Various theorems in the theory of spinless particle scattering by a real central potential are derived directly from the Kohn variational principle

Epstein, S. T.

English

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ltlf*TITHHEORET1CAL CHEMISTRY INS
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THEOREMS SATISFIED BY VARIATIONAL WAVE FUNCTIONS IN SCATTERING THEORY
by
Saul T. Epstein
Theoretical Chemistry Institute and Physics Department
University of Wisconsin
Madison, Wisconsin 53706
*
Supported by National Aeronautics and Space Administration Grant
NGL 50-002-001 and the Department of Physics.
I. INTRODUCTION
This report is a revision of the secorid.-ifjart of an earlier 'report.
Our interest in the subject was reawakened by reading the paper of
2Heaton and Moiseiwitsch which, among other things, led us to the reali-
zation that many of our earlier results were derived under unnecessarily
severe restrictions.
Our goal is to derive various theorems in the theory of scattering
of a spin-less particle by a real central potential, directly from the
4
Kohn variation principle. Such an approach provides a unified view of
the theorems and also, as a by product, yields sufficient conditions
under which an optimal variational wavefunction will satisfy analogous
theorems.
II. THE KOHN VARIATION PRINCIPLE4
Some Definitions :
Let Yo be a trial function satisfying the asymptotic condition
-S L /^ &-*-- AST) ±\ Q* GL«— J5LTT
where \ is independent of Then the Kohn variation
principle states that
(5)
where
and where, from (4) the variations of
condition
satisfy the asymptotic
In A moment we will derive the Kohn variation principle. That is we will
show that if the QTjj. are otherwise unrestricted then (5) implies
and is implied by the Schrodinger equation
d E- (8)
Therefore since, when (8) is -satisfied, we have . -s, )^  we see
that provide a variational approximation to the true
w
^
ere
 ^JL ^-s t^16 phase shift).
Written out more explicitly (5) is evidently
o =
(9)
Further by an integration by parts, and by use of (1) and (6) it is easy
to show that
0 =
Thus we may write (9 ) in the same form as the familiar bound state
8
variation principle, namely
0= (§vv,. o~ tt,^-\ JL. rn=-«N v.v «J ^ do)
in which form it is clear that with S*"^ ^ arbitrary, the Kohn varia-
tion principle implies and is implied by the Schrodinger equation.
Finally we make a simplification - clearly thg£gjs no need to
introduce complex wave functions, so we won't do so. /dince with M^ j,
real the two terms in (10) are equal to one another, we can replace (9 )
and do) by
0-
and
0 •=• C£%} Ce-^c) ^ (12)
The^ e. together with (6), the asymptotic conditions (4) and (7), and the
*v(l) - (3) are then our basic equations.
III. THE GENERALIZED HELLMANN-FEYNMAN THEOREM
Let 1U. be a real parameter in V . Then diff erentrating (6)
with respect to f»* we find
*t -f-
We now note,that
**jL*
satisfies (7) with
whence we see'from (ll) and (12) that the sum of the last three terms in
(13) vanishes to that
which by analogy with the bound state case, we call the generalized
Hellmann-Feynman theorem. For exact wave functions it is a well known
g
result. In a variation calculation (14) will be a possible variation of
^. , and hence (16) vrill be satisfied by the optimal trial function,
*-*•-
provided that the set of trial functions is independent of /U.
IV. INTEGRAL HELLMANN-FEYNMAN THEOREM
Let ^V be an optimal trial function for a potential V
Then since
we see that if o is a real constant then
is of the .form (7) with
Thus we can choose,
whence we find from (11) , using (17) and (6) that
or
then we find from (12) [i.e., (tt) with '*s/everywhere]
0 -a
 (
We now note that if we choose, as we may
(20)
(21)
Thus we can write (19) as
whence from (6) we have finally
(22)
which,for exact wave functions, is a well-known result. Also it.is
clearly the continuum analogue of the so-called integral Hellmann-Feynman
12Theorem for bound states. ,
In a variation calculation (18) and (20) will be possible varietiOYii
of *A% an^ i\t, respectively, and hence (22) will be satisfied
the optimal trial functions provided that . \^_ — y and Wg,"^
any given function such that
13
are chosen from a common linear space of trial functions.
V. A SIMILAR THEOREM
The theorem of the preceding section describes what happens .if we
change "V at \si^A **^« . Now we investigate the effects
of changing JL at fi*«JI VI . First let J^ "> Ji be of
the form
j£~ JL f «*AI :
where /v\ is an integer. Then we can write
*T* *. JL
whence
(24)
satisfies (7) with
(25)
From (11) and (6) we then find, after some rearranging
0 =
Further from (23) we see that we may
(27)
which, when inserted into (12) yields
Thus we can write (26) as
<»>
which ,. is a known theorem.14
_
For the case
 a\JL —
Jfo except that now we can write
we proceed in a similar
whence we can choose
with
&"V^ CVi — >-,e) £>& ' (32)
and also we can use
(33)
The result is then
" * ~~ l-'U^ .i. ' ^- ' —<e} ~V7<_ ~ *7«.j *fy) ' oU OX^L
14
which *&• again for exact wave functions is a known result.
In a variation calculation (24) and (27) will be possible variations
of ^ and ^ y and therefore (29) will be satisfied if
^k _^ ^ C and « x^ ^.^ C , Q/vHv X defined as in the previous section,
are chosen from a common linear space of trial functions. Similarly one
will have (34) if T^fc,— "X anQ" Hi.'4-/£ are chosen from a common
linear space.
Next we consider the
Here we can write
(35): *-
and
_ ,
 (36)
~-\>
then we see that
r C (37)
satisfies (7) w>"Ht
*> ~^ } ££- (38)
and also we see that
(39)
satisfies (7)'. Proceeding in the standard way one then findsArather
messy result ^
0- (40)
which, in case the Schrodinger equation is satisfied on av.erage i.e., so
that CvArJL) Ce
14
reduces to the form of the known result for exact functions. However
of itself (40) does not seem especially interesting. Also" it is not at
10
all clear to us how to ensure that (37) and (39) are possible variations
so we will not pursue the matter further. The case Jl-y-JH- ^ /^ 4 "2
yields similar results. In Section VIII we will derive rather nicer
theorems for these cases.
VI. THE VIRIAL THEOREM
We now observe, essentially following Robinson and Hirschfelder
that
dip-
Thus
& w Mi, (42) \
satisfies (7) with
(43)
Inserting these results in (11) and (12) we therefore have
'-) (&- lit)\A/ ^K-) (44)
and
> ^" *-' (45)
.Since ._"_ L^£—\U-) Hlj,— vanishes asymptotically-one-may freely inte-
grate -w* parts in (45) in order to transfer the */»£* to the (tulvt
<f *
11
Also one may transfer the to the rigfit by noting
that dif ferentations of (6) yields
Then using these results one finds, after a bit of rearranging, that
(45) can be written as
0,
Subtracting this from (44) then yields
0
 - -
(48)
which become^j on evaluating the commutator,
0= - J
dk
which for exact wave functions is the virial theorem (Note that in
(49) V can also be replaced by O~ ) •
In a variation calculation one can ensure that (42) is a possible
variation of H^ by introducing a variation (scaling) parameter
as follows? Let
conditions. Then one readily sees that
satisfy the asymptotic
(50)
12
will also satisfy the conditions, and further one sees that
$& (51)
frfe
where . -.. 0 vi , as desired. The triai^fitractions used in the^ vi
first of references 2 are (aside from a difference ingnotation ) precisely
of the class (50) . More generally if l<^ times the set of trial
functions is invariant to fc.-*^ .^  fcl~?> |=- then (49) will be sat-
isfied.
VII. HYPERVIRIAL THEOREMS
See the second of References 1.
VIII. GENERALIZED TIETZ THEOREMS
Returning to the case
which we considered in Sec. V we now note from (35) that we can make a
Otl
rather simp lei" connection between <4k/ an<* TJ » namely
(52)
£*>•<. »" " " *"'
so that
(53)
satisfies (7) with' " "
13
(54)
Thus we find from (11) that
O * V _ £>* 1 V 2j? 6% Ce-HU ^j (55)
fc-
However from (36) we also have
<5W
SO
^
yj •—» I *-* *—
r1"
 K\ «l > / (57)
satisfies (7)' whence (12)' yields
or, integrating by parts
(59)
Multiplying this by *2,nAA. and adding to (55) then yields the result
<«).
Unhappily it is not clear to us how one could arrange for (;60) to be
satisfied in a variation calculation, i.e., for (53) and (57) to be possible
variations .
14
Equation (60) is for exact wave functions, in fact essentially the
same as results given already by Tietz (in the case ^  -s-^ M )
18
more generally by Fradkin and Calogero. To see this one first notes that
in carrying out the operations in the integrand of (60) one encounters
derivatives of ^ \ > . However thefe can be eliminated as follows:
Evidently
(61)
satisfy (7) and (71) respectively with
.
 (62)
From (11) and (12') then we find
0- C^ S ClS-HO .^^ j (63)
(64)
which by subtraction yields
(65)
Carrying out the operations in the integrands of (60) and (65) and com-
bining theWft.yields
7 (66)
15
which, for exact wave functions, is the result given by Fradkin ,a:hd
18
Calogero. The case
can be dx&ytw-^ ) in a similar way and yields the result given in Reference
18 but with tL>Vs instead of y '.s again. For £-, J? ->-ty--*v and
^2— the straightforward results are more complicated (involving
both. *)> ^ and Il^ Tl^  as in (40)) and presumably even less inter-
esting (to repeat, we really don't know how, in a practical way, to
guarantee any of the. /results of this section variationally) so we will
not go into details.
16
FOOTNOTES AND REFERENCES
1. WIS-TCI-216, January 1967. This report, in turn was an- extension
of S. T. Epstein and P. D. Robinson, Phys. Rev. 129, 1396 (1963).
2. M. Heaton and B. L. Meiseiwitsch, J. Phys. B _4, 322 (1971). See
also 0*T>» 6-0 McWhirter and B. L. Moiseiwitsch, J. Phys. B j>,
1439 (1972).
3. Namely that, in the notation Ma' will be using, (4
CB -Hfe9 tyi') =- O > etc> ~ something which is rarely true
in variational calculations' based on the Kohn variation principle.
4. W. Kohn, Phys. Rev. J74, 1763 (1948).
5. No doubt also our methods can be extended to other situations -
theorems satisfied by the full scattering amplitude (we will deal
'!
only with individual partial waves), electron-atom collisions etc.
See Epstein and Robinson, Reference If also Reference 2$ and also
M. B. McElroy and J. 0. Hirschf elder, Phys. .Rev. 131 , 1589 (1963).
6. In the References 2 a slightly different notation is used. Their
ti is K times our Hj_ , their \f is K times our
A£, and their Ay is K times our
7. To derive (9) one also needs
0
We will assume that this is true without further comment.
8. . This form is of interest in connection with the work of K. R. Brownstein,
and W. A. McKinley, Phys. Rev. 170, 1255 (1968). See also H. J. Kolker,
J. Chem. Phys. _53, 4697 (1970).
17
9. W. A. McKinley and J. H. Macek, Phys. Lett. 10, 210 (1964)^  and
R. Sugar and R. Blankenbecler, Phys. Rev. 136, B472 (1964). See
also W., E. Smith, N. dm. 39, 216 (1965).
10. A. C. Hurley, Proc. Roy. Soc. A22-.6,, 1179 (1959).
11. See for example A. Messiah, "Quantums'Mechanics" Vol. I (North Holland
Publ. .Co., Amsterdam, 1961) page 404.
12. R. G. Parr, J. Chem. Phys. 40, 3726 (1964).
13. S. T. Epstein, A. C. Hurley, R. E. Wyatt, R. G. Parr, J. Chem. Phys.
47, 1275 (1967).
14. See for example D. M. Fradkin and F. Calogero, Nuc. Phys. 75, 475
(1966) Equation (14). Note that their M.^  is £*C#o §j^
times our *T^
15. P. D. Robinson and J. 0. Hirschfelder, Phys. Rev. 129, 1391 (1963).
16. Y. N. Demkov, Doklady Akad. Nauk. (USSR) 138, 86 (1961).
17. T. Tietz, Nuclear Physics 44, 633 (1963). This theorem is also
equivalent to Gerjuoy's momentum transfer cross-section theorem
(see E. Gerjuoy, J. Math. Phys. 6_, 993 (1965), especially Equation (23))
18. Reference 14, Eqn. (15).


Theorems satisfied by variational wave functions in scattering theory

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