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The sign-change argument revisited
Volume 76, number 1
CHEMICAL
THE SIGN-CHANGE
PHYSICS
1.5November 1980
LETTERS
ARGUMENT REVISITED
Kamal BHATTACHARWA and Sankar Prasad BHATTACHARYYA I7leov Group of the Department of Physical Chemstry. Indran Assoclntlon for the Ciiltwanon of Scletrce. Jadmpur,
Recewed
Calcurr~-700032, Zndra
8 August 1980,
m fmal form 16 September
1980
Recent examples suggestmg the mapphcablty of the “sign-change” argument III perturbation ysed. It IS shown that the arguments leadmg to such a conch&ton are not entirelyJusttiisd_
1. Introduction
Eo@)=
In perturbation problems convergence lfficulties are often encountered and m those sltuatlons physzcal arguments are frequently mvoked to justify the observed behaviour. Of these, the rather well-known one is the “sign-change” argument [ 1 ,z]. It starts with the basx
prermse
that
rf a perturbation
senes
converges
for tlhl, the couplingparameter associated wth the perturbing harrultonian, convergence of the same se-
theory arc critically anal-
1 +3X/4-~21X~/16+liIX3/21
f...,
to more realistic sltuatlons, e.g. the Stark effect case where only metastable states correspond to the hamiltonian H@)=_5724-1 and one
_k,
rarely
goes beyond
second-order
@oIarizabil-
ity calculations)
to note that the associated (ground state) energy senes
ries for -1X( is guaranteed by the existence of a circle of convergence with #zotz-zeroradnts. But if It so happens that for -IX1 there exist no bound-state solutions to the corresponding S&r&linger equatron, it is unhkely that the energy perturbation series, E(A) would converge to a well-defmed pomt (value). Hence, the series will have a zero ra&us of convergence for either sign of 1x1,assunzing that perturbation theory must converge, if at all, to some physkaliJJ meaningful region. Essentially, the same argument was used by Dyson [3] (changmg e2 + -e2) to trace the ongin of the fanuhar &vergence problem in perturbationtheoretical approach to quantum electrodynamics. In fact, the sign-change argument works well m a wde variety of problems startmg from the “mnocent-
Recently, Sunon [7] cited a “simple’* Coulomb problem as a counter example exposing the vulnerabiiity of the mherent assumption of this “folk theorem” which, at the same tune, amounts to refuting Dyson’s celebrated argument also. More recently, Killingbeck [8] found another example, a polynomial perturbatlon acting on the hydrogenic S-state hamiltonian, where also the sign-change argument apparently fails. In what follows, we analyse these examples to show that contrary to the implicit assumpfiun of these
looking”
authors
wth
test problem
of the anharmonic
oscillator.
the hamdtonian
f _..
IS even more dangerous, though it was pointed out in the hterature [6] long ago (for a recent review, see ref.
PI)-
[7.8]
(see section
2) none of their examples
represent proper perturbation problems in the regions considered (section 3).
H(x)=-@+x2+Ax4 and the correspondmg dangerous gy series [4,5]
Eo(X) = -4 - 9X?-14 - 3555X4/64
(ground state) ener-
117
Volume
76, number
2. Counter
1
CHEMICAL
PHYSICS
examples
Simon’s
counter
LETTERS
15
turbation and apply Raylergh-Schrodmger tion theory (RSPT), we get example
proceeds
with the ham.&
toman
Eo(A) =
c
$A”
November 1980 perturba-
>
m=O
H~)=-;V2_.-1
+b-l
For srmphcrty, only, so that @u =e-',
we shall consrder
EO
=_- :,
The correspondmg are
However,
= E,@l4,(X)
c I,,
=
Eo
when k-1 is considered Srmon used the equahty oz
state
-A)‘-.
have the form
e;=
e;X”‘,
0
(2)
.
since Eo(A) should m
Eo(X)=
(S) state
for the perturbed
Eo(A)=-$(l
H(h)\lro(h)
(1)
the ground
ffOoO=EOQO -
quanttttes
*o(A)=e-(l-A)r,
_
=H,+&--l
,
as a perturbatron
to Ho.
(3 and, comparing
cluded
coefficrents
of K (L = 0, 1,2,
and noting a discrepancy m the magnitudes of A(+X) and A(A) (typrcally, A = 0.00000 for X = +O.l, and .I = 0.034 17 for X = -0.1) we are led to the mescapable conclusron that for X < 0, a component of Eo(A) is nussing III the RS series, although the srgn-change argument apphed to the hamrltoruan H(X) in (7) does not suggest the invahdtty of RSPT m this problem. Krllingbeck’s fine analysts [8] shows that bound states e.xrst for -1X(, yet the wavefunction [see (7)] becomes unnormahzable, rather mystenously. Thus, here again, the sign-change argument seems to be m trouble.
._.), con-
+I,
+_;,
E;;’ (I?I 2 3) = 0,
(6)
denving certam rreru sum-rules, as by-product. Argumg further that stnce m the problem ctted perturbation theory would also converge even for A(rea.l) > 1 where the state becomes clearly mphyncal, he concluded that the mearung of the formal ergenvalue equatron (3) IS difficult to mterpret and such convergence to an “antr-bound” state contradicts the underlymg assumptron of the ngn-change argument. klhngbeck’s mteresting observation [s] was the result of “choosmg some wavefunctron fast and fmding the correspondmg hanultoruan later” [2] type of approach. Then It rs easy to see that for = e-(r+hr’)
HO)=-iv’-r-l
,
+2?,_ri-33-$,
Eo(h)=3A-$. If we consrder 118
produced within the first order), one encounters a rather disturbmg situation unme&ately. Computing the quantity A(A) [= E, (numencal) - E, (RSPT)]
that
Eo=-&
*,@)
Along with Killmgbeck, if it is nowasamcd that RSPT works for X > 0. wth all h&her order (nz > 2) corrections (!) varushmg e_xactly (for the exact energy is re-
(7) the X-dependent
part in H(x) as a per-
3. Re-investi&ion imphcit in Srmon’s argument is the assumptron of the apphcabdity of perturbation theory over the entrre range of ]A] used and the existence of a properly convergent senes even for ]A] > 1. But IS thx assumptron well-founded? We may first note that, although the perturbmg operator mvolved m (1) IS bounded, the Kate-Relhch theorem [2,9,10] s6.U demands that IAl be suffclemZy small to ensure the convergence of RSPT. One might wonder if It 1s possible at all to pro-de a reasonable magnitude of the “smallness” parameter h, based on purely physical grounds. In fact, we have m possession a “rule of thumb” [lo] whrch says that perturbation theory is hkely to work when energy shifts are small relative to the spacings of unperturbed levels (on the average). Such a consideration immediately rules out the possrbllity of letting IAl + 1 in (1). However, tis naive and qua& tative rule might not be quote appealmg. But as we shall presently see, a more precise analysis of the prob-
15 November 1980
CHEMICAL PHYSICS LETTERS
Volume 76, number 1
I
lem only worsens the sltuahon.
Let us accept that (6)
1s true
w-101ET (nz > 2) = 0. If it were so, the zeroth-order
for any 1x1.This at once forces us to conclude that the perturbed wavefunction for small coupling
wavef
wdi be
e-(’ = @o + WA ,
Jl@SPT)
where 0: is the first-order obeys the equation (HrJ - Eo)Qb + (r-1
(9) correction
- Wolr-‘I
function
90))90
that
= o ,
(10)
with
= E; = 1 .
(11)
To solve (IO), It is common practice to follow the -‘F-function techruque” [ 11,12] and tlus gives 0; = (r - $,C+) ,
(Q&)>
=0 -
= (1 -I-Xr - 3A/2)00
(12)
.
(13)
Needless to say this does not agree with the exact perturbed function in (3) so(A)
= e-(1-&)’
= e-/(1
+ Xr + . ..) _
(14)
In fact, (13) and (14) agree only when h 1s mfimtesunally small_ although. wth enerw [see (S)] , a parallel relation m the wavefunction also was expected for any A. Thus, we conclude that m Sunon’s example (1) either the set of equatrons
IS correct
whxh
demands h to satisfy
(1 +Xr-33)\/2)e-‘=e-(*-“)‘,
= e-r
9
wluch IS obviously true only m the linut IAl + 0 (and not when ]hl = 0.1). Essentially, the situation here is sinular to some kmd of “adiabatic swicritching-on”, so to say, that mamtams statlonarity- It is, on the other hand , quite likely that non-vanishing higher-order terms are contained in expansion (S), in whrch case the inapphcability of RSPT is yet to be shown. OtherLvlse, as we have pomred out, RAPT in this case is true
only in an “adrabatrc”
Usmg the ekpresnon for 0; and standard RSPT equations, it can easdy be checked that E; = -$ and E; = 0. Now from (9) and (12), we have \k,(RSPT)
+ ti2)
(15)
wtuch, as we hawe seen, IS true OIZ& m the nelghbourhood of X equal to zero so that for erther sign of IAl the states corresponrhng to H(X) are ante physlcal; (u) or his proposed “sum rules” are mcorrect and h&her-order fite ener,q corrections exist in expansion (S), in whch case It requires further scrutmy to find the hmitmg value of A (= ho) for whxh the RS senes still converges (whxh 1s quite unlikely to be greater than 1) and only then 1t.s stand agauxt the sign-change argument can be gven any credence. Regarding the harmltonian in (7), we comment that RSPT in this case does not work at all for any X
to the perturbed have
sense.
4. Conclusion
The above analysis reveals certain subtle pomts of RSPT which were somehow rmssed in earlier works and led to fallacrous sltuatlons. We hke to emphasize that an esact matching of the RAPT ener,T series truncated at fmite order with the true energy of the perturbed state does not necessarily imply that still h&her-order terms are all vanishing. The important pomt IS that a conclusion based on perturbatlontheoretical arguments is meannmgful only if the convergence of the corresponding perturbation s&es is ensured. Thus simply means that one must have a ngorous basis for treating a given problem from a perturbatlve vlewTomt_ To be wise after the event, another crucial point in thus issue, which might otherwxe be misunderstood, IS that the sign-change argument does Not ensure convergence. Its domain of apphcation lies only in predrctmg divergence. In any case, our re-mvestigntion on the doubts cast so far pomts to the conclusron that this apparently naw2 [ZJ argument still retains its full glory. W2 also happily note, in passing, that Dyson’s
argument
remams
vahd quite clearly_
References [l] [2] [3] [4]
J.B. Krieger, J. htath. Phys. 9 (1968) 432. J. Krllmgbeck, Rept. Progr. Phys. 10 (1977) 963. F-J. Dyson, Phys. Rev. 8.5 (1952) 631. C. Bender and T.T. Wu, Phys. Rev. 184 (1969) 1231.
119
Volume 76, number 1
CHEMlCALPHYSICSLETTERS
[ 5f F.T. Hioe, D. MacM~en and E.W.Montroll,whys. Rcpt. 43 (1978) 305, [ 6 ] A. Dalgamo, in: Quantum theory, Vol. 1. ed, D.R. Bates (AcademicPress, New York, 19611pp. 171-209. 171 B. Simon, Ann, Phys. NY 58 (1970) 76. 181 J. KilUngbeek,Phys.Letters 67A (1978) 13. i9j T. Kato, J. Fat. Sci. Univ. Tokyo Sec. 1 6 (1951) 14%
15 November 1980
[ 101 J-0. Hirschfelder,W. Byers Browa and S.T. Epstein, Advan.Quantum C!hem.1 (1964) 255. f 111A, D&amo and J.T. Letis, Proc. Roy. Sot. A233 (1955) 70, [I21 A. Da&no and A.J.. Stewart, Proc. Roy. Sot. A257 (1960) 534.
CHEMICAL
THE SIGN-CHANGE
PHYSICS
1.5November 1980
LETTERS
ARGUMENT REVISITED
Kamal BHATTACHARWA and Sankar Prasad BHATTACHARYYA I7leov Group of the Department of Physical Chemstry. Indran Assoclntlon for the Ciiltwanon of Scletrce. Jadmpur,
Recewed
Calcurr~-700032, Zndra
8 August 1980,
m fmal form 16 September
1980
Recent examples suggestmg the mapphcablty of the “sign-change” argument III perturbation ysed. It IS shown that the arguments leadmg to such a conch&ton are not entirelyJusttiisd_
1. Introduction
Eo@)=
In perturbation problems convergence lfficulties are often encountered and m those sltuatlons physzcal arguments are frequently mvoked to justify the observed behaviour. Of these, the rather well-known one is the “sign-change” argument [ 1 ,z]. It starts with the basx
prermse
that
rf a perturbation
senes
converges
for tlhl, the couplingparameter associated wth the perturbing harrultonian, convergence of the same se-
theory arc critically anal-
1 +3X/4-~21X~/16+liIX3/21
f...,
to more realistic sltuatlons, e.g. the Stark effect case where only metastable states correspond to the hamiltonian H@)=_5724-1 and one
_k,
rarely
goes beyond
second-order
@oIarizabil-
ity calculations)
to note that the associated (ground state) energy senes
ries for -1X( is guaranteed by the existence of a circle of convergence with #zotz-zeroradnts. But if It so happens that for -IX1 there exist no bound-state solutions to the corresponding S&r&linger equatron, it is unhkely that the energy perturbation series, E(A) would converge to a well-defmed pomt (value). Hence, the series will have a zero ra&us of convergence for either sign of 1x1,assunzing that perturbation theory must converge, if at all, to some physkaliJJ meaningful region. Essentially, the same argument was used by Dyson [3] (changmg e2 + -e2) to trace the ongin of the fanuhar &vergence problem in perturbationtheoretical approach to quantum electrodynamics. In fact, the sign-change argument works well m a wde variety of problems startmg from the “mnocent-
Recently, Sunon [7] cited a “simple’* Coulomb problem as a counter example exposing the vulnerabiiity of the mherent assumption of this “folk theorem” which, at the same tune, amounts to refuting Dyson’s celebrated argument also. More recently, Killingbeck [8] found another example, a polynomial perturbatlon acting on the hydrogenic S-state hamiltonian, where also the sign-change argument apparently fails. In what follows, we analyse these examples to show that contrary to the implicit assumpfiun of these
looking”
authors
wth
test problem
of the anharmonic
oscillator.
the hamdtonian
f _..
IS even more dangerous, though it was pointed out in the hterature [6] long ago (for a recent review, see ref.
PI)-
[7.8]
(see section
2) none of their examples
represent proper perturbation problems in the regions considered (section 3).
H(x)=-@+x2+Ax4 and the correspondmg dangerous gy series [4,5]
Eo(X) = -4 - 9X?-14 - 3555X4/64
(ground state) ener-
117
Volume
76, number
2. Counter
1
CHEMICAL
PHYSICS
examples
Simon’s
counter
LETTERS
15
turbation and apply Raylergh-Schrodmger tion theory (RSPT), we get example
proceeds
with the ham.&
toman
Eo(A) =
c
$A”
November 1980 perturba-
>
m=O
H~)=-;V2_.-1
+b-l
For srmphcrty, only, so that @u =e-',
we shall consrder
EO
=_- :,
The correspondmg are
However,
= E,@l4,(X)
c I,,
=
Eo
when k-1 is considered Srmon used the equahty oz
state
-A)‘-.
have the form
e;=
e;X”‘,
0
(2)
.
since Eo(A) should m
Eo(X)=
(S) state
for the perturbed
Eo(A)=-$(l
H(h)\lro(h)
(1)
the ground
ffOoO=EOQO -
quanttttes
*o(A)=e-(l-A)r,
_
=H,+&--l
,
as a perturbatron
to Ho.
(3 and, comparing
cluded
coefficrents
of K (L = 0, 1,2,
and noting a discrepancy m the magnitudes of A(+X) and A(A) (typrcally, A = 0.00000 for X = +O.l, and .I = 0.034 17 for X = -0.1) we are led to the mescapable conclusron that for X < 0, a component of Eo(A) is nussing III the RS series, although the srgn-change argument apphed to the hamrltoruan H(X) in (7) does not suggest the invahdtty of RSPT m this problem. Krllingbeck’s fine analysts [8] shows that bound states e.xrst for -1X(, yet the wavefunction [see (7)] becomes unnormahzable, rather mystenously. Thus, here again, the sign-change argument seems to be m trouble.
._.), con-
+I,
+_;,
E;;’ (I?I 2 3) = 0,
(6)
denving certam rreru sum-rules, as by-product. Argumg further that stnce m the problem ctted perturbation theory would also converge even for A(rea.l) > 1 where the state becomes clearly mphyncal, he concluded that the mearung of the formal ergenvalue equatron (3) IS difficult to mterpret and such convergence to an “antr-bound” state contradicts the underlymg assumptron of the ngn-change argument. klhngbeck’s mteresting observation [s] was the result of “choosmg some wavefunctron fast and fmding the correspondmg hanultoruan later” [2] type of approach. Then It rs easy to see that for = e-(r+hr’)
HO)=-iv’-r-l
,
+2?,_ri-33-$,
Eo(h)=3A-$. If we consrder 118
produced within the first order), one encounters a rather disturbmg situation unme&ately. Computing the quantity A(A) [= E, (numencal) - E, (RSPT)]
that
Eo=-&
*,@)
Along with Killmgbeck, if it is nowasamcd that RSPT works for X > 0. wth all h&her order (nz > 2) corrections (!) varushmg e_xactly (for the exact energy is re-
(7) the X-dependent
part in H(x) as a per-
3. Re-investi&ion imphcit in Srmon’s argument is the assumptron of the apphcabdity of perturbation theory over the entrre range of ]A] used and the existence of a properly convergent senes even for ]A] > 1. But IS thx assumptron well-founded? We may first note that, although the perturbmg operator mvolved m (1) IS bounded, the Kate-Relhch theorem [2,9,10] s6.U demands that IAl be suffclemZy small to ensure the convergence of RSPT. One might wonder if It 1s possible at all to pro-de a reasonable magnitude of the “smallness” parameter h, based on purely physical grounds. In fact, we have m possession a “rule of thumb” [lo] whrch says that perturbation theory is hkely to work when energy shifts are small relative to the spacings of unperturbed levels (on the average). Such a consideration immediately rules out the possrbllity of letting IAl + 1 in (1). However, tis naive and qua& tative rule might not be quote appealmg. But as we shall presently see, a more precise analysis of the prob-
15 November 1980
CHEMICAL PHYSICS LETTERS
Volume 76, number 1
I
lem only worsens the sltuahon.
Let us accept that (6)
1s true
w-101ET (nz > 2) = 0. If it were so, the zeroth-order
for any 1x1.This at once forces us to conclude that the perturbed wavefunction for small coupling
wavef
wdi be
e-(’ = @o + WA ,
Jl@SPT)
where 0: is the first-order obeys the equation (HrJ - Eo)Qb + (r-1
(9) correction
- Wolr-‘I
function
90))90
that
= o ,
(10)
with
= E; = 1 .
(11)
To solve (IO), It is common practice to follow the -‘F-function techruque” [ 11,12] and tlus gives 0; = (r - $,C+) ,
(Q&)>
=0 -
= (1 -I-Xr - 3A/2)00
(12)
.
(13)
Needless to say this does not agree with the exact perturbed function in (3) so(A)
= e-(1-&)’
= e-/(1
+ Xr + . ..) _
(14)
In fact, (13) and (14) agree only when h 1s mfimtesunally small_ although. wth enerw [see (S)] , a parallel relation m the wavefunction also was expected for any A. Thus, we conclude that m Sunon’s example (1) either the set of equatrons
IS correct
whxh
demands h to satisfy
(1 +Xr-33)\/2)e-‘=e-(*-“)‘,
= e-r
9
wluch IS obviously true only m the linut IAl + 0 (and not when ]hl = 0.1). Essentially, the situation here is sinular to some kmd of “adiabatic swicritching-on”, so to say, that mamtams statlonarity- It is, on the other hand , quite likely that non-vanishing higher-order terms are contained in expansion (S), in whrch case the inapphcability of RSPT is yet to be shown. OtherLvlse, as we have pomred out, RAPT in this case is true
only in an “adrabatrc”
Usmg the ekpresnon for 0; and standard RSPT equations, it can easdy be checked that E; = -$ and E; = 0. Now from (9) and (12), we have \k,(RSPT)
+ ti2)
(15)
wtuch, as we hawe seen, IS true OIZ& m the nelghbourhood of X equal to zero so that for erther sign of IAl the states corresponrhng to H(X) are ante physlcal; (u) or his proposed “sum rules” are mcorrect and h&her-order fite ener,q corrections exist in expansion (S), in whch case It requires further scrutmy to find the hmitmg value of A (= ho) for whxh the RS senes still converges (whxh 1s quite unlikely to be greater than 1) and only then 1t.s stand agauxt the sign-change argument can be gven any credence. Regarding the harmltonian in (7), we comment that RSPT in this case does not work at all for any X
to the perturbed have
sense.
4. Conclusion
The above analysis reveals certain subtle pomts of RSPT which were somehow rmssed in earlier works and led to fallacrous sltuatlons. We hke to emphasize that an esact matching of the RAPT ener,T series truncated at fmite order with the true energy of the perturbed state does not necessarily imply that still h&her-order terms are all vanishing. The important pomt IS that a conclusion based on perturbatlontheoretical arguments is meannmgful only if the convergence of the corresponding perturbation s&es is ensured. Thus simply means that one must have a ngorous basis for treating a given problem from a perturbatlve vlewTomt_ To be wise after the event, another crucial point in thus issue, which might otherwxe be misunderstood, IS that the sign-change argument does Not ensure convergence. Its domain of apphcation lies only in predrctmg divergence. In any case, our re-mvestigntion on the doubts cast so far pomts to the conclusron that this apparently naw2 [ZJ argument still retains its full glory. W2 also happily note, in passing, that Dyson’s
argument
remams
vahd quite clearly_
References [l] [2] [3] [4]
J.B. Krieger, J. htath. Phys. 9 (1968) 432. J. Krllmgbeck, Rept. Progr. Phys. 10 (1977) 963. F-J. Dyson, Phys. Rev. 8.5 (1952) 631. C. Bender and T.T. Wu, Phys. Rev. 184 (1969) 1231.
119
Volume 76, number 1
CHEMlCALPHYSICSLETTERS
[ 5f F.T. Hioe, D. MacM~en and E.W.Montroll,whys. Rcpt. 43 (1978) 305, [ 6 ] A. Dalgamo, in: Quantum theory, Vol. 1. ed, D.R. Bates (AcademicPress, New York, 19611pp. 171-209. 171 B. Simon, Ann, Phys. NY 58 (1970) 76. 181 J. KilUngbeek,Phys.Letters 67A (1978) 13. i9j T. Kato, J. Fat. Sci. Univ. Tokyo Sec. 1 6 (1951) 14%
15 November 1980
[ 101 J-0. Hirschfelder,W. Byers Browa and S.T. Epstein, Advan.Quantum C!hem.1 (1964) 255. f 111A, D&amo and J.T. Letis, Proc. Roy. Sot. A233 (1955) 70, [I21 A. Da&no and A.J.. Stewart, Proc. Roy. Sot. A257 (1960) 534.