- Email: [email protected]
Analysis of the Schrödinger energy series
ANNALS
OF
3, 292-303
PHYSICS:
Analysis
(1958)
of the
Schrsdinger EUGENE
Wayman
Crow
Laboratory
of Physics,
Energy
Series*
FEENBERG Washington
University,
St. Louis,
Missouri
The rate of convergence of the Brillouin-Wigner and Schrodinger energy series can be modified by using elements of freedom present in the formulation of the perturbation problem. Two such elements of freedom are (a) a uniform displacement of the zeroth order energy spectrum and (b) a uniform change of scale in the spacing of the zeroth order energy levels. Transformation a is used to generate the Schrodinger energy series from the corresponding Brillouin-Wigner series. The effect of transformation b on the former series is worked out and exhibited explicitly. The problem of defining a criterion for determining the scale factor is bypassed by the observation that continued fraction approximants to the Schrodinger energy series are invariant under transformation b. Explicit formulas are given for the first three invariant forms. INTRODUCTION
The Schrodinger perturbation procedure begins with the decomposition of a given Hamiltonian operator H into a sum of two operators, a zeroth order component Ho and a perturbation operator W. Elements of freedom are present in the choice of Ho and the corresponding W. Two simple, but physically important, elements of freedom are (a) a uniform displacement of the zeroth order energy spectrum (b) a uniform change of scale in the spacing of the zeroth order energy levels (for free particles in a box this transformation is equivalent to the introduction of an effective mass). The same elements of freedom occur in the formulation of the BrillouinWigner perturbation procedures. Here the extremum property of the odd order energy approximants determines optimum magnitudes for the uniform displacement and also, in the original formulation of BW procedure, optimum scale values for the zeroth order level spacing (I). The recentIy developed continued fraction formulation of the BW procedure (I, 2) gives odd order approximants to the energy eigenvalues which are invariant under transformation b. In this paper transformation a is used to generate the Schrodinger energy * This research was supported by the United Office of Scientific Research of the Air Research tract So. AF18(603)-108. 292
States Air Force and Development
through the Air Force Command, under con-
SCHR6DINGER
ENERGY
293
SERIES
series from the BW approximants to the energy eigenvalues. The application of transformation b to the Schrbdinger energy series brings with it the problem of defining an adequate criterion for determining the scale factor. This problem is bypassed by the observation that continued fraction approximants to the Schrodinger energy series possess the invariance property mentioned earlier in connection with the BW formulation. In nuclear problems computational difficulties limit applications of the BW procedure to the first order correction to the wave function and second and third order corrections to the energy (3, 4). The range of usefulness of this approximation is restricted to light nuclei, possibly extending to the end of the first p shell. Valuable results have been found at mass numbers 6 and 7 by methods closely related to the first order BW procedure (5, 6). Wigner and others have pointed out that the number of terms in the BW energy series required to attain physically meaningful results increases with t’he number of particles A. This number of terms, usually quoted as of the order A, has possibly been overestimated because of failure to utilize transformations a and b in discussing the many particle problem. Under conditions of constant particle density and interact,ions of finite range one expects an interaction energy proportional to the number of particles. The usual treatment gives the result that inclusion of the first order correction to the wave function in the energy calculation improves the energy by an additive term independent of A when A is large. Using transformations a and b I show that the energy is improved by an additive term proportional to A Ii2 when the first order correction to bhe wave function is included in the energy calculation. This result is still yuit.e inadequate, but does suggest that t.he number of t,erms in the BW energy series required to attain meaningful accuracy may be smaller t’han A in the limit of large A when proper use is made of transformations a and b or alternatively of transformation a and the invariant continued fraction approximants. I)ERIVATION
OF
THE
SCHRdDINGER
ENERGY
SERIES
Let E, and #, denote the eigenvalues and the normalized, orthogonal eigenfunctions generated by Ho . Consideration may be limited to specific values of the appropriate constants of motion wit’hout explicit introduction of symbols to denote t,hese constants. The matrix elements of TY are denoted by W,, . Some trivial complications in the st.atement of the perturbation formulas may be avoided by incorporating the zero-zero matrix element of the perturbation operator into Ho . One has then Woo = 0 and (H),, = Eo . The general BW energy term for the state generated from $0 in zeroth order approximation is
dE
- Eo , r’)
=
WdWnm - mm,>. . . (Wgz- is,,) w.?o C’ mn”.vz (E - Em - U)(E -E,, - U)...(E - EC- i?)’
(1)
294
FEENBERG
in which the prime on the summation symbol means the s - 1 variable indices m, n, . . . y, z do not take on the value 0. Note that E~is a sum of terms each having s factors in the numerator and s - 1 factors in the denominator. Transformation a is incorporated in the formalism by introducing t,he parameter I:. The nth approximation to the wave function yields an approximation to the eigenvalue determined by the lowest root of the implicit equation 2 n+1
E = Who +
c2 dE
n = 1,2, . . . .
- Eo , I:),
(2)
From the lowest root E one goes to the minumin value of E as a function of U, thus determining an optimum value of U and the lowest possible value of E. The Schrbdinger energy series is generated by the choice, not the optimum,
U = E - E,, .
(3)
Eq. (2) reduces to an algebraic equation of order 2n tity U:
1 in t’he unknown
quan-
2n+1 u
=
c
f,?(U,
(4)
U).
2
Xow writing
E,(U) = E,(U, Cr), Eq. (4) is equivalent li =
A suitable notation
facilitates
the further
Q
(s; t1tz ... in which
2n+1 8-2 l Tk {& z k% jij L
=
f,(~J)]“~o. discussion.
1 =
(5)
Let (m) := Eo - E, and
C’ W”?nW?m . . . w,,w,o mn’-‘tJz (m)yn)t2 . . . (Z)t8-l ’
,,&
--1 (s;
(6)
and ti = 1, 2, 3, . . . . Next define
W occurs s times in each numerator fs,
to
t1t2
(‘7)
. . . Ll)
8
the sum ranging over all distinct constraint t1 + tz +
sets of integers tl , t2 , . . . , t,-l subject to the . . + 2,-l = t 2 s -
1.
(8)
Thus for s = 4, t = 6, the sum in Eq. (7) includes all distinct permutations 4, 1, 1; 3, 2, 1; and 2, 2, 2. Examination of Eq. (1) defining es(U) = es(Ur W) now yields t,+,(U)
=
fs+l,s
-
ufs,8
+
U2f8+-1,,
I-
* ’
1
+
(-l)S-ll+t:,,r
.
of
(9)
SCHRGDINGER
EiXERGY
SERIES
295
Eqs. (4) and (9) combine to give 2n
2n+1 T.. =
=
c
s=2
fs,s-l
-
2n-1
r: 2 es, s + r2 c Es.s+1 + . . . n=2
P-‘f2,
3,, ) (10)
a more explicit statement of Eq. (5). At this point it is helpful to use Eq. (iL) again in the form
= exp The first line of EC]. (I I) is Smply t.he complete IW series for the energy with a particular choice for t,he displacement parameter. The second line introduces the Taylor’s series for E~(I:, 0) as a function of C’ in t*erms of ~,~-l and its derivatives with respect to E, . The last named function appears because it is t,he same function of IS, t,hnt, c~(C:, 0) is of & + I:. Equat.ing coefficientIs of Crk in Eqs. (5), (lo), and (11) we get.
Thus, for example, at s = 4: f43 = (4; 111) f4$ = (4; 211) + (4; 121) + (4; 112)
E45
= (4; 311) + (4; 131) + (4; 113) + (4; 221) + (4; 212) + (4; 122)
zzz---i a
2 aE0
E44 =
9
1 a2 m
(13)
643 ,
f(fi = (4; 411) + (4; 141) + (4; 114) + (4; 321) + (4; 13’2) + (4; 213) + (4; 312) + (4; 123) + (4; 231) + (4; 222)
= -- 1 a
1 a3
Eqs. (10) and (11) are implicit
equations which
retain the extremum
property
296
FEENBERG
of the BW formalism from which they are derived. In one special case (n, = 1) Eq. (10) reduces to an explicit relation: zr
=
E21 + -7 1 +
e32
(14
t22
The Schrodinger energy series is a sequence of explicit approximations for U obtained from Eqs. (10) or (11) by the formal device of associating an expansion parameter X with W, expressing U as a power series in X and fmally setting X = 1. Introducing
u = 2 X”U,
(13
2
into Eq. (11) and equating coefficients of X” generated by the left and right hand members we obtain u2
=
E21 ,
u3
=
E32 ,
u4
=
e43
-
U2t22
us
=
664
-
u3e22
-
l72e33
US
=
c65
-
U4E22
-
U3t%3
-
u2644
+
u2’E21
cT4E33
-
U3E44
-
U2e65
f
u22E34
u5c33
-
u4c44
-
U3E65
-
u2m
,
UT = 676- U&2 us
=
ta7
-
U6C22
-
+ 2 U&&4 ug
=
tgjj
-
U?e22
-I-
1332634
-
f
-
)
U32e23
+
2 U2U4e23
u6e33
-
USC44
i723C35
+
2u,u,C,,
-
,
-
u23c24
u4t56
-
-,-
2u:,u4e34
+
2u2u3C23
$
u22e45
-
u2c77
06)
,
,
U3f66
+
+
2u,u&!,
U22e66
-,-
2utu4C23
3U,“U,E,4.
The extremum approximants
property
is lost in this formal
development;
thus the finite
2p+1
u(p) = -& u, )
p = $6, 1, 35, *. *
07)
may be larger or smaller than the true value depending on p and the details of the problem. Despite the complicated appearance of the Schrodinger energy terms as exhibited in Eq. (16), these quantities behave in an analytically simple manner
SCHG~DINGER
with respect to two physically of the following two sections. REMARKS
mvmd2~
important
ON
LARGE
297
SERIES
variables.
UNIFORM
This behavior is the subject
SYSTEMS
I consider a sequence of many particle systems, the 4th system composed of .4 identical particles in a box of volume Au. Short range interactions produce a mean interaction energy proportional to A. Questions of co-existent phases and of saturation or nonsaturation are excluded by the assumption of constant mean particle density along the sequence. Under these conditions the individual terms in the Schrbdinger energy series must each approach a limiting form proportional to the number of particles; i.e.,
4’~ +
U,/A
= f*(v).
(18)
Brueckner (7) appears to have been the first to state Eq. (18) as a general theorem; he also verified the statement for s = 4 and 5 by explicit detailed consideration of the matrix elements. The explicit verification is by no means trivial. For example Ud involves both ~3 and Uzep3, both quadratic functions of A when A is large. The proper linear behavior of Uq results from an exact cancellation of quadratic terms contributed by ~43and U~EZZ. A general proof has been given by Riesenfeld and Watson (8)‘. Equation (14) illustrates the failure of the BW procedure when A is large. Since ~21, t32 , and ~22are all linear functions of A the limiting value of I: is a constant (independent of A). For fixed A, the order n in Eq. (10) may be taken large enough to yield satisfactory results for U, but it is clear that the required lower limit on n must increase steadily with A. Thus an essential nonuniformity is involved in the passage from the implicit BW form of Eqs. (10) and (11) to the explicit Schrodinger form of Eq. (17) in all applications to large uniform systems. In particular, the connection between the approximate wave functions of the BW procedure and the successive terms in the Schrbdinger energy series is lost as A increases without limit. For example, in this limit, Us = cpl has no simple connection with the first order wave function of the BW procedure, but must be correlated with a description of the system in which many independent pairs of particles occupy excited orbitals. Equations (10) and (11) are convenient for the derivation of the Schrbdinger energy series, but they are not the best forms within the range of the BW procedure. Continued fraction forms give better results with only a trivial increase in the amount of computational labor (I, 2, 9). 1 Equation energy series
(22’) of ref. 8 provides a starting equivalent to Eqs. (11) or (A9)
point of the
for the present
derivation paper.
of the Schr6dinger
298
FEE1\TUEIiG
To illustrate
a quadratic
this statement
1 discuss the form corresponding
to Eq. (14) :
equation for 1: with t’he solution
(20) Now the limiting
form for sufficieutly
large values of &4 is
I: 2 f&.l”’
(21)
of order Al” [corn * p are with t’he constant given by Eq. (l&)1. The statistical weight of the first order correction to the wave function is given by [t2/(e2 - Q)]‘E:,~ with the limiting value 1 as .4 increases without limit. A numerical example may also be useful. Calculations in nuclear 2, 3, and -k particle problems (3) with specific forms of exchange and tensor forces give ez2 N A/50. Using this relation and also set’ting arbitrarily e3? = ( -$$)tgl , Eq. (20) reduces to t! = p&J
1 + ( 1 + ,g&A
y-‘.
At -4 = 50, IT = (l$)eE:‘, . Under the same conditions much. THE
SCALE
TRANSFORMATI(
(22)
Eq. (la) gives only half as )?;
The p transformation introduced earlier (1, 2, 4) in studies of the BW formalism substitutes p(E - 3,) for E - RF:,,. In the present context I wish to replace E. - E, by ~1(E. - E,). This objective is accomplished by the t,ransformation: H
= Ho’ + W’,
Ho' = Ho - (P - 1) (Eo - Ho),
(23)
W’ = W + (p - 1) (E, - Ho), which has the desired consequence Eo’ - Ho = /L(E, - Ho).
(24)
SCHRtiDINGER
ENERGY
299
SERIES
Equations (4) to (10) still hold with W’ substituted for W, p(m) for (m), and primes placed on all other symbols. Thus, for example (3; 21)’ = ; [(3; 21) + (/.L The transformation the correspondence
formula for es, S--l follows with a binomial expansion:
1)(2; 2)1.
readily
from the definitions
and
(25) More generally 4, t = ; ;g C,t(lc)(fi
-
l)l’e-k,
t--B .
06)
To determine the coefficients CSt(6) let p(s, t) denote the number of solutions Eq. (8). Equation (12) with k = t - s + I yields p(s t) = (s 7
l)s(s + 1) . . . (t (t - s + l)!
of
1) (27)
= (t 4,: The condition
determining
(l,,(k)
I).
is
C,,(k)p(s - I;,t - k)= (s; 2>Ph,
t>
taking irlto arcount the binomial structure of t’he expansions in powers of p - 1 and the difference in the number of round bracket terms (defined by Eq. (6)) making up r,st and &k, t--k . Equations (27) and (28) yield
C,,(k)= (t ;’ l > with the resulting
explicit form for Eq. (26) : 1 8--y t - 1 Est’ = ~ c b-4 pt k=O ( k >
In the limit form
n
l)kc,-k,
t--t .
-+ m the primed version of Eq. (10) contains coefficients
(30) of t.he
300
FEENBERG
These are in fact equal to the corresponding unprimed quantities. The proof follows readily from Eq. (30). The primed versions of Eqs. (10) and (15) now yield formulas for U,’ as in Eq. (16) with primes on all quantities occurring in the right hand members. These yield the linear approximants U’(P)
2p+1 F
=
u,‘,
p
=
corresponding to Eq. (17). I find by trial terms are related by a linear transformation
y2,
1, Fi,
. . . )
(31)
that the primed and unprimed U, identical in form with Eq. (25); i.e., (32)
Equation lustrating
(32) has been verified for .Y = 2, 3, . . . up to s = 10. An example ilthe verification may be helpful. At s = 5 Ug) = E&4’-
p4u6’
=
EC4
+ 3(P
-
l)E43
+
3(/J -
-
U3)E22’ -
+
lYe32 E22[U3
+
(l.4 (p
Regrouping
terms, the last equation becomes
/L4U6’
-
=
[ES4
U3t22
U2’633’
-
l)"CZl 1) c21
-
C[2[t33
+
2(p
-
l)e221.
- U2E331 + 3(/J - 1)
[e43
-
U,c,]
+ :3(/J -
U2U3 + (/J -
1)“Uz )
which verifies Eq. (32) for s = 5. The actual proof of Eq. (32) can be developed from a theorem proved in the Appendix. The first three approximants given by Eq. (31) are
(33)
The sequence of primed approximants converges at a rate dependent on the numerical value assigned to CL.We would like to choose p so that’ even t’he first
SCHRBDINGER
ENERGY
SERIES
301
few approximants are close to the limit of the sequence. Since the limit is not known, a definite choice must be based on assumptions, implicit or explicit, about the behavior of high order approximants beyond the small number which, in any given case, have actually been evaluated. One choice deserving :I brief discussion is determined by 11; = 0; i.e.,
This yields
These expressions have the form of the original linear approximants C’(l) and u ‘2’2), plus, in each case, a characteristic remainder which is small if j CT?/CT21 << 1. The reader can easily verify that I r2 7.:z2 112 [j,’ - j-ja’ = __772 - I::$ (37) /j;y iy2/ (-j4’ - [,T,‘Z) - U3a) =- u,“(u,r~, ( IT2 - 1!3’)3 ( cr2 - Cl,)3 . Thus J);“’ a,ld uji(3’2) are unchanged by the substitution of LTs’ for t’, . The logical situation may appear slightly confusing since we start from a special value for p and arrive at invariant forms for C’i”’ and I:2(3’2). Whether or not these forms are useful extrapolation formulas may be left for experience to decide. Invariant forms for higher order approximants are known. In particular the form for 7~ = 2 is
The proof of invariance using Eq. (32) is straightforward; also if L’, is replaced by X”I’, and UzC2’ expressed as a power series in X one finds [Ti(2) = X”[Ts + X”[:a + Xh[r, + xj[.:5 + o(p)
(39)
verifying the connection with the linear approximant I:“’ (or IT’(‘)). The general problem of invariant extrapolation forms is best discussed in the language of continued fractions (2). Speisman (10) in a recent paper gives a convergence condition for the Schrsdinger energy series, expressed as an upper limit on the strength of the interaction operator. The convergence condition is readily adapted to the scale trans-
302
FEENBERG
formation and then provides a possible criterion for determining the best value of p. I am indebted to Dr. S. I?. Rosen for a useful suggestion and to Mr. John Clark for a careful reading of this manuscript. APPENIlIS
Given two sets of quantities
A, , B, subject to the transformation
law
6.i- l)“A,-n) (cl
-
l)“B,-k.
I define c, = c As--kBk , k c,’ = c
As-k’&’
k
and show that (/A Proof of Eq. (A4) : Eqs. (Al),
l>“C,-, .
(A2), and (A3) yield
Let h = 1 + I;, m = s + k: and make the substitutions
(“-;-‘)-(ii”r’,‘,). Equation CLI’ =$,&P = $2 Equation
(A5) becomes (P -
l,“~L~(s+;::,ljh F
(A4) follows
l)(i/
&mBm-h directly
F (m “,
y;pil !
l) (;
1> As+k--h&-s--k 1 ;---‘,)
from Eq. (A7) and the binomial
. identity
(A7) (11) (MU
SCHR6DINGER
ENERGY
303
SERIES
This theorem may he applied to Eq. (10) in the limiting case 71--f 03 which reads u
=
2 s=l
Es+1,s
-
u
2 ex,,? s=?, LW
+
r-?fLs
s=3
+
. . . +
(-1yup
&-pCl,r P+l
+
. . . .
One seesthat the partition of L~‘P~~s’-p+I, a into a series by the device of a formal expansion parameter generates the same terms as the scale transformation. The transform of the product series is identical with the product of the transformed series. IIECEIVED:
November 8, 1957 REFERENCES
1. P. GOLDHAMMER.~ND IX:. FEENBERG, Phys. f&v. 101, 1233 (1956); E. FEENBERG, Phys. IZw. 103, 1116 (1956). 2. It. C. YOUKG, 1,. C. BIEDENHARN, AND E;. FEENBERG, Phys. Rev. 106, 1151 (1957). 3. nl. BOLSTERLI AND I<. FEEKBERG, Phys. Rev. 101, 1349 (1956). 4. 15. FEENBERG AND P. GOLDHAMMER, Phys. Rev. 106, 750 (1957). 5. I). H. LYOSS AND A. M. FEINGOLD, Phys. Rev. 96, 606 (1954). 6. A. nl. FEINGOLU, Phys. Rev. 101, 258 (1956). '7. K. A. BRUECKNER, Phys. Rev. 100, 36 (1955). 8. W. B. RIESENE'ELD AND K. M. WATSON, Phys. Rev. 104, 492 (1956). 9. B. A. LIPPMAN, Phys. Rev. 103, 1149 (1956). IO. G. SPEISMAN, Phys. Rev. 107, 1180 (1957). 11. G. RACAH, Phys. Rev. 61, 186 (1942).
OF
3, 292-303
PHYSICS:
Analysis
(1958)
of the
Schrsdinger EUGENE
Wayman
Crow
Laboratory
of Physics,
Energy
Series*
FEENBERG Washington
University,
St. Louis,
Missouri
The rate of convergence of the Brillouin-Wigner and Schrodinger energy series can be modified by using elements of freedom present in the formulation of the perturbation problem. Two such elements of freedom are (a) a uniform displacement of the zeroth order energy spectrum and (b) a uniform change of scale in the spacing of the zeroth order energy levels. Transformation a is used to generate the Schrodinger energy series from the corresponding Brillouin-Wigner series. The effect of transformation b on the former series is worked out and exhibited explicitly. The problem of defining a criterion for determining the scale factor is bypassed by the observation that continued fraction approximants to the Schrodinger energy series are invariant under transformation b. Explicit formulas are given for the first three invariant forms. INTRODUCTION
The Schrodinger perturbation procedure begins with the decomposition of a given Hamiltonian operator H into a sum of two operators, a zeroth order component Ho and a perturbation operator W. Elements of freedom are present in the choice of Ho and the corresponding W. Two simple, but physically important, elements of freedom are (a) a uniform displacement of the zeroth order energy spectrum (b) a uniform change of scale in the spacing of the zeroth order energy levels (for free particles in a box this transformation is equivalent to the introduction of an effective mass). The same elements of freedom occur in the formulation of the BrillouinWigner perturbation procedures. Here the extremum property of the odd order energy approximants determines optimum magnitudes for the uniform displacement and also, in the original formulation of BW procedure, optimum scale values for the zeroth order level spacing (I). The recentIy developed continued fraction formulation of the BW procedure (I, 2) gives odd order approximants to the energy eigenvalues which are invariant under transformation b. In this paper transformation a is used to generate the Schrodinger energy * This research was supported by the United Office of Scientific Research of the Air Research tract So. AF18(603)-108. 292
States Air Force and Development
through the Air Force Command, under con-
SCHR6DINGER
ENERGY
293
SERIES
series from the BW approximants to the energy eigenvalues. The application of transformation b to the Schrbdinger energy series brings with it the problem of defining an adequate criterion for determining the scale factor. This problem is bypassed by the observation that continued fraction approximants to the Schrodinger energy series possess the invariance property mentioned earlier in connection with the BW formulation. In nuclear problems computational difficulties limit applications of the BW procedure to the first order correction to the wave function and second and third order corrections to the energy (3, 4). The range of usefulness of this approximation is restricted to light nuclei, possibly extending to the end of the first p shell. Valuable results have been found at mass numbers 6 and 7 by methods closely related to the first order BW procedure (5, 6). Wigner and others have pointed out that the number of terms in the BW energy series required to attain physically meaningful results increases with t’he number of particles A. This number of terms, usually quoted as of the order A, has possibly been overestimated because of failure to utilize transformations a and b in discussing the many particle problem. Under conditions of constant particle density and interact,ions of finite range one expects an interaction energy proportional to the number of particles. The usual treatment gives the result that inclusion of the first order correction to the wave function in the energy calculation improves the energy by an additive term independent of A when A is large. Using transformations a and b I show that the energy is improved by an additive term proportional to A Ii2 when the first order correction to bhe wave function is included in the energy calculation. This result is still yuit.e inadequate, but does suggest that t.he number of t,erms in the BW energy series required to attain meaningful accuracy may be smaller t’han A in the limit of large A when proper use is made of transformations a and b or alternatively of transformation a and the invariant continued fraction approximants. I)ERIVATION
OF
THE
SCHRdDINGER
ENERGY
SERIES
Let E, and #, denote the eigenvalues and the normalized, orthogonal eigenfunctions generated by Ho . Consideration may be limited to specific values of the appropriate constants of motion wit’hout explicit introduction of symbols to denote t,hese constants. The matrix elements of TY are denoted by W,, . Some trivial complications in the st.atement of the perturbation formulas may be avoided by incorporating the zero-zero matrix element of the perturbation operator into Ho . One has then Woo = 0 and (H),, = Eo . The general BW energy term for the state generated from $0 in zeroth order approximation is
dE
- Eo , r’)
=
WdWnm - mm,>. . . (Wgz- is,,) w.?o C’ mn”.vz (E - Em - U)(E -E,, - U)...(E - EC- i?)’
(1)
294
FEENBERG
in which the prime on the summation symbol means the s - 1 variable indices m, n, . . . y, z do not take on the value 0. Note that E~is a sum of terms each having s factors in the numerator and s - 1 factors in the denominator. Transformation a is incorporated in the formalism by introducing t,he parameter I:. The nth approximation to the wave function yields an approximation to the eigenvalue determined by the lowest root of the implicit equation 2 n+1
E = Who +
c2 dE
n = 1,2, . . . .
- Eo , I:),
(2)
From the lowest root E one goes to the minumin value of E as a function of U, thus determining an optimum value of U and the lowest possible value of E. The Schrbdinger energy series is generated by the choice, not the optimum,
U = E - E,, .
(3)
Eq. (2) reduces to an algebraic equation of order 2n tity U:
1 in t’he unknown
quan-
2n+1 u
=
c
f,?(U,
(4)
U).
2
Xow writing
E,(U) = E,(U, Cr), Eq. (4) is equivalent li =
A suitable notation
facilitates
the further
Q
(s; t1tz ... in which
2n+1 8-2 l Tk {& z k% jij L
=
f,(~J)]“~o. discussion.
1 =
(5)
Let (m) := Eo - E, and
C’ W”?nW?m . . . w,,w,o mn’-‘tJz (m)yn)t2 . . . (Z)t8-l ’
,,&
--1 (s;
(6)
and ti = 1, 2, 3, . . . . Next define
W occurs s times in each numerator fs,
to
t1t2
(‘7)
. . . Ll)
8
the sum ranging over all distinct constraint t1 + tz +
sets of integers tl , t2 , . . . , t,-l subject to the . . + 2,-l = t 2 s -
1.
(8)
Thus for s = 4, t = 6, the sum in Eq. (7) includes all distinct permutations 4, 1, 1; 3, 2, 1; and 2, 2, 2. Examination of Eq. (1) defining es(U) = es(Ur W) now yields t,+,(U)
=
fs+l,s
-
ufs,8
+
U2f8+-1,,
I-
* ’
1
+
(-l)S-ll+t:,,r
.
of
(9)
SCHRGDINGER
EiXERGY
SERIES
295
Eqs. (4) and (9) combine to give 2n
2n+1 T.. =
=
c
s=2
fs,s-l
-
2n-1
r: 2 es, s + r2 c Es.s+1 + . . . n=2
P-‘f2,
3,, ) (10)
a more explicit statement of Eq. (5). At this point it is helpful to use Eq. (iL) again in the form
= exp The first line of EC]. (I I) is Smply t.he complete IW series for the energy with a particular choice for t,he displacement parameter. The second line introduces the Taylor’s series for E~(I:, 0) as a function of C’ in t*erms of ~,~-l and its derivatives with respect to E, . The last named function appears because it is t,he same function of IS, t,hnt, c~(C:, 0) is of & + I:. Equat.ing coefficientIs of Crk in Eqs. (5), (lo), and (11) we get.
Thus, for example, at s = 4: f43 = (4; 111) f4$ = (4; 211) + (4; 121) + (4; 112)
E45
= (4; 311) + (4; 131) + (4; 113) + (4; 221) + (4; 212) + (4; 122)
zzz---i a
2 aE0
E44 =
9
1 a2 m
(13)
643 ,
f(fi = (4; 411) + (4; 141) + (4; 114) + (4; 321) + (4; 13’2) + (4; 213) + (4; 312) + (4; 123) + (4; 231) + (4; 222)
= -- 1 a
1 a3
Eqs. (10) and (11) are implicit
equations which
retain the extremum
property
296
FEENBERG
of the BW formalism from which they are derived. In one special case (n, = 1) Eq. (10) reduces to an explicit relation: zr
=
E21 + -7 1 +
e32
(14
t22
The Schrodinger energy series is a sequence of explicit approximations for U obtained from Eqs. (10) or (11) by the formal device of associating an expansion parameter X with W, expressing U as a power series in X and fmally setting X = 1. Introducing
u = 2 X”U,
(13
2
into Eq. (11) and equating coefficients of X” generated by the left and right hand members we obtain u2
=
E21 ,
u3
=
E32 ,
u4
=
e43
-
U2t22
us
=
664
-
u3e22
-
l72e33
US
=
c65
-
U4E22
-
U3t%3
-
u2644
+
u2’E21
cT4E33
-
U3E44
-
U2e65
f
u22E34
u5c33
-
u4c44
-
U3E65
-
u2m
,
UT = 676- U&2 us
=
ta7
-
U6C22
-
+ 2 U&&4 ug
=
tgjj
-
U?e22
-I-
1332634
-
f
-
)
U32e23
+
2 U2U4e23
u6e33
-
USC44
i723C35
+
2u,u,C,,
-
,
-
u23c24
u4t56
-
-,-
2u:,u4e34
+
2u2u3C23
$
u22e45
-
u2c77
06)
,
,
U3f66
+
+
2u,u&!,
U22e66
-,-
2utu4C23
3U,“U,E,4.
The extremum approximants
property
is lost in this formal
development;
thus the finite
2p+1
u(p) = -& u, )
p = $6, 1, 35, *. *
07)
may be larger or smaller than the true value depending on p and the details of the problem. Despite the complicated appearance of the Schrodinger energy terms as exhibited in Eq. (16), these quantities behave in an analytically simple manner
SCHG~DINGER
with respect to two physically of the following two sections. REMARKS
mvmd2~
important
ON
LARGE
297
SERIES
variables.
UNIFORM
This behavior is the subject
SYSTEMS
I consider a sequence of many particle systems, the 4th system composed of .4 identical particles in a box of volume Au. Short range interactions produce a mean interaction energy proportional to A. Questions of co-existent phases and of saturation or nonsaturation are excluded by the assumption of constant mean particle density along the sequence. Under these conditions the individual terms in the Schrbdinger energy series must each approach a limiting form proportional to the number of particles; i.e.,
4’~ +
U,/A
= f*(v).
(18)
Brueckner (7) appears to have been the first to state Eq. (18) as a general theorem; he also verified the statement for s = 4 and 5 by explicit detailed consideration of the matrix elements. The explicit verification is by no means trivial. For example Ud involves both ~3 and Uzep3, both quadratic functions of A when A is large. The proper linear behavior of Uq results from an exact cancellation of quadratic terms contributed by ~43and U~EZZ. A general proof has been given by Riesenfeld and Watson (8)‘. Equation (14) illustrates the failure of the BW procedure when A is large. Since ~21, t32 , and ~22are all linear functions of A the limiting value of I: is a constant (independent of A). For fixed A, the order n in Eq. (10) may be taken large enough to yield satisfactory results for U, but it is clear that the required lower limit on n must increase steadily with A. Thus an essential nonuniformity is involved in the passage from the implicit BW form of Eqs. (10) and (11) to the explicit Schrodinger form of Eq. (17) in all applications to large uniform systems. In particular, the connection between the approximate wave functions of the BW procedure and the successive terms in the Schrbdinger energy series is lost as A increases without limit. For example, in this limit, Us = cpl has no simple connection with the first order wave function of the BW procedure, but must be correlated with a description of the system in which many independent pairs of particles occupy excited orbitals. Equations (10) and (11) are convenient for the derivation of the Schrbdinger energy series, but they are not the best forms within the range of the BW procedure. Continued fraction forms give better results with only a trivial increase in the amount of computational labor (I, 2, 9). 1 Equation energy series
(22’) of ref. 8 provides a starting equivalent to Eqs. (11) or (A9)
point of the
for the present
derivation paper.
of the Schr6dinger
298
FEE1\TUEIiG
To illustrate
a quadratic
this statement
1 discuss the form corresponding
to Eq. (14) :
equation for 1: with t’he solution
(20) Now the limiting
form for sufficieutly
large values of &4 is
I: 2 f&.l”’
(21)
of order Al” [corn * p are with t’he constant given by Eq. (l&)1. The statistical weight of the first order correction to the wave function is given by [t2/(e2 - Q)]‘E:,~ with the limiting value 1 as .4 increases without limit. A numerical example may also be useful. Calculations in nuclear 2, 3, and -k particle problems (3) with specific forms of exchange and tensor forces give ez2 N A/50. Using this relation and also set’ting arbitrarily e3? = ( -$$)tgl , Eq. (20) reduces to t! = p&J
1 + ( 1 + ,g&A
y-‘.
At -4 = 50, IT = (l$)eE:‘, . Under the same conditions much. THE
SCALE
TRANSFORMATI(
(22)
Eq. (la) gives only half as )?;
The p transformation introduced earlier (1, 2, 4) in studies of the BW formalism substitutes p(E - 3,) for E - RF:,,. In the present context I wish to replace E. - E, by ~1(E. - E,). This objective is accomplished by the t,ransformation: H
= Ho’ + W’,
Ho' = Ho - (P - 1) (Eo - Ho),
(23)
W’ = W + (p - 1) (E, - Ho), which has the desired consequence Eo’ - Ho = /L(E, - Ho).
(24)
SCHRtiDINGER
ENERGY
299
SERIES
Equations (4) to (10) still hold with W’ substituted for W, p(m) for (m), and primes placed on all other symbols. Thus, for example (3; 21)’ = ; [(3; 21) + (/.L The transformation the correspondence
formula for es, S--l follows with a binomial expansion:
1)(2; 2)1.
readily
from the definitions
and
(25) More generally 4, t = ; ;g C,t(lc)(fi
-
l)l’e-k,
t--B .
06)
To determine the coefficients CSt(6) let p(s, t) denote the number of solutions Eq. (8). Equation (12) with k = t - s + I yields p(s t) = (s 7
l)s(s + 1) . . . (t (t - s + l)!
of
1) (27)
= (t 4,: The condition
determining
(l,,(k)
I).
is
C,,(k)p(s - I;,t - k)= (s; 2>Ph,
t>
taking irlto arcount the binomial structure of t’he expansions in powers of p - 1 and the difference in the number of round bracket terms (defined by Eq. (6)) making up r,st and &k, t--k . Equations (27) and (28) yield
C,,(k)= (t ;’ l > with the resulting
explicit form for Eq. (26) : 1 8--y t - 1 Est’ = ~ c b-4 pt k=O ( k >
In the limit form
n
l)kc,-k,
t--t .
-+ m the primed version of Eq. (10) contains coefficients
(30) of t.he
300
FEENBERG
These are in fact equal to the corresponding unprimed quantities. The proof follows readily from Eq. (30). The primed versions of Eqs. (10) and (15) now yield formulas for U,’ as in Eq. (16) with primes on all quantities occurring in the right hand members. These yield the linear approximants U’(P)
2p+1 F
=
u,‘,
p
=
corresponding to Eq. (17). I find by trial terms are related by a linear transformation
y2,
1, Fi,
. . . )
(31)
that the primed and unprimed U, identical in form with Eq. (25); i.e., (32)
Equation lustrating
(32) has been verified for .Y = 2, 3, . . . up to s = 10. An example ilthe verification may be helpful. At s = 5 Ug) = E&4’-
p4u6’
=
EC4
+ 3(P
-
l)E43
+
3(/J -
-
U3)E22’ -
+
lYe32 E22[U3
+
(l.4 (p
Regrouping
terms, the last equation becomes
/L4U6’
-
=
[ES4
U3t22
U2’633’
-
l)"CZl 1) c21
-
C[2[t33
+
2(p
-
l)e221.
- U2E331 + 3(/J - 1)
[e43
-
U,c,]
+ :3(/J -
U2U3 + (/J -
1)“Uz )
which verifies Eq. (32) for s = 5. The actual proof of Eq. (32) can be developed from a theorem proved in the Appendix. The first three approximants given by Eq. (31) are
(33)
The sequence of primed approximants converges at a rate dependent on the numerical value assigned to CL.We would like to choose p so that’ even t’he first
SCHRBDINGER
ENERGY
SERIES
301
few approximants are close to the limit of the sequence. Since the limit is not known, a definite choice must be based on assumptions, implicit or explicit, about the behavior of high order approximants beyond the small number which, in any given case, have actually been evaluated. One choice deserving :I brief discussion is determined by 11; = 0; i.e.,
This yields
These expressions have the form of the original linear approximants C’(l) and u ‘2’2), plus, in each case, a characteristic remainder which is small if j CT?/CT21 << 1. The reader can easily verify that I r2 7.:z2 112 [j,’ - j-ja’ = __772 - I::$ (37) /j;y iy2/ (-j4’ - [,T,‘Z) - U3a) =- u,“(u,r~, ( IT2 - 1!3’)3 ( cr2 - Cl,)3 . Thus J);“’ a,ld uji(3’2) are unchanged by the substitution of LTs’ for t’, . The logical situation may appear slightly confusing since we start from a special value for p and arrive at invariant forms for C’i”’ and I:2(3’2). Whether or not these forms are useful extrapolation formulas may be left for experience to decide. Invariant forms for higher order approximants are known. In particular the form for 7~ = 2 is
The proof of invariance using Eq. (32) is straightforward; also if L’, is replaced by X”I’, and UzC2’ expressed as a power series in X one finds [Ti(2) = X”[Ts + X”[:a + Xh[r, + xj[.:5 + o(p)
(39)
verifying the connection with the linear approximant I:“’ (or IT’(‘)). The general problem of invariant extrapolation forms is best discussed in the language of continued fractions (2). Speisman (10) in a recent paper gives a convergence condition for the Schrsdinger energy series, expressed as an upper limit on the strength of the interaction operator. The convergence condition is readily adapted to the scale trans-
302
FEENBERG
formation and then provides a possible criterion for determining the best value of p. I am indebted to Dr. S. I?. Rosen for a useful suggestion and to Mr. John Clark for a careful reading of this manuscript. APPENIlIS
Given two sets of quantities
A, , B, subject to the transformation
law
6.i- l)“A,-n) (cl
-
l)“B,-k.
I define c, = c As--kBk , k c,’ = c
As-k’&’
k
and show that (/A Proof of Eq. (A4) : Eqs. (Al),
l>“C,-, .
(A2), and (A3) yield
Let h = 1 + I;, m = s + k: and make the substitutions
(“-;-‘)-(ii”r’,‘,). Equation CLI’ =$,&P = $2 Equation
(A5) becomes (P -
l,“~L~(s+;::,ljh F
(A4) follows
l)(i/
&mBm-h directly
F (m “,
y;pil !
l) (;
1> As+k--h&-s--k 1 ;---‘,)
from Eq. (A7) and the binomial
. identity
(A7) (11) (MU
SCHR6DINGER
ENERGY
303
SERIES
This theorem may he applied to Eq. (10) in the limiting case 71--f 03 which reads u
=
2 s=l
Es+1,s
-
u
2 ex,,? s=?, LW
+
r-?fLs
s=3
+
. . . +
(-1yup
&-pCl,r P+l
+
. . . .
One seesthat the partition of L~‘P~~s’-p+I, a into a series by the device of a formal expansion parameter generates the same terms as the scale transformation. The transform of the product series is identical with the product of the transformed series. IIECEIVED:
November 8, 1957 REFERENCES
1. P. GOLDHAMMER.~ND IX:. FEENBERG, Phys. f&v. 101, 1233 (1956); E. FEENBERG, Phys. IZw. 103, 1116 (1956). 2. It. C. YOUKG, 1,. C. BIEDENHARN, AND E;. FEENBERG, Phys. Rev. 106, 1151 (1957). 3. nl. BOLSTERLI AND I<. FEEKBERG, Phys. Rev. 101, 1349 (1956). 4. 15. FEENBERG AND P. GOLDHAMMER, Phys. Rev. 106, 750 (1957). 5. I). H. LYOSS AND A. M. FEINGOLD, Phys. Rev. 96, 606 (1954). 6. A. nl. FEINGOLU, Phys. Rev. 101, 258 (1956). '7. K. A. BRUECKNER, Phys. Rev. 100, 36 (1955). 8. W. B. RIESENE'ELD AND K. M. WATSON, Phys. Rev. 104, 492 (1956). 9. B. A. LIPPMAN, Phys. Rev. 103, 1149 (1956). IO. G. SPEISMAN, Phys. Rev. 107, 1180 (1957). 11. G. RACAH, Phys. Rev. 61, 186 (1942).