Fortschritte der Physik / Progress of Physics: Volume 31, Number 3 [Reprint 2021 ed.] 9783112497401, 9783112497395

Citation preview

FORTSCHRITTE DER PHYSIK PROGRESS OF PHYSICS

Volume 31 1983 Number 3

Board of Editors F. Kaschluhn A. Lösche R. Rompe

Editor-in-Chief F. Kaschluhn

Advisory Board

A. M. Baldin, Dubna J . Fischer, Prague G. Höhler, Karlsruhe K. Lanius, Berlin J . Lopuszanski, Wroclaw A. Salam, Trieste D. V. Shirkov, Dubna I. Todorov, Sofia J . Zinn-Justin, Saclay

CONTENTS: K . BOSE a n d D . N . TRIPATHY

Studies of Coupled Anharmonic Oscillator Problem Using Coherent States and Path Integral Approaches

131 — 163

H . STÒCKEL

Linear and Nonlinear Generalizations of Onsager's Reciprocity Relations. Treatment of an Example of Chemical Reaction Kinetics

165—184

AKADEMIE-VERLAG • BERLIN ISSN 0015-8208

Fortschr. Physik., Berlin 31 (1983) 3, 131-186

EVP 1 0 , - M

Instructions to Authors 1. Only papers not published and not submitted for publication elsewhere will be accepted. 2. Manuscripts should be submitted in English, with an abstract in English. Two copies are desired. 3. Manuscripts should be no less than 30 and preferably no more than about 100 paged in length. 4. All manuscripts should be typewritten on one side only, double-spaced and with a margin 4 cm wide. Manuscript sheets should be numerated consecutively from " 1 " onwards. Footnotes should be avoided. 5. The titel of the paper should be followed by the authors name (with first name abbreviated), by the institution and its address from which the manuscript originates. 6. Figures and tables should be restricted to the minimum needed to clarify the text. They should be numbered consecutively and must be referred to in the text and on the margin. Figures and tables should be added to the manuscript on separate, consecutively numerated sheets. The tables should have a headline. Legends of figures should be submitted on a separate sheet. All figures should bear the author's name and number of figure overleaf. Photographs for half-tone reproduction should be in the form of highly glazed prints. Line drawings should be in a form suitable for reproduction. The lettering should be sufficiently large and bold to permit reduction. If requested, original drawings and photographs will be returned to the author upon publication of the paper. 7. Formulae should not be written to small and not with pencil. Separate lines for formulae are desirable. Si-units should be used. Letters in formulae are normally printed in italics, numbers in ordinary upright typeface. Underlining to denote special typefaces should be done in accordance with the following code: Italics: wavy underlined with pencil (only necessary for type written symbols in the text) Boldface italics (vectors): wavy underlined twice Upright letters (all abbreviations like all units (cm, g, ...), all elements and particles (H, He, ..., n, p, ...), elementary mathematical functions like Re, Im, sin, cos, exp, ...): green underlined Greek letters: red underlined Boldface Greek letters: red underlined twice Upright Greek letters (symbols of elementary particles): red and green underlined Large letters: underlined with pencil twice Small letters: overlined with pencil twice (This will be necessary for handwritten letters that do not differ in shape, as c C, k K, o O, p P, s S, u U, v V, w W, x X, y Y, z Z). I t will help the printer if the position of subscripts and superscripts is marked with pencil in the following way: at, 6», -MiJ, My, PPnA* Please differentiate between following symbols a, a ; a, a , oc; 00 is solved exactly without using W K B techniques. The exact solution agrees with the W K B solutions to zeroth order. This agreement is most impressive and testifies to the accuracy and utility of W K B method. SIMON [10] employed the theory of singular perturbations in Hilbert space to prove rigorously many of the properties conjectured by B E N D E R and W U and generally to present a rigorous study of the analyticity properties of the energy eigenvalue in the complex /t-plane. He has also shown that the abovesaid perturbation series is asymptotic in an open domain around the origin. In particular, it has been shown in all above works that the energy levels of Ax4 anharmonic oscillator satisfy the condition / 1 \ 4/3 1 / 3 En{X) ~ in + y) ¿

for complex A.

Numerical analysis starting from the perturbation series using Pade and Borel Pade [11] sums confirmed the branch point nature of the essential singularity. L O E F F E L et al. have also demonstrated the use of Pade approximants in extracting from the Rayleigh Schrodinger series a converging sequence of approximation for the eigenvalue. A more practical calculational method is that of B I S W A S et al. [12], who developed a nonperturbative approach for the calculation of eigenvalues of Xx2m type oscillators, using the infinite Hill determinant. The method being nonperturbative, allowed the calculation of energy eigenvalues for arbitrary values of the coupling to confirm the asymptotic nature of perturbation series and singularity of the coupling constant at the origin. In the same serie3 of work B I S W A S et al. [ 1 3 ] extended the work of anharmonic oscillator of the type ax2 + bx4 + ex9 using the theory of continued fraction and introducing a new set of coupling constant. They wrote Green's function in terms of an infinite continued fraction of the Stieltjes type, whose poles give the energy eigenvalue. I t has been proved that continued fraction converges where the corresponding perturbation series in the dominant coupling diverges. They also obtained the analytic structure of the Green's function in the complex plane of coupling constant. Following this approach H I O E and M O N T R O L L [14] employed another recursion approach wherein an iterative scheme is set up for the n-the eigenvalue. This scheme is based on the straightforward expansion of the truncated characteristic determinant of the Hamiltonian matrix (in a suitably chosen harmonic oscillator basis), the truncation being around the ( n , n) element. More recently G R A F F I and G R E E C H I [15] have developed continued fraction approximants to the Hill determinants arising in the study of Xxim type oscillators. Besides these methods, which in practice, can be carried to any accuracy, there is also a formula due to M A T H E W S [16] for quick estimation of energy eigenvalue. Mathews very recently deve-

Coupled Anharmonic Oscillator Problem

137

loped a new iterative diagonalization method which enables several eigenvalues to be calculated simultaneously with little more labour than for a single eigenvalue. An approximate analytical formula for the energylevels is also presented b y M A T H E W S et al. I n another approach K . B A N N E R J I [17] recently developed a very simple, b u t of high degree of accuracy, method based on construction of moments for obtaining the energy eigenvalues of a anharmonic oscillator system nonperturbatively. He showed t h a t the multiple moments for any transition between anharmonic oscillator eigenstates satisfy an exact linear recurrence relation. The result is valid for any polynomial potential. Still another powerful method for calculating the large orders of perturbation theory in quantum field theories has been discussed by L I P A T O V [18]. E . B R E Z I N et al. [19] have shown t h a t the same method applied to anharmonic oscillators in quantum mechanics allows one to rederive and generalise results previously obtained by B E N D E R a n d W u . He also verified and. generalised Lipatov's results to the case of an internal 0(n) symmetry. Using the Lipatov approach ATJBERSON and M E N N E S S I E R [20] studied the large q u a n t u m number behaviour of the nth coefficient of energy value i.e. En in the perturbation expansion for the ground state energy E(g) of the anharmonic oscillator. Their calculations do not reply on the analyticity property of E(g) in the coupling constant g. They compute the 1 jn correction to the leading term and recover the result of B E N D E R and W u . A general renormalization procedure is conjectured, which has been checked upto the order 1/n 3 . [21] studied the quartic anharmonic oscillator. Generalised Bose operators are used to reduce the general anharmonic oscillator into a harmonic oscillator with a frequency which depends on the strength of the anharmonicity (A) and on the q u a n t u m number. The analytic properties of the energy levels as functions of A are very transparently exhibited. A simple method of higher order calculation in perturbation theory (PT) for an arbitrary gv(r) anharmonic oscillator is presented by DOGLOV and P O P O V [22]. The method is based on using the Riccati equation instead of the Schrodinger one. The structure of the perturbation series for the energy eigenvalues is investigated. If v(r) exp (brv) as r -> oo, then the perturbation series is not Borel summable for 0 < v < 2. If v 2 the P T expansion in powers of g does not exist. A new convergent version of PT, valid for any g, 0 < g < oo is presented. Very recently E . CALICITY, S . G R A F F I and M . M A I O L I [23] studied the perturbation theory of odd anharmonic oscillators. They studied the perturbation theory for H = p2 + x2 + fix2n+1 (n = 1, 2 ...). I t is proved t h a t when I m /? 4= 0, H has discrete spectrum. Any eigenvalues is uniquely determined by the (divergent) Rayleigh-Schrodinger perturbatioiexpansion, and admits an analytic continuation to I m /? = 0 where it can be interpreted as a resonance of the problem. JACOB KATRIEL

II. Different Approaches Applied to the Problem We shall discuss, in this chapter, two different approaches which we used in our investigation. I n the following section 2.1. we develope the concept of coherent states, as in our major part of the investigation coherent state representation approach has been used. Section 2.2. describes the Feynman path integral approach. 2.1. Coherent State Representation A set of coherent states was first used by Schrodinger in 1926 to describe nonspreading wave packets of an oscillator. The concept of coherent states was introduced by G L A U B E R [24], who showed t h a t the use of a set of coherent states makes it possible to give a n adequate quantum description of a coherent beam of laser light.

138

S. K. B o s e and D. N. T b i p a t h y

The coherent state of a harmonic oscillator system is introduced with the help of annihilation and creation operators. The annihilation and creation operators a and a+ are defined by a=^={x

+ ip),

a+=-^={x-ip),

(2.1)

where x and p are the position and momentum variables respectively. The operators satisfy the well kown commutation relation [a, a+\ — 1.

(2.2)

These operators, with the number operator (N = a+a), satisfy the following commutation relations [a, N]=

a,

[N, a+] = a+

(2.3)

and have the property a+ | n) =|/w + 1 \n +

1),

(2.4)

a |w) = "/m \n — 1); where \n) is the 'n quanta' state of the harmonic oscillator. The coherent state is characterized by a complex number —1 t

^

f

where b'j = b^ and 6f;- = (i.e., we take the coefficients b\j to be symmetrical). The x's represent displacements, the p's momenta, co's frequencies, m^ the masses, a the coupling parameter, a { is and 6 y 's are the relatives trengths of anharmonicities and the strengths of coupling of oscillators respectively. With the similar Hamiltonian given in (3.4) with more degrees of freedom, HIOE et al. [41] tried to study the molecular dynamics of complex nonlinear system (of which these are among the simplest examples) or in nonlinear field theories. The understanding of model (3.4) is a prerequisite for more realistic and complex systems. In above Hamiltonian (3.4) the potential energy term of the system is given by 2M

1

V = Z TT »^¡WiXi2 T+ »=i 2

(X

2m

4p

¡=1

—

1 + -

2m 2p-l £

"

1

i,j = l s = 1

(3.5)

the last term represents the interaction of two oscillators and first two terms represent the individual motion of the oscillators. The solution of anharmonic oscillator can be obtained by systematic use of coherent states of the oscillators in describing their motion. Hence, at first we shall find out the equation of motion for the system of coupled oscillators with the help of Lagrangian equation of motion. The Lagrangian equation of motion is expressed as

dt V8± ) ~ ~dx ~

L = Lagrangian for the system. In our system '1 L

= T - V =

^ - x ? -

2

'

- mpnW 4

'

i)t V (