Document Text Contents
Page 1
Lecture Notes in
Chemistry
Edited by G. Berthier M.J.S. Dewar H. Fischer
K. Fukui G. G. Hall J. Hinze H. H. Jaffe J. Jortner
W. Kutzelnigg K. Ruedenberg J. Tomasi
53
G.A. Arteca F.M. Fernandez
E.A. Castro
Large Order Perturbation
Theory and Summation Methods
in Quantum Mechanics
Springer-Verlag
Berlin Heidelberg New York London
Paris Tokyo Hong Kong Barcelona
Page 2
Authors
G.A.Arteca
F. M. Fernandez
E.A. Castro
Divisi6n Ouimica Te6rica
Instituto de Investigaciones Fisicoquimicas
Te6ricas y Aplicadas (INIFTA)
Facultad de Ciencias Exactas
Universidad Nacional de La Plata
Sucursal 4, Casilla de Correo 16
1900 La Plata, Argentina
ISBN-13: 978-3-540-52847-0 e-ISBN-13: 978-3-642-93469-8
001: 10.1007/978-3-642-93469-8
This work is subjectto copyright. All rights are reserved, whether the whole or part of the material
is concerned, specifically the rights of translation, reprinting, re·use of illustrations, recitation,
broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication
of this publication or parts thereof is only permitted under the provisions of the German Copyright
Law of September 9, 1965, in its current version, and a coPyri9ht fee must always be paid.
Violations fall under the prosecution act of the German Copyright Law.
e Springer-Verlag Berlin Heidelberg 1990
2152/3140-543210 - Printed on acid-free paper
Page 327
317
L(l) {(K+1) E (l)E (O)K+(l_K)E (l)E (0) (K-2)_C! } E(0)2 (39.38b)
n n n n n
D' (0) (39.38c)
As an example, we consider the ground state of the quartic anharmo-
nic oscillator (K=2, n=O) and compare the numerical results obtained
from Eqs. (39.34) and (39.37). Coefficients E (0) and E (1) are well
known (Eqs. (36.20» and coefficients e (0) an~ e (1) we~e determined
n n
from the published data in Ref./17/:
1.060362090 0.362022634
Table 12.1
Quartic anharmonic oscillator ground state as a function of the
parameter A.
A Ea ) Eb ) EC )
10-5 1.00000750 1.00000681 1.00000750
10-4 1. 00007499 1.00006813 1. 00007499
10-3 1.00074893 1.00068055 1.00074869
10-2 1.00739525 1.00673406 1.00737367
10-1 1.06638663 1.06140633 1,06528550
1 2.46022754 2.44711146 2.44917407
10 2.46022754 2.44711146 2.44917407
10 2 5·00609714 4.99872897 4,99941754
103 .10.64308123 10.63950624 10.63978071
104 22.863.15714 22.86143152 22.86160887
a) Eq. (39.37)
b) Eg. (39.34)
c) Exact results /18/
(39.39)
Page 328
318
computed results are compared with "exact" ones /13/ in Table 12.1.
The agreement is quite acceptable in the whole range of A, in spite of
the simplicity of the employed expressions.
§. 40. VFM and JVIKB integrals for 10 systems with potentials without
defined parity and central field systems.
In this section the geometrial relations studied in §.39 are ex-
tended to potentials without defined parity and central field poten-
tials.
Let H = p2+V (x) the Hamiltonian corresponding to a 10 system and
Vex) a potential bounded from below:
dV > >
V(xo ) ~ (V(x) ; dx < 0 whenever (x-xo ) < 0 (40.1)
The classical momentum take values 0 ::: p(x)2 ~ E-V(Xo )' for a given
state with total energy E. For such a state, it is possible to build a
rectangle of sides 2p and q2 - q1' where
2
E-p (40.2)
under the condition of being inscribed within the phase space trajec-
tory (Fig.~2.2). The rectangular area is
S (40.3)
and has its maximum value when as/ap = O. Since E remains fixed, q1
and q2 are interdependent through p; then, it is possible to replace
the difference q2 - ql by q - qo' where q describes the p-variation
and qo is a constant(to be used in order to have a unique eigenvalue
for each quantum number). On the basis of these considerations, we
find the following functional
Page 653
643
R
Rayleigh-Schrodinger perturbation theory
and geometrical relations
and VFM
Rellich theorem
Regular perturbations
RKR method
Renormalized series
Rescaling relationship
Saddle-point method
Scaling laws
Scaling variational method
and FM
Schiff-Snyder Hamiltonian
Schwartz inequality
Screening Coulomb potential
Semiclassical
approximation
functional energy expressions
relations
Sensitivity .rules
Spin g factor
Stark
effect in hydrogen atom
resonances
Stieltjes theorem
Stirling approximation
Symanzik theorem
Thomas-Fermi potential
S
T
45
324-327
273-303
73
79
561, 632-635
330-344, 376
363-364
597-602
581-583
157-160
353-355
621
9
491
23
151-156
330-344
351-352
614
534-551
534-551
112-113
602
502
491
Page 654
Variational functional method
and Jl'lKB method
and RSPT
644
V
application to 1D Systems
application to the Zeeman effect
bounded harmonic oscillator
central field systems
finite boundary conditions
scaling laws
semiclassical behavior
systems with confining potentials
systems with DBC
translation of coordinates
Vibrational potentials of diatomic
molecules
Virial theorem
Wick ordering
Yukawa potential
Zeeman effect in hydrogen atom
w
Y
z
141-163
305-327
272-303
165-188,190-213
254-270
208-213
179-181
190-200
267-270
267-270
131-138
200-203
176-179
554-573, 632-635
19, 191-192
131-136, 605-609
490
217-249, 494-532, 613-625
Lecture Notes in
Chemistry
Edited by G. Berthier M.J.S. Dewar H. Fischer
K. Fukui G. G. Hall J. Hinze H. H. Jaffe J. Jortner
W. Kutzelnigg K. Ruedenberg J. Tomasi
53
G.A. Arteca F.M. Fernandez
E.A. Castro
Large Order Perturbation
Theory and Summation Methods
in Quantum Mechanics
Springer-Verlag
Berlin Heidelberg New York London
Paris Tokyo Hong Kong Barcelona
Page 2
Authors
G.A.Arteca
F. M. Fernandez
E.A. Castro
Divisi6n Ouimica Te6rica
Instituto de Investigaciones Fisicoquimicas
Te6ricas y Aplicadas (INIFTA)
Facultad de Ciencias Exactas
Universidad Nacional de La Plata
Sucursal 4, Casilla de Correo 16
1900 La Plata, Argentina
ISBN-13: 978-3-540-52847-0 e-ISBN-13: 978-3-642-93469-8
001: 10.1007/978-3-642-93469-8
This work is subjectto copyright. All rights are reserved, whether the whole or part of the material
is concerned, specifically the rights of translation, reprinting, re·use of illustrations, recitation,
broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication
of this publication or parts thereof is only permitted under the provisions of the German Copyright
Law of September 9, 1965, in its current version, and a coPyri9ht fee must always be paid.
Violations fall under the prosecution act of the German Copyright Law.
e Springer-Verlag Berlin Heidelberg 1990
2152/3140-543210 - Printed on acid-free paper
Page 327
317
L(l) {(K+1) E (l)E (O)K+(l_K)E (l)E (0) (K-2)_C! } E(0)2 (39.38b)
n n n n n
D' (0) (39.38c)
As an example, we consider the ground state of the quartic anharmo-
nic oscillator (K=2, n=O) and compare the numerical results obtained
from Eqs. (39.34) and (39.37). Coefficients E (0) and E (1) are well
known (Eqs. (36.20» and coefficients e (0) an~ e (1) we~e determined
n n
from the published data in Ref./17/:
1.060362090 0.362022634
Table 12.1
Quartic anharmonic oscillator ground state as a function of the
parameter A.
A Ea ) Eb ) EC )
10-5 1.00000750 1.00000681 1.00000750
10-4 1. 00007499 1.00006813 1. 00007499
10-3 1.00074893 1.00068055 1.00074869
10-2 1.00739525 1.00673406 1.00737367
10-1 1.06638663 1.06140633 1,06528550
1 2.46022754 2.44711146 2.44917407
10 2.46022754 2.44711146 2.44917407
10 2 5·00609714 4.99872897 4,99941754
103 .10.64308123 10.63950624 10.63978071
104 22.863.15714 22.86143152 22.86160887
a) Eq. (39.37)
b) Eg. (39.34)
c) Exact results /18/
(39.39)
Page 328
318
computed results are compared with "exact" ones /13/ in Table 12.1.
The agreement is quite acceptable in the whole range of A, in spite of
the simplicity of the employed expressions.
§. 40. VFM and JVIKB integrals for 10 systems with potentials without
defined parity and central field systems.
In this section the geometrial relations studied in §.39 are ex-
tended to potentials without defined parity and central field poten-
tials.
Let H = p2+V (x) the Hamiltonian corresponding to a 10 system and
Vex) a potential bounded from below:
dV > >
V(xo ) ~ (V(x) ; dx < 0 whenever (x-xo ) < 0 (40.1)
The classical momentum take values 0 ::: p(x)2 ~ E-V(Xo )' for a given
state with total energy E. For such a state, it is possible to build a
rectangle of sides 2p and q2 - q1' where
2
E-p (40.2)
under the condition of being inscribed within the phase space trajec-
tory (Fig.~2.2). The rectangular area is
S (40.3)
and has its maximum value when as/ap = O. Since E remains fixed, q1
and q2 are interdependent through p; then, it is possible to replace
the difference q2 - ql by q - qo' where q describes the p-variation
and qo is a constant(to be used in order to have a unique eigenvalue
for each quantum number). On the basis of these considerations, we
find the following functional
Page 653
643
R
Rayleigh-Schrodinger perturbation theory
and geometrical relations
and VFM
Rellich theorem
Regular perturbations
RKR method
Renormalized series
Rescaling relationship
Saddle-point method
Scaling laws
Scaling variational method
and FM
Schiff-Snyder Hamiltonian
Schwartz inequality
Screening Coulomb potential
Semiclassical
approximation
functional energy expressions
relations
Sensitivity .rules
Spin g factor
Stark
effect in hydrogen atom
resonances
Stieltjes theorem
Stirling approximation
Symanzik theorem
Thomas-Fermi potential
S
T
45
324-327
273-303
73
79
561, 632-635
330-344, 376
363-364
597-602
581-583
157-160
353-355
621
9
491
23
151-156
330-344
351-352
614
534-551
534-551
112-113
602
502
491
Page 654
Variational functional method
and Jl'lKB method
and RSPT
644
V
application to 1D Systems
application to the Zeeman effect
bounded harmonic oscillator
central field systems
finite boundary conditions
scaling laws
semiclassical behavior
systems with confining potentials
systems with DBC
translation of coordinates
Vibrational potentials of diatomic
molecules
Virial theorem
Wick ordering
Yukawa potential
Zeeman effect in hydrogen atom
w
Y
z
141-163
305-327
272-303
165-188,190-213
254-270
208-213
179-181
190-200
267-270
267-270
131-138
200-203
176-179
554-573, 632-635
19, 191-192
131-136, 605-609
490
217-249, 494-532, 613-625
