##### 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