Recurrence Relations and Closed Formulas ...

Recurrence Relations and Closed Formulas Connecting Franck-Condon Factors and Squeezed States ALEJANDRO

PALMA

Instituto de Física (BUAP), AP J-48, Puebla 72570 and Instituto Nacional de Astrofísica, Optica y Electrónica (INAOE), AP 51 Y 216, Puebla, Pue. 72000, Mexico Received 28 March 1996; revised 7 August

1996; accepted 9 September 1996

ABSTRACT: Recurrence relations and closed formulas which are common to

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Franck-Condon factors and squeezed states are presented in a unified formulation. Both concepts present a certain parallelism that is explained by formally showing their becoming particular cases of more general concepts. This treatment sets forth the mathematical structure that unifies the two concepts. © 1997 John Wiley & Sons,Inc. Int J Quant Chem 63: 229-232, 1997

Introduction

R

ecurrence relations and formulas in closed form are derived to calculate some properties that bear physical and chemical relevance. Behind the derivation of these mathematical relations, in the case of Franck-Condon factors, lie an interesting historical development. The earliest results can be traced back to the pioneering work of Hutchisson [1], later followed by those of Manneback [2] and Wagner [3], until full solution was Dedicated to the memory of [ean-Louis Calais. Contract grant sponsor: Sistema Nacional de Investigadores (SNISEP, Mexico), Contract grant sponsor: CONACyT. Contract grant sponsor: 2110E9303. © 1997 John Wiley & Sons, Ine.

attained by Ansbacher [4], who resorted to analytical methods. Shortly after Ansbacher's work was published, another line of research on this subject was addressed by Koide [5]. He made the first attempt to obtain Ansbacher's relation by means of second quantization methods. Apparently unaware of the latter, Katriel and Adam [6] undertook the same problem 10 years later, reporting closed formulas that were different from those obtained by Koide. Nishikawa [7] gave general solutions based on properties of coherent states. Palma and Morales [8] obtained Ansbacher's recurrence relations and closed formulas by using complex variable and second quantization techniques. This work led to the calculation of properties possessing a more general structure than those found in Franck-Condon factors [9, 10]. Later, an

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PALMA important step on this line was reached when squeezed states or the so-called two-photon coherent states [11] were incorporated into a unified formulation [12, 13].

From this standpoint, the squeezed state and the Franck-Condon factors thus become unified in one concept. The similarities between these two quantities were previously pointed out by Wheeler [15].

The Nonabelian Two-dimensional Algebra

Overlap Recurrence Relations

Although the six operators: a+2, a2, aí a, «', a, and 1, are closed under the commutator, it can be shown that the operators associated with the calculation of the Franck-Condon integral spans a twodimensional algebra. To this end, it is convenient to consider the mapping between two harmonic oscillator states where each is supported by a different well. By following a notation similar to that given in [14], one can write (1)

1m) = &,Im), where

&' =

e-y(d/dx)e1n

¡3x(d/dx),

and 'Y and

f3 are

the spectroscopic constants defined in [8]. The operator &' consists of a translation operator times a scaling (or distortion) operator. The operator &' can be written in terms of the boson operators using the well-known relations

d -

=

dx

1 -(a ti

- a+).

(2)

By defining ft = x[ d/(dx)] and Q = x, a nonAbelian, two-dimensional algebra is spanned, i.e.:

[ft,Q]=Q.

(3)

On the other hand, squeezed states are defined through two operators acting on the vacuum:

From the above discussion, it follows that recurrence relations which are valid for squeezing as well as for Franck-Condon factors can be derived. We set out by considering the operator

The result of operating &' on the vacuum state leads to the well-known definition of squeezed states, whereas its operation by making z = +ln f3 and a = 'Y/ti results into a Franck-Condon sta te. The operator &' connects the creation and annihilation operators of the two sets of states:

These are very important expressions, which lead directly to recurrence relations. Indeed, there is a correspondence between the algebraic expressions of the two sets of operators and the associated recurrence relations. Explicitly, these read z* A+= -A8tanhlzl

+ a+sechlzlz

a

=

Asechl z]

+ a+ 8tanhlzl

+a

a*sechlzl

(8)

z - a*8tanhlzl. (9)

la, z ) where

D(a)5(z)10),

(4)

and 5(z) = It is worth noting that the Franck-Condon would seemingly become a particular case of squeezing if z = - In f3 and a = 'Y/ti. However, this, in fact, is not the case, since the Franck-Condon overlap is defined between two arbitrary states In) and 1m), whereas the corresponding overlap related to squeezed states is the one of the operator D(a)5(z) amid vacuum states. Bearing this in mind, we consider a more general matrix element out of which the Franck-Condon factors and the squeezed overlap become particular cases. Such a matrix element is 2

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