Computer Physics Communications

ELSEVIER Computer Physics Communications I15 ( 1998) 183-199

Symbolic calculations of unitary transformations in quantum dynamics

Nam-Anh Nguyen ‘, T.T. Nguyen-Dang Dhparternent de Chimie, Fucultk des Sciences et de Ginnie, Universife Luval, GIK 7P4, Qrkbec, QC, Canada

Received 28 February 1998

Abstract

The present paper deals with the development of a Mathematics package of programs for handling quantum mechanical equations involving commutators and unitary transformations. One program implements general commutators handling rules, Another evaluates a series of nested commutators representing a general unitary transformations. We apply these new symbolic routines to test-transformations, ranging from simple translations in both spatial and momentum spaces to transformations involving nonlinear generators and denoting non-inertial changes of reference frames in the phase space. We illustrate with a simple example how these tools can be used to solve formally a molecular dynamical problem. @ 1998 Published by Elsevier Science B.V.

Keywords: Commutator algebra; Unitary transformation

PROGRAM SUMMARY

fide ofprogram: NCComAlgebra

Cutulogue identifier: ADJL

Progrcrm Summary URL: http://www.cpc.cs.qub.ac.uk/cpc/summaries/ADJL

Program obtainable from: CPC Program Library, Queen’s Univer- sity of Belfast, N. Ireland, and from the authors by e-mail

Opruting systems under which the program bus been tested: Win- dows NT 4.0 and Windows 95

Programming language used: Mathematics v.3.0

’ E-mail: anh@chm.ulaval.ca 2 Available at http: //auclid.ucsd. edu/-ncalg/download. html

OOIO-465.5/98/$ - see front matter @ 1998 Published by Elsevier Science B.V. All rights reserved. PUSOOlO-4655(98)00129-5

Memory required to execute with typical data: Default Mathemat- ica requirement

Instal/ation: The package NCAlgebra 2 must be installed together with the present package following the instructions provided in an included installation guide.

No. of bytes in distributed progrum. including test data, etc.: 16891

Distribution fiwmat: uuencoded compressed tar file

Keywwrd.~: Commutator algebra, unitary transformation

Nature qf physical problem Computer algebra implementation of the definition of a commuta- tor, of commutator handling rules. Symbolic evaluation of unitary

184 N.-A. Nguyen, 7IT N~rryen-Dan~/C[)rruter Physim Communications 115 (1998) 183-I 99

transformations for use in quantum dynamical problems.

Restrictions on the complexity of the problem Unitary transformations are represented by a truncated, finite se- ries of nested commutators involving the generator. The maximum number of terms presently allowed in this series is 32. This can

be extended by the user.

Unusual features of‘ the program Uses an extension of Mathematics language offered by the pack- age NCAlgebra. This must therefore be installed as a Mathematics add-on package.

LONG WRITE-UP

1. Introduction

Theoretical research in molecular physics has been supported mainly by numerical computations, although the power of symbolic computation in this field has been illustrated on many occasions [ l-41. In addition, the recent trends in molecular dynamics and spectroscopy do point to the important role that symbolic computation may play in the future. Some of the intensive efforts which are being made in theoretical spectroscopy to simulate complex, high resolution spectra use ideas and/or approaches, such as the guided Lanczos procedure [5], or the time-dependent Bloch wave-operator approach [6], which may profit from some symbolic analysis. In this respect, it is to be noted that artificial intelligence techniques have been used in some initiatives to deal with certain aspects of the wave-operator approach [7]. In the context of intense field molecular dynamics, the need to describe the strong laser-molecule interactions non-perturbatively calls for the developments of new representations. Often, these result from unitary transformations of the laser-driven molecular system [8,9], and the design of new representations using this type of approach can benefit from some symbolic modelling. In the same context, the exploration of control scenarios, in particular those of the ‘optimal-control’ [ lo] and ‘tracking-control’ [ 1 I] types, can be assisted by symbolic programming.

In the theory of intense field/molecule interaction simple unitary transformations have been used to switch from one gauge or representation to another [ 12,131. Each of these simple transformations is relatively straight- forward to evaluate, requiring no symbolic assistance, and serves to remove a particular form of radiative couplings. The price to be paid, the introduction of a new form of couplings, generally as strong as the original ones, scarcely justifies the application of the unitary transformation itself, except in circumstances defining particular constraints on the field attributes such as its frequency or its intensity. To obtain a new representation that does offer an authentic reduction of couplings, even their complete removal in certain cases, a unitary transformation of indeterminate form is introduced [ 14-l 6,8]. Given a molecular system, the unitary transfor- mation is required to be such that an adiabatic invariant, called Lewis invariant [ 171, can be found for the transformed system. In this type of procedure, which involves a double search having as objects the unitary transformation itself and the adiabatic invariant of the transformed Hamiltonian [ 161, the operator calculus can become prohibitively complicated. This is mainly due to the tremendously complex transformation rules that one would obtain when the unitary transformation is highly nonlinear with respect to the dynamical variables of the system. Also, one may want to repeat the whole double-search procedure within an iterative approximation scheme. All these aspects point to the timeliness of an efficient symbolic package to perform various tasks needed to make the systematic handling of unitary transformations and commutators more accessible.

The present paper deals with the development of a Mathematics 3 program package, named NCComAlgebra, for handling quantum mechanical equations involving commutators and unitary transformations. When dealing with quantum mechanics using a computer algebra language, such as Mathematics or Map]e4, one inevitably encounters problems with non-commutative products of operators. In the presently unique textbook on quantum

’ MATHEMATICA is a registered trademark of Wolfram Research, Inc 4 MAPLE is a registered trademark of Waterloo Maple Software, Inc.

N.-A. Nguyen, 27: Njiuyen-DunR/C~~rnl)uter Physics Communicutrons 115 (1998) 183-199 185

mechanics [ 181 with Mathematics, one can find a number of codes augmenting the set of rules for non- commutative products. More complete rules allowing the manipulation of commutators, and the simplification of commutator-containing expressions, are needed however for the evaluation of unitary transformations. Starting with the NCAlgebra package [ 191, we develop a program to implement genera1 commutators handling rules. We then use this routine in another program which evaluates, as operator-valued operations, a series of nested commutators representing a genera1 unitary transformation. The package contains a third program which handles operator series resummation.

The paper is organised as follows: In the next section, we recall elementary facts on unitary transformations and on their use in quantum dynamics. The methodology employed and the package of programs developed presently is described in Section 3. An appendix completes this section by giving the detailed methodology used for the analytical resummation of operator series. Section 4 illustrates, on a number of simple and typical examples, the use of the new programs to evaluate commutator-involving expressions and unitary transformations in quantum dynamics. We close the paper with a summary and a number of relevant remarks in Section 5.

2. Symbolic representation of unitary transformations

In quantum mechanics, unitary transformations can help uncover new forms of Hamiltonians, or new visions of strongly interacting systems. To be useful, a unitary transformation must be such that the new vision it generates be simpler than the original one. By this, we mean that the new vision, in comparison with the old one, employs fewer basis states for instance, or yields a more direct physical insight for the problem at hand [ 201. A unitary operator is defined as one satisfying

where 6’+ is the Hermitian conjugate (the adjoint) of 0. Generally, 0 can be written in the exponential form

(rj = ,ie ) (2)

6 being a Hermitian operator, called the generator of the unitary transformation. This generator is a function of canonical positions and momenta of the system, here denoted collectively by (P, J?). In the subsequent sections, we specialise to a one-dimensional system, and (i, ~5) represents an actual pair of one-dimensional position and momentum variables.

In the case of a conservative system described by a time-independent Hamiltonian fi, the basic problem is to solve the eigenvalue problem fi IP) = E, IP).

Under the unitary transformation 0, the Hamiltonian becomes

and the eigenstates IV) are mapped into I!@) = i? ]P). Often, the original Hamiltonian fi is complicated and

difficult to deal with. The unitary operator 0 is then considered appropriate if the transformed Hamiltonian, 8, is more diagonal in some basis than the original one, i.e. I@) is described by a smaller expansion in this basis,

In the case of a system described by an explicitly time-dependent Hamiltonian such as

A(t) = l&J + Q(t) (

in which & denotes a zero-order, unperturbed molecular system, and p(t) the interaction of the molecular system with an external field’, the basic problem is to solve the time-dependent Schrodinger equation or to

5 This could for instance represent the set of b.E( I) or A(t) .fi interaction terms for a system of electronic and nuclear charges in u laser field

186 N.-A. Nguyen, T 7: Nguyen-Dang /Computer Physics Communications 115 (I 99U) 183-I 99

describe the time-evolution operator p( t, to), itself a unitary transformation satisfying

i&f( t, to) = A( t)F( t, to) . (5)

One may search to transform the original problem, Eq. (5), into a time-independent one, using a time- dependent unitary transformation

O(t) = e-im , (6)

in which the generator 6(t) is to be found such that

R = ij+AQ - &+a,0 (7)

is independent of time. The dynamics then becomes simpler because it can be expressed in terms of the stationary-state type transport of the eigenstates ]k) of k, in terms of which the transformed time-development operator is diagonal,

Requiring k to be independent of time turns out to be too restrictive. Another possibility is to require o(t) to be such that /? admits an adiabatic, or Lewis invariant i(t) [ 17,16,11],

[i&i-t [R,i],f] =o, (9)

in which case the time evolution is exactly adiabatic in the eigenbasis, { Ik, t) ] i(r) ]k, t) = ~k( t) Ik, t)}, of i(t), i.e. the transformed time-development operator is also diagonal in this basis,

o’er= Ce-;S’dr’r*(r’)/k,~)(k,tl.

k

(10)

Given either one of the two preceding objectives, there is practically no approach which can be used to systematically tailor the unitary transformation 0, i.e. the generator 6 to suit that objective. Often, one proceeds by trial and error, relying more or less on intuition and on experiences with simple prototypes, or with infinitesimal transformation6. The difficulty is twofold and appears at two levels. On the first level, given a specific form of 0 (or of e), it is generally not a simple matter to extract from it the images of the basic canonical position/momentum pairs (i,p). On the other level, a condition such that the non-adiabatic decoupling condition, Eq. (9)) requires delicate handling of commutators, their generations, their comparisons and their resummations.

To see how difficult it may be to describe correctly the images of the basic canonical position/momentum pairs (i, p) under the unitary transformation c, it suffices to recall a formal approach in which these images are obtained by solving a differential system [ 201. Consider the one-parameter class of transformations {@(a) = eiad(‘)l, E R}, of which the unitary transformation .!? is a member, l? = @((Y = l), and let the image of an operator 2 under @((Y) be denoted g(a), i.e.

8(a) = ~+(a)8~(a).

Then, the images of i and p^ can be obtained by solving

(11)

‘See for instance the constructions in Refs. [ 161

N.-A. Nguyen, ‘l’YT Ngrryen-Dang/C[~mprrrer Physics Communicutions 115 (1998) 183-199 187

&i(a) = -iiv+(a)[&F]kv(a)) ( 12)

&$(a) = -itif(a>[~,g]lv(a). (13)

One then realises that unless e is of a certain simple form, for instance a quadratic form in i and p^, or a function of one and only one of these two variables, this set of differential equations represents a nonlinear, difficult problem, the solution of which, corresponding to the operator-valued “initial conditions” i(0) = i, J?(O) = p^, is not evident.

Another approach which will be used in the symbolic package presented in the following sections consists of writing the images of i and ~5 in the forms of series of successive commutators, as described in Eq. (25) below. In both approaches, the basic problem in symbolic computation is the accurate handling of multiple commutators.

3. Methodology

Present computer algebra systems come with an operator representing non-commutative product. For instance, Maple uses &* and Mathematics ** non-commutative operators to represent non-commutative products. In addition, extensions of the existing languages such as the NCAlgebra package in Mathematics [ 191 or the Grassmann package [ 211 in Maple allow a distinction to be made between non-commutative (quantum mechanical operators) and commutative (c-numbers) objects, and already define operations such as taking the inverse or the adjoint of an operator, solving operator-valued equations, simplifying expressions which contain both types of variables, including their derivatives.

However, these packages generally suffer from not being able to manipulate commutators [A, h] between two quantum mechanical operators ff , h. In Maple, the concept of commutator is defined, and noted as c( A, B), A, B being two objects governed by the &* product, but it is difficult to simplify complex, multiply nested commutators, although some of the rules shown below do exist. In Mathematics, even the notion of commutator was not internally defined. With an explicit definition of the commutator, such as

[AJ] =AB-B/i+--+ (14)

Commutator [ a-, b-1 := a * *b - b * *a

as found in Ref. [ 181, Mathematics would just be able to logically recognise a relation such as

(15)

[A,B+e] = [AJ] + [A,C]) ( 16)

tested in the form

Commutator[a,b+c] ==Commutator[a,bl +Commutator[a,c]

as true. However, it is totally unable to exploit this relation, and other important identities, such as the relations of Eqs. ( 17), ( 18) below, to simplify complex expressions,

[d,BC] =B[A,Q + [R,B]C, (17) Lk.fm = mw,f(B) if [A, [A,&]] =0 and [B, [A,&]] =O. (18)

These relations must therefore be implemented as working rules (Mathematics’s delayed replacement rules). This is done in the NCComAlg program of our NCComAIgebra package. Note that all the programs presented here are designed as add-ons of the NCAlgebra package, and make use of the extensions of the Mathematics language offered by that package. The main NCAlgebra commands used in our programs are identified in Table 1. A complete list of the new commands defined in our programs can also be found in this table.

188

Table 1

N.-A. Nguyen, TT Nguyen-Dan~/Computeer Physics Communications 115 (1998) IRS-199

Main commands used in the examples of Section 4. Each command is outlined in terms of the action it produces, and in terms of its origin. In the description field, NC stands for Non-Commutative, UT stands for Unitary Transjiwmation, 2 denotes an arbitrary operator.

Command’s name Action Origin

NCComSimplifyl exl? 1 NCComExpand 1 exl’ I NCComExpand2[ en]? 1 NCComCollect [ exl? 1 NCComCollect2lexp 1 NCComInit NCUTransform[J?,ij Polynom2CF( srrirs. z 1 SetCommutative[n, bl SetNonCommutative[ (I, hi NCExpand[ ex,, 1 NCCollect I exp, List ] NCUnMonomial I r.rp I CommutativeQ [rx/>l

Write exp in fidly sunpli’ed jiwm Expand commutators in exp

Expand commutator.s into basic commutator.s Collect NC products into commutators Reverse the action of NCComExpand2 Initialize commutation relations PerJi,rm on2 a UT (or an anti-VT) given its generator .? Perform the resummation c$a series in a variable z

Set variables a,b to be commutative Set a,b to be non-co,n,,zutative Expund expressions into NC products

Collect NC products in terms oj vuriables in List Replace NC products x * ~rx * *... (n terms) by x^n Give True f exp is commutative and False if’ exp is non-commututive

NCComAlg NCComAlg NCComAlg NCComAlg NCComAlg NCComAlg NCUTransform Polynom2CF NCAlgebra NCAlgebra NCAlgebra NCAlgebra NCAlgebra NCAlgebra

In NCComAlg, the command NCComExpand defines the expansion rule of an elementary commutator [A, 81, i.e. Eq. (14) read from left to right, through

Com[ a-, b-1 :> a * *b - b * *a

It also defines, for two operators A and 9 whose commutator is a c-number, the useful rule

Literal[x-* *y-l :> y* *x - Com[y,xl/;CommutativeQ[Com[y,x]]

allowing one to cast a non-commutative product into a predetermined form (the normal product of annihilation and creation operators for example). The reverse operation of regrouping a * *b with -b * ca to yield the commutator Com[a, b], with the implication that this is replaced by its symbolic value whenever available, requires another rule, which is invoked by the NCComCollect command and is defined by

Literal[P-.R-.x- * *y- + Q-.R-.y- * *x-l :> (P * R)Com[x, y]

with the condition that P, Q and R are c-numbers and 9 = -P. Similarly, the distributivity property of Eq. ( 16), the identities of Eqs. ( 17), ( 18), all read from left to right,

are set as rules which are invoked by the NCComExpand2 command and are coded as follows:

Com[a-,b-+c-] :> Distribute[Com[a,b+c]] ( 19)

Com[ a-, b- * *c-l :> b * *Com[ a, c] + Com[ a, b] * *c

Com[a,f[b]] :> Com[a,bl * *f’[b]/;

(20)

CommutativeAll~[Com[a,Com[a,b]]]&&CommutativeAll~[Com[b,Com[a,b]]] (21)

The same relations read in the reverse direction (from right to left), corresponding generally to the replacements of expanded forms by more compact, regrouped forms, require the definition of another set of rules, invoked by the NCComCollect2 command. Corresponding to the expansion rule of Eq. (19) for example, is the following rule:

N.-A. Nguyen, ET N~uym-DanR/Compurer Physics Comtnunicuiions I I.5 (1998) 183-199

Literal[ P-Com[ x-, y..] + &.Com[ x-, z-1 :> Com[ x, Py + qz ] /;

CommutativetJ[P]&&CommutativeCj[Q]

while to the rule of Eq. (20) corresponds

189

(22)

Literal[P-.(y-* *Com[x-,z-] + Com[x-,y-] * *z-)1 :> PCom[x,y* *z]/;Commutativeq[P] (23)

Two more commands are defined by NCComAlg. Applied to a given expression, the command NCComSimpIify attempts to fully expand it (using NCComExpand first then NCComExpandZ), to regroup its terms as monomial noncommutative products and to cast the expanded expression into a compact final form; it is the composition of the two new commands NCComExpand and NCComExpand2 (defined by NCComAlg) with the NCUnMonomial and ExpandNonCommutativeMultiply commands of NCAlgebra and is defined by

NCComSimplify[expr-]:=expr//NCComExpand//NCComExpand2//

ExpandNonCommutativeMultiply//NCUnMonomial//NCComExpand2//NCComExpand (24)

The command NCComInit is introduced for convenience, in view of the potential use of the program in a molecular dynamics context. This command merely initializes by defaults r, p or Q, P as a canonically conjugate coordinate-momentum pair, i.e. non-commutative variables satisfying the elementary commutation rule

By extension, any indexed pair of variables of the form i [ i] , fi [ i] or & [ i] , p [ i] is recognised, upon invoking the NCComInit , as a canonically coordinate-momentum pair. It is to be emphasized that the initialization of basic commutation rules are necessary for the functioning of the unitary transform routine to be described below, although this initialization can be entered during a session by the user, when needed, without invoking the NCComInit command.

As will be illustrated by a number of examples found in the next section, the program NCComAlg by itself is a useful tool for general quantum mechanical problems involving commutators. In the context of symbolic representation of unitary transformations, this program is called as a subroutine by another program, NCUTransform, which actually seeks to evaluate the operator-valued images of observables 2 under a given unitary transformation.

In the previous section, we have mentioned one approach, the differential method, embodied in Eq. (12), to obtain the images of the canonical position/momentum pair (i,p^). Another approach to determine the image of an operator 2 is based on the following relation [ 201:

By expressing 8( cr) in this form of a series of successive, multiply nested commutators [ , ] ,I = [ [ , ] n- t , &] , we avoid the problem of solving nonlinear differential equations involving non-commutative operators. Moreover, the implementation of Eq. (25) permits the direct evaluation of the image of any operator, as 8 is not restricted to i or 8. Due to its recursive form, this relation also presents a practical advantage; namely that its implementation in a symbolic language is straightforward. The program NCUTransform implements in fact the more general relation

e-%eS = 2 + $ [J&S] + & [ [ri&?] + ; [ [ [T?,$,Q,Q +. . . ,

190 N.-A. Nguyen, L7: Nguyen-Dang/Cornl,uter Physics Communications 115 (1998) 183-199

thus permitting both unitary and non-unitary transformations to be described. When invoked through NCUTransf orm[x, s], where x represents the operator 2 in the above and s the generator 3, NCUTrans- form makes as many calls to the two commands NCComExpand and NCComExpand2 of NCComAlg (applied successively in this order), as needed to generate the successive multiply nested commutators [, I,, in the sim- plest, i.e. maximally reduced, symbolic form. In certain cases, the generator e is such that after a number nmax of iterations, subsequent commutators [ , ] n,+, vanish identically, in which case, the program will automatically terminate, and return an exact result. But more often than not, the sequence of successive commutators con- tinues indefinitely, and I is represented by an infinite power series in cy, the terms of which are generally non-commutative, operator-valued functions of (?,p^). In practice, this series must be truncated at a certain order, kmax, of LY, even though 1, lk,,,,+ I remains nonzero. In the present form of the program, this truncation is decided by using a convergence criteria. The program keeps track of all operator products generated up to the level k, and compares the coefficients of these operators with the corresponding ones computed at the level k + 1. The program considers the series to have converged at k,,, = k + 1 only when the variations in those coefficients are less than lo-’ in absolute value. An upper limit to k,,, corresponding to 32 iterations has been set in all the calculations shown below.

In the most general case where c denotes a non-linear unitary transformation, we expect the above commutator series to be highly complex and resummation into a simple, closed-form expression may be impossible. The resummation of such a series would be possible if the successive terms in the series commute with each other, which is generally not the case. To handle this situation, the terms containing commuting operators of a given type, e.g. functions of p^ only, must be collected together, giving a subseries which could be resummed analytically, using the procedure shown in the appendix. A routine, named Polynom2CF, has been written to implement this series resummation procedure. The syntax for its call is PolynomlLC[exp,var] ,exp being the series to be resummed with respect to the variable var.

4. Examples

The examples shown in the present section are organised as follows. Each of the Mathematics v.3.0 input lines is presented left-flushed, followed (when applicable) by the corresponding output line centred. To distinguish Mathematics expressions from other mathematical expressions, we have tried to keep the Mathematics native font for input/output. The Mathematics input uses either NCAlgebra commands or new commands defined in the NCComAlg and NCUTransform programs. Table 1 outlines the commands invoked in terms of their origins and of the actions they produce.

4. I. Commutator algebra

We start by testing the commutator algebra as handled by NCComAlg. First,

< < NCComAlg‘NCComAlgebra‘

loads the NCAlgebra package along with the three programs described in the previous section: NCComAlg, NCUTransform and Polynom2CF. The basic commutator rules are well obeyed, as attested by the followings: entering

Com[ a, b + c] //NCComExpand2

returns

Com[ a, b] + Com[ a, c]

while

N.-A. Nguyen, T.T Nguyen-Dan~/Computer Physics Communications 115 (1998) 183-199

Com[ a * *d, b + c] //NCComExpand2

gives

191

a**Com[d,b]+a**Com[d,c]+Com[a,b]**d+Com[a,c]*td

Now, a more complex relation, the Jacobi identity, is verified, as

Com[Com[a,b],c] +Com[Com[b,c],a]+Com[Com[c,a],b]//NCComSimplify

returns

0

After checking the functioning of NCComAlg with respect to these basic rules, we illustrate the use of this program for the derivation of commutators of the type

uul, (27)

appearing in various forms of the hypervirial theorem. To this end, define a one-dimensional Hamiltonian of the following general form:

Z? = g + ki* + (YE? + i/(r) = g + R,, , (28)

as encountered in some representation of a laser-driven system, which in field-free condition is described by the potential function Q(r). In this Hamiltonian, E represents the electric field, ai the dipole moment of the system, and ki’ a second-order response function. A specific form of Q(r) needs not be given. After declaring fi to be an operator-valued variable, while cy, E, k, M, and n are c-numbers with

] ; SetCommutat SetNonCommutative[H

we enter

ive[cy,c,k,m,n];

NCComInit

to set [F, p^] = i, and

H= (l/(2 m))(p2) +V[r] fcu E r+k r2

to produce

2

5 +kr2+r&e+V[r]

With R = QY, the following commutator is involved in the demonstration of the virial theorem:

[A, ;(F$ +/??)I = [A,?p] .

As is well known, this should give twice the kinetic energy operator plus (-i.&&,,). Entering

Com[H,r**p]//NCComSimplify

produces

-ip2 -+2ikr2+ira~+ir**V’[r]

(29)

192 N.-A. Nguyen, TT Nguyen-Dan~/Cornl?uter Physics Cmmunicarions 115 (1998) 183-199

which is indeed the expected result. With A = i2p^, one expects a term linear in p to appear in conjunction with PO2 and to arise from the commutator of A with the kinetic energy part of the Hamiltonian, while the potential part should give ( -i2&&). Indeed, entering

Com[H, r * *r * *p] //NCComSimplify

we obtain

-E2+2ikr3+ir2,e- 2ir**p2

+ i r2 * *V’[ r] m m

As a generalization of the above, we have

Com[ H, rn * *p] //NCComSimplify

producing

4.2. Unitary transformations

In this section, the program NCU’lkansform is tested. To this end, images of the basic canonical posi- tion/momentum pairs (i, p^) under various types of the generator, 6( P, p^), are determined. It is well known [ 201 that the unitary transformation exp( ia@), (Y being independent of i and p^, is a translation in the configuration space. This unitary transformation, corresponding to e = p^ in Eq. (2), displaces the position operator by -cry, while it leaves the momentum operator invariant. Entering

{NCUTransform[r,Ipa],NCUTransform[p,Ipa]}

returns the transformed pair (i( cu) , ~3 ( a) ),

{= - ff,PJ

which is the expected result. In general, if the generator is a pure function of a, then

P%p^l =o, [&,?I = -idpG’,

and the system of Eqs. ( 12) predicts

P(a) = ? - af?/j~, /5((Y) =p^.

Thus, with 8 = g2, entering

{NCUTransform[r, I p”2 a],NCUTransform[p, I p^2 a]}

we obtain

whereas with 6 = p^“, and

{NCUTransf orm[ r, I p*n cu] , NCUTransf orm[p, I p*n cy] }

we have

{r - np-l+na,p)

(30)

(31)

N.-A. Nguyen. 27: N~lryen-DanR/C~~mputer Physics Communications I15 (1998) 183-199 193

With e = i + p”, the unitary transformation denotes the time-development operator for a particle in a linear field, and

[&I =i, [&‘. i] = -2@ ) (32)

so that the system of equations ( 12) predicts

i(a) = i - 2&z - a* ) @(a) =/?+a. (33)

This is indeed obtained, as entering

{NCUTransf orm[ r, I (r + p-2) (~1, NCUTransf orm[p, I (r + p-2) (Y] ]

gives

{r-2pa-Cu2,p+a}

An interesting, nontrivial transformation is the simultaneous scalings of i and p^ generated by 6 = (it + pi) /2 = ip^ - i/2. This transformation o( cy) = exp[ iae] is called a scaling as it gives

i(a) = Fe--“, @(a!) =$e”. (34)

NCUTransform[r, I r * *p cu]

the following result is found:

2 3 4 5 6 r-rLY+~-~+z&-~+E& rff’ rcy8 rcyg -+-----

5040 40320 362880 ’

which, treated by Polynom2CF,

Polynom2CF[ %, cu]

outputs

E-“r

Similarly, we have

NCUTransf orm[ p, I r * *p (Y]

(35)

(34)

Polynom2CF[ %, a]

E” p

A more challenging case is given by the choice 6 = @.( l/i). With (i,p^) denoting the dynamical variables of a quantized laser-field, this type of seemingly improbable transformation has in fact appeared in the context of the transformation of a field-dressed molecular Hamiltonian to give a high-order adiabatic separation between the intense field and the molecule [ 8 1. Using

194 N.-A. Nguyen, L 7: Nguyen-Dang/ Computer Physic.s Communications I15 (1998) 183-l 99

the differential equations ( 12) predicts

P(a) = i* - 2ff, f(ff) /T(a) =,I?- i

With the program NCUTransform, entering

NCUTransf orm[r, I p * *(i/r) a] //Polynom2CF[#, cu]&

produces

(38)

/

r2-2cK r-----

r2

while

NCUTransform[p, I p * *(l/r) a]//NCCollect[#,pl&// Polynom2CF]#] [211,a1#[[11 I&

provides

r2-22a J----- P- r2

which are the correct results. Interestingly, entering

NCUTransf orm[ r-2, I p * *( i/r) a]

gives

2 r -2a

which is the expected, nontrivial result of Eq. (38). This is the first illustration of what has been said above, namely that the program can handle directly any operator .%(?,a). Generalizing to the case IZ? = $.( l/i”), for which Eqs. ( 12) predict

i”+‘(a) =?“+I -(n+ l)a, pl(a)=pI - ( i(a) n+l i 1 .

Inputting

NCUTransform[r^(n+ i),I p**(l/r^n) a]//NCExpand

we indeed find

4.3. Unitary transformations in molecular dynamics

Insofar as the image J?(Q) of an operator J? is concerned, no particular care need be exercised as to the possibility that parameters in the generator e are time-dependent. In fact, in all the above examples, e is independent of time and the time can at most appear in the global parameter (Y. In these cases, &fi commutes with l?+ and -ii?&l? simply reduces to d,a. In the case of an explicitly time-dependent generator, such as the following:

e = ((t)p^ + r](t)T’ + K(t) , (40)

N.-A. Nguyen, 7: 7: Nguyen-Dung/Computer Physics Communications I15 (1998) 183-I 99 195

the time-derivative, ate’, of the generator generally does not commute with (? and evaluating the term -&+a,0

needed in the construction of the dynamically transformed Hamiltonian k, Eq. (7), is not trivial. This problem can be handled by introducing t^ and it = -id, as new canonical, non-commutative variables with

[P,j$l = i, (41)

analogously to the (3,@) pair. Since i is now an operator, the parameters 5, v, and K of (? must also be defined as operators. We thus set

SetNonCommutative[t,pX,7j,l,K];

followed by

Com[t,pt]=I;Com[t,r]=O;Com[t,p]=O;Com[p~,rl=O;Com[p~,pl=O;

Once this is done, entering

NCUTransform[t,I(~[t]**p+~[t]**r+K[t])]

gives

t

as it should (t^ obviously must be invariant), while

uptu=NCUTransform[pt,I(l[t] **p-t-v[t] **r+K[t])]

gives

5’rtl * *p + $?[tl * *v[tl + r]‘[tl * *r - $‘[t] * *l[t] i- pt + K’[t]

We can now evaluate -ii?a,ir using the exact relation

-iO+d,O = iIf [f?,, il] = O+j?,il - $, .

Thus

(42)

uptu - pt

gives

!J’[tl * *p + $?[tl * *?)[tl + d[tl * *r - +t1 * *l[t] + K’[t] .

It can be verified that this is the exact result for this simple transformation. Note that with the introduction of the pair (f,p^,), we have here a first example of a multidimensional problem. In this case, the system ( 12) consists of four equations, two for the actual (P,b) pair and two for the artificially introduced (f, j$) pair. The generator being independent of J’$, fi( a) leaves t unaffected and merely displaces ~3, by a,G( F( a), j?( a), r), which should be read here as a function of the displaced i and p^, which are described by the following results:

uru=NCUTransform[r,I(l[t]**p+r][t] **r+K[t])]

r - 5[tl

and

upu=NCUTransform[p,I([[t]**p+v[t] **r+K[t])]

p + rl[tl

196 N.-A. Nguyen, T.T. NRlryen-DanR/Cornl,lrrer Physics Communications 115 (1998) 183-199

The constant terms in the above output for -ii?+@ arise from the use of these displaced spatial coordinate and momentum. We now illustrate how the above results can be used symbolically to solve a specific dynamical problem. Let us require the parameters 4’. 7, and K of tZ? to be such that

B = irfAO - ipa,fi = j& ( (43)

where

R()=;(p^2+?2), (44)

fzA,+e(t)?, (45)

define a harmonic oscillator (a molecular vibrational mode), of unit frequency forced by a pulsed laser, whose electric field is e(t). This definition is implemented as follows:

hO=1/2(p**p+r**r)

$p**p+r**r)

H=hO+e[t]**r

;(P**P+ r**r)+e[t]**r

We then implement the definition of the dynamically transformed Hamiltonian, I%, as given in Eq. (7),

k=(H/.r- > uru,p- >upu) + (uptu-pt)//NCExpand

yielding

P* *P 2

+ ip * *v[t] + 7 - $r**l[t]+e[t]**r

-e[t] * *[[t] - il[tl * *r + ki[tl * *C[tl + iv[tl I

**p+s77

+f[tl * *p-t ii’ltl * *77[tl + rl’[tl * *r - $qt1 * *(It] + K’

[tl * *rl[tl

[tl

Construct now

k-hO//NCExpand//NCComCollect//NCComExpand2

which gives

p**r][t]+p***J'[t]+r**e[t]-r**S[t]+r**77'[t]-ee[t]**I[t]

++I * *5[tl - +I * *d[tl + +I * *rl[tl + $[t1 * *[‘It] + K’[t]

or, further collecting terms,

NCCollect[%,{r,p}]//NCUnMonomial

p * *(rl[tl + S’[tl> + r * *(e[tl - 5[tl + v’[tl) - e[tl * *5[tl

-+I * *d[tl + $1 * *l’[t] + q + 77[tl2 + ‘(‘[t] 2

Ignoring the set of terms which are independent of i and p in this result, terms which can be removed by an appropriate choice of I, the condition in Eq. (43) reads

N.-A. Nguyen. T.T. Nguyen-DanK/Cotnl~uter Physics Comnunications II.5 (I 998) 183-l 99 197

{%[[1,2]] ==0,%[[2,211 ==O}

{rl[tl + I’[tl = o,e[tl - 5[tl + rl’[tl = 0)

Solving this by invoking for instance

DS~l~~~~,{~~~l,~~~l},~l

would give the final exact solution of the dynamical problem, a solution which links the time-evolution operator for the laser-driven oscillator to that of its field-free system [ 111,

5. Summary and conclusions

The above examples demonstrate the feasibility of using computer algebra to symbolically evaluate unitary transformations of quantum mechanical operators. Each of these examples is simple enough to allow an exact result to be obtained by hand (some of these can even be handled by a ‘back-of-the-envelope’ type of derivations), against which the result of the symbolic computation is to be opposed. Yet, this group of tests is sufficiently rich and general to warrant the correctness and reliability of the positively tested algorithms. The programs developed presently can now be included as modules or subroutines in a more specialized, high- level program. This may be a package which handles the search of a Lewis invariant of a dynamical system, or one which symbolically evaluates the time-evolution operator of a time-dependent system, this operator being expressed in some prefactorized form such as the form resulting from a split-operator formula of some order [ 22,231, or the form obtained by a factorization of the Wei-Norman type [ 24,111. In their present forms, the two programs NCComAlg and NCUTransform combined together can handle any system consisting of a finite number of canonically conjugate position/momentum pairs. These are internally defined by the generic notations (r[ i] , p [i] ) or (Q [ i] , P [ i] ), which are automatically recognised through an initialization using the NCComInit command. As illustrated in the example given in the last section, where the time is introduced as a dynamical variable, one can also externally initialize the commutation relation of any pair of dynamical variables, In addition, this commutation relation needs not be of the form characterizing a canonically conjugate position/momentum pair. For instance, it can denote the cyclic commutation property of angular momenta, in which case NCUTransform can evaluate the action of elementary rotations either in a real configuration space or in a spin space.

It is to be kept in mind that the attention focused here on symbolically represented unitary transformations and on their use in quantum dynamics, in particular in the context of the study of molecular excitations by intense laser fields, by no means implies that calculations of wavefunctions can be bypassed. Rather, by manipulating the time-evolution operator viewed as a unitary transformation, which may be factorized in some approximation, and/or the Hamiltonian which may become more separable in some non-inertial frame, we merely prepare the ground for more accurate and more efficient calculations of wavefunctions. These calculations presently belong to the realm of numerical computations. The present paper is intended to show how symbolic computations can assist us in the first task, the formal study of dynamical representations for strongly coupled systems.

Acknowledgements

The authors wish to thank Drs. A. Chakak and H. Abou-Rachid for stimulating discussions and for valuable comments on the present manuscript. Financial supports of this research by the Natural Science and Engineering Research Council of Canada (NSERC) and by the Quebec’s Fonds pour la Formation de Chercheurs et I’Aide 2 la Recherche (FCAR) are gratefully acknowledged.

198 N.-A. Nguyen, TT Njil/~len-DanR/C~/rnputrr Physics Cornmrmicnfions 115 (I 998) 183-199

Appendix A. Series resummation algorithm

The following algorithm, due to Roach [ 251, is used for the resummations of series of operators generated by NCUTransform. Suppose that a function y(a) can be represented in the following form of a truncated series:

,,(a> = CYd, n=o

such that the ratios y,,+i /y,, exist. Let 4 such that 2q + 1 < n,,,, and define

P,,=(n+l)+% O<n<2q+l,

(A.11

then there exists a set of parameters ai, b;, (i = 1 to q), and c independent of n in terms of which one can write

(A.3)

The function y(a) can then be expressed in terms of a hypergeometric function yFy,

.,‘(a) = YO rfq [;;;;:-:;:;;;ca]. (A.4)

Thus, the task of finding the closed form of y(a) reduces to that of determining the a;, bj, and c parameters. To this end, it suffices to find two polynomials P(n) and Q(n) both of degrees q such that

P(n) PI1 = ecn, 3

and

P(n) = &nk, k=O

(A.5)

(A.6)

(A.71

The u;, b;, and c parameters are therefore determined by the zeros of P(n) and Q(n), which are in turn governed by the values of Yk and Tk. These can be obtained by inverting the following set of 2q + 1 algebraic equations (which are linear with respect to the 2q + 1 unknowns Yk and vk):

Y

Pn = xykn” - pn 2 77kak. k=O k=I

(A-8)

Note, however, that the whole procedure presented here relies on the assumption that the ratios p,, = y,,+, /y,, exist, which implies, among other things, that y, behaves as c-numbers. In the present application where Eq. (A. 1) is to represent a series or a subseries of commutators originating from Eq. (25), this remark means that the series resummation scheme presented here is applicable only to subseries of commuting terms.

199

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