial is utilized. It is also ob­served that the coefficient of e in Eq. (50) consists

of contributions arIsmg from the Hn+N and Hn+N- 2

Hermite polynomials present in the first-order perturba­tion theory correction of Eq. (41).

In order for the matching condition [Eq. (48) J to be satisfied, if;2( E, 1;) must be equivalent to the perturba­tion expansion through order em. We now demonstrate that y,2(e, 1;) [Eq. (50)J is equivalent to first-order perturbation theory for a special case. For the ground state (n=O) with N =4, if;2 in Eq. (50) becomes

y,2(e, 1;) = exp( -1;2/2)[1+ (-0.251;4-0.751;2)E+0.031251;8e2J. (51)

The perturbation theory wavefunctions [Eq. (34) J were found to be n+Ni

'!F= exp( -1;2/2) [Hn+ L ( L A;.;Hj)e'J. (52) i-1 i-n-Ni

For n=O, N=4, Eq. (52) reduces to (see Appendix)

'!F(E, 1;) = exp( -1;2/2) [1 + (-0.251;4-0. 751;2)e+ (O.031251;8+0.27083f;6+0.9687l;4+ 2.62491;2) e2+O(e3)]. (53)

Comparing Eqs. (51) and (53), we see that m=1. In carrying the basic expansion to second order in w-1,

we have accounted for one order of the perturbation theory solution. It then follows for this nonsingular

problem that the perturbation theory result to any order can be obtained by keeping higher-order terms in the basic asymptotic expansion and re-expressing them in terms of stretched variables {this may be ex-

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TABLE 1. Coefficients Ai,; and Ei.

nO n1 n2 n3 n4 n" n6 n7

Ni

A. Tabulation of A.; coefficients of Eq. (54) as a power series in n of the form Ai;= ~ 'Y;ni.

A 1•n-3 o . ()()()()()() 0.66666666 A 1•n_1 0.0000 0.0000000 A 1•n+! -0.750000 - 0.7500000 A 1•n-f,'! -0.40416666

A 2.n-6 0.00000000 -6.6666666 A 2•n-4 0.00000000 2.2500000 A 2•n_2 0.00000000 -0.1250000 A 2•n+2 0.84375000 1.0312500 A 2."+4 0.05468750 0.0312500 A 2•n+6 0.00086805

A 1•n-4 0.00000000 -1.5000000 A 1•n- 2 0.00000000 0.5000000 A'.n+2 -0.37500000 -0.2500000 A'.n+4 -0.01562500

A 2 ... -8 0.00000000 -157.50000000 A 2•n_6 0.00000000 55. OOOOOOOO A •. n-4 0.00000000 -10. 500000OO A 2.n-2 0.00000000 2.06250000 A2.n+2 2.34375000 2.80468750 A2."+4 2.81250000 0.20312500 A2.n+6 1.10677080 0.00390625 A 2."+6 0.00012207

B. Tabulation of Ei coefficients of Eq. (55)

N=3,0=)"/,,,6/2 o

- (1.375+3. 75n+3. 75n')

o

;-Ni

N=3 k=1 1.000000 0.3333333 1.5000000

N=3 k=2 15.222222 -12.500000 4.7222222 - 0.83333333 0.05555555

-7.1250000 7.750000 -3.3750000 0.50000000 2.5000000 -3.250000 0.8750000 0.2187500

N=4 k=1 2.7500000 -1.5000000 O. 25()()()()(x)

-1.5000000 1.0000000

N=4 k=2 408.375000 -410.375000 211.531250 -61.250000 10.0625000 -0.8750000

-155.58333 171. 625000 -95.208333 28. 125000 -4.2083333 0.2500000 32.7500000 -38.250000 20.750000 -5.250000

-5.4062500 7.375000 -3.968750 -0.062500 0.9608375 -0.015625 0.0312500

N=4,0=)"/,,,3 +(0.75+1.5n+1.5n2

)

- (2.625+7.375n+6.375n2+4.25n3)

+ (20. 8125+65. 0625n+88. 5n2+46. 875n3+23. 4375n4 )

E,

&

Ea

E4 E6

- (14.53125+51. 09375n+66.09375n2+44.0625n3)

o - (241. 289062+872. 632812n+1253. 67187n2+ 1114. 14062n3+417 . 539063n4+167 . o 15625n6 )

+ (3580. 98047+13944.058611+24 146.0391n2+21 771. 9141n3+14 305. 8398n4+4103 .85936116

& - (310. 226563+1268. 78906n+2173.125n2

+ 1808. 67188n3+904. 335938n4)

+ 1367. 95313n6 )

- (63 982.8135+270 874. 704n+493 994.355112+553 004. 617n3+356 925.454114+185595.949116

+42825. 7676n6+ 12235.9336117 )

..... ..... 0 00

n8

P

i":I

0 UJ

(')

::I: t'1i

;:.. Z t::;

p 0.031250

'-<

..... ;:.. "tj

i":I ;:.. o-,J

t'1i

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A S Y M P TOT ICE X PAN S ION FOR H ARM 0 N leo SCI L L AT 0 R 1109

plicitly seen by considering the perturbation theory contributions that arise due to including the term a3w-3 [see Eq. (23)] in the basic asymptotic expansion). Although we have demonstrated matching in this special case (n=O, N=4), the matching condition can be satisfied for any nand N as well.

Now that the matching condition has been estab­lished, we again focus attention on Eq. (36). Letting N =3,4 in Eq. (36), we obtain 20th-order perturbation results for the wavefunction and energy of the nth quan­tum state. However, for purposes of brevity, we only present sixth-order results for the energy and second­order results for the wavefunction. In this regard, for a given Nand n, the wavefunction and energy are ex­pressed as

6 n+Ni weN, n) = exp( -~2/2) L [ L Ai,jHj(~) ]e'eN)

i-o ,i-n-Ni

(54) and

setting the coefficient of ek to zero k-l

bk"-2~b/+2nbk= L (Bk_w~k-w-L2Ek_w)bw. (61) w-O

The solution to Eq. (61) can be written as n+3k

bk= L Ak.;Hj(~), .i=-n-3k

where Ao.;=hn.

(62)

Substituting Eq. (62) into Eq. (61), we obtain an analogous expression to Eq. (36) as n+3k k-l n+3w L 2(n-J)Ak.jH j= L B k - w L Aw.i

.;"n-3k W=O ;i-n-3w

i+k+2-w k-l

X L r(j, 2, k-w+2)Hr-2 L E k - w r-.i+w-k-2

n+3w X L Aw.iHj • (63)

;""'n-3·",

(55) For k=l andj=n, Eq. (63) gives

with e(N) =X/W(I!2)(N+2). The coefficients A i.j and Ei can be found in Table I.

It is also interesting to note that for N = 2, the exact energy is known to be

E(2, n) /w =n[l + (2A/mw2) ]I!2(n+t).

Expanding Eq. (56) in powers of X/w2, we obtain

E(2, n)/w

(56)

= (n+t) [1+ (fi/m) (X/w2) _t(fiX2/m2w4)+O(X3/w6) J. (57)

It is thus observed that the coefficient of

X/w2[(fi/m) (n+t)]

in Eq. (57) corresponds to that correction in Eq. (24) which removes logarithmic divergences from the basic asymptotic expansion when N = 2. Moreover, El (4, n) in Eq. (55) accomplishes the same purpose when N =4 in Eq. (24).

Similar results can be obtained for the potential of Eq. (2). Let

ld2w 12~ .. E - -- + -~ "-' Bie'ew= -w. 2dr 2 i-O W

(58)

In looking for a solution of the forms

(64)

where Al,n=O. Since the sum of the indices of r is odd, r(n, n, 3) =0 (see Appendix A) and El =0. For j.,t.n, we obtain

A 1•j = (n-j)-lr(n,j, 3), where

j=n+3, n+2, ••• , n-2, n-3. (65)

For k=2,j=n in Eq. (63), we obtain for the energy correction

E 2=t(B2r(n, n, 4)+iB12[r2(n-3, n, 3)

-r2(n+3, n, 3)]+tB12[r2(n-l), n, 3

-r2(n+l,n,3)]). (66)

In particular, the energy expansion obtained to eighth order for B(Xx) = cosXx is found to be

E/w=n+t- (0.1875+0.375n+0.375n2)e2

- (0.125+0.35677n+0.320312n2+O.213541n3)e4

- (0.2554117+0.8522101n+ 1.271004n2+O.88519626n3

+0.499727037n4+ 0.0714111328n5+0.0238037n6) e6

- (0.669808706+ 2.6477737n+4.48114411n2

(59) +4.79360178n3+2.9000519n4+ 1.473242n5

and

'" w= exp( -r/2) L b;e; (60) :i-O

and substituting them into Eq. (58), we obtain, by

+ 0.317299498n6+ 0.0906422933n7 ) e8• (67)

It is clear that a high-order perturbation solution can thus be obtained for the general class of potentials [Eqs. (1) and (2)J considered by utilizing Eqs. (36) and (63), respectively.

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1110 G. R. OSCHE AND G. J. IAFRATE

TABLE II. The first 20 terms of the ground-state energy expansion for the symmetric anharmonic oscillator

E.X lo--n n

O. 7500000000 0 2 -0.2625000000 3 0.2081250000 2

4 -0.2412890625 3 5 0.3580980469 4 6 -0.6398281347 5

7 0.1329733727 7 8 -0.3144821469 8 9 0.8335416033 9

10 -0.2447894070 11 11 0.7893333160 12 12 -0.2773877696 14

13 0.1055646658 16 14 -0.4326810684 17 15 0.1900817198 19

16 -0.8912101754 20 17 0.4442550890 22 18 -0. 2345464308 24

19 0.1309150261 26 20 -0.7693999854 27

IV. EXPECTATION VALUES AND PADE APPROXIMANTS

In this section, high-order perturbation theory and PaM approximants are utilized to obtain accurate expectation values of various observables. For purposes of discussion, we consider the energy and mean-square displacement associated with the ground state of the symmetric anharmonic oscillator. The energy is ex­pressed as

(68)

where (69)

and the 20 E;'s are listed in Table II. The mean-square displacement can be found6 •

14 by applying the Hellman-Feynman theoreml6 to the ca­nonical double perturbationl6 problem (where gx2 is the second perturbation) with Hamiltonian

X(A, g) = -Hd2/dx2)+!W2X2+Ax4+gx2. (70)

In defining a canonical frequency 0 as

O=w[l+ (2g/w2) J1I2, (71)

the above Hamiltonian [Eq. (70) J can be rewritten as

X= -Hd2/dx2)+!02X2+ Ax4. (72)

The energy perturbation expansion for X can then be written as

20 (A )i E(A, g) =0 I: Ei - , i~O 03

(73)

where the E;'s are discerned from Eq. (68). In utilizing the Hellman-Feynman theorem, one

obtainsl7

aE(A g) I 20 (A )i (X2)= ' =w-1 I: (1-3i)E i - • ag g=O i~ 03

(74)

The behavior of the coefficients in Eq. (68) indicates that the energy expansion (as well as the expansion for (x2» is divergent but asymptotic, a result which is not uncommon in perturbation theoryP In order to have a convergent expansion within some radius of convergence EO, bound states for the perturbed problem must exist for any E< 1 Eo I. Since a large class of per­turbing potentials with E= -I E 1 do not give rise to bound states (tunneling states occur), it then follows that any energy expansion about E=O for these po­tentials will diverge. Thus, for large values of coupling constant and for high-lying quantum states, perturba­tion theory is usually not useful.

We now utilize a method which proves to be effec­tivel2 ,18 in obtaining accurate results for the energy and mean-square displacement even for values of the cou­pling constant [E=A/W2 ;;::0.2J that are nonperturbativr. We re-express the original perturbation theory expan­sion (to kth order) as a series of rational fractions, PaM approximants, of the form

k m n

F;:::;jI: FiEi"'Rm,n(k)=I: a".,jEi/I: bn,IE I (75) ~o pO ~

subject to the condition that

m+n=k. (76)

The array Rm,n(k) comprises a Pade table. Generally, it is found12 ,19 that superior accuracy is obtained by considering the diagonal elements of the table; that is, those entries for which the degree of the polynomials in both the numerator and denominator are the same. In Table III, we show the results obtained by PaM approximating the perturbation theory expansions for the energy and mean-square displacement. Comparison with continued fraction techniques and extremely ac­curate variation calculations shows excellent quantita­tive agreement even for values of A which far exceed the perturbation theory criteria.

V. SUMMARY AND CONCLUSION

A large-w asymptotic solution to the Schrodinger equation is obtained for a general class of one-dimen­sional anharmonic oscillators. It is shown that the

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ASYMPTOTIC EXPANSION FOR HARMONIC OSCILLATOR 1111

TABLE III. EIO,IO(W) and (X2)IO,IO(W) represent the diagonal Pade approximants of 20th-order perturbation theory for the ground­state energy and mean-square displacement of a symmetric anharmonic oscillator.

2Eef •

0,125 1.14190184 1.14190179

0.250 1.2418540 1.241848

0.375 1.322873 1.322811

0.500 1.392352 1.39210

• Eef and (x2 )el refer to the continued-fraction result of Reid (Ref. 5),

"matching condition" for the basic asymptotic and perturbation theory solutions is automatically satisfied, provided the energy values are selected so that the coefficients of lnx terms in the basic solution vanish. Indeed, for any nonsingular perturbation problem, the matching condition, with the appropriate largeness parameter, is automatically satisfied since the basic asymptotic expansion is a uniform asymptotic expan­sion of the perturbation theory solution. Thus, the large-w expansion technique provides an alternative method for obtaining high-order perturbation theory energy and wavefunction corrections. Preliminary in­vestigations indicate that the large-w approach is also helpful in finding perturbation theory results for three­dimensional anharmonic oscillators with even parity; that is, anharmonic perturbations of the form 'Ar2N.

These results will be discussed elsewhere. The usefulness of high-order perturbation theory

results was markedly extended by re-expressing the perturbation theory expansions, for various observables, as diagonal Pade approximants. The utilization of Pade approximants is effective in greatly accelerating the convergence of many slowly convergent sequences and in inducing convergence in many divergent se­quences.12 ,18,19

APPENDIX A: PROPERTIES OF r(n, j, N)

Hermite polynomials obey the recursive relation

(Al)

Successive application of this formula indicates that i+N

~NHi= L r(i,j, N)H j • (A2) j=i-N

Formally, r(i,j, N) is defined as

r(i,j, N) = L: Hj~Hi exp( -nd~ / L: Hl

X exp( -e)d~. (A3)

In making use of the normalization properties of

1.14190184 0.39970246 0.39970246

1. 241854 0.3548403

1.322873 0.3263625

1.392352 0,3058136 0.30596

b Eo. refers to accurate variational calculation of Reid (Ref,6) .

Hermite polynomials, Eq. (A3) reduces to

r(i,j, N) = (21!1l')-1 L: Hj~NHi exp( -e)d~

:= (21!1l')-IS(i,j). (A4)

Since S(i,j) obeys a reciprocity relation with respect to interchange of i andj, it then follows that

r( .. N) -2i - i ( 'lj'l)r(' . N) J,$, - J. L ~,J, . (AS)

In addition, due to the parity properties of Hermite polynomials,

r(i,j, N) =0, (A6) if

i+j+N=1, 3, S, .•. (a positive odd integer).

In using Eqs. (A4)-(A6), we explicitly evaluate the r's through N =4:

r(n, n+1, 1) =i, r(n, n-1, 1) =n,

r(n, n+2, 2) =t, r(n, n, 2) =n+t,

r(n, n-2, 2) =n(n-1),

r(n, n+3, 3) =t,

r(n, n+1, 3) =i(n+1)

r(n, n-1, 3) =!n2,

r(n, n-3, 3) =n(n-1) (n-2),

r(n, n+4, 4) =l6'

r(n, n+2, 4) =t(2n+3),

r(n, n, 4) =i(2n2+2n+1),

r(n, n-2, 4) =2n3-3n2+n,

r(n, n-4, 4) =n(n-1) (n-2) (n-3).

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1112 G. R. OSCHE AND G. J. IAFRATE

APPENDIX B: REDUCTION OF PERTURBATION THEORY WAVEFUNCTION TO

STRETCHED VARIABLES

The kth-order perturbation theory correction to the wavefunction was found to be [see Eq. (34) ]

n+Nk

The results obtained to fourth order for n=O and N=3,4 are:

for N=3,

bo=l,

b1 = -~-0.33e,

bk = :E Ak"HfW, (Bl) b2 = 1.37S~2+0.4S9~4+0.0SSS~6, j=n-Nk

In order to relate the basic asymptotic wavefunction to the appropriate perturbation theory corrections, it is necessary to express bk as a polynomial series in the stretching variable ~. Since Hj(~) is a polynomial of degree ~j, it then follows from Eq. (Bl) that bk can also be expressed as

b3 = -0.00617e-0.0972e-0.6833~L2.1243e-4.99n,

b4 = + 14.S312Sr+ S.lS88S~4+ 1.0843 n6+ 0.lS9201~8 +0.0131173~1O+0.000S144e2;

for N=4,

bo=l, n+Nk

bk = :E Bk ,8e· (B2) bl=-0.2S~LO.7Sr, 8=0

In general, to relate the B's to the A's, we equate Eq. (Bl) to Eq. (B2) and utilize the orthogonality properties of Hermite polynomials to obtain

n+N~

:E Bk ,8am •• =A k .m , 8=0

=0, where

if 05:m<1l-.Yk, (B3)

An alternate procedure for obtaining the proper B's would be to substitute Eq. (B2) directly into the dif­ferential equation of Eq. (30) to obtain n+Nk-2 n+Nk

:E (s+2) (s+l)B~,8+2~8+ :E SBk,8~8 8=0

n+Nk n+Nk +2n :E SBk,8~8=2 :E Bk-l,8-2~'

k n+N(k-J) X - 2 :E E j :E Bk_j,.~8. (BS)

.fo-l 8=0

By collecting powers of ~, this recursive relation pro­duces Nk-n+l equations in B k ,8 and Ek which can be solved in each order of k simultaneously.

b2 = 0.0312S~8+0.27083~6+0.968n4+ 2.6249r,

b3= -0.OO2604e2-0.04427~lLO.36718~L 2.1172~6

- 7 .S937e- 20.812S~2,

b4=0.OOO1627~16+0.004SS7~14+0.06499S~12+0.637369~1O

+4.489n8+ 24.0377~6+86. 7812~4+ 241.289r.

* Present address: U.S. Army Electronics Command, Elec-tronics Components Laboratory, Fort Monmouth, N.J. 07703.

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