reference - International Atomic Energy Agency

REFERENCE INTERNATIONAL CENTRE FOR

THEORETICAL PHYSICS

CANONICAL TRANSFORMATIONS IN QUANTUM MECHANICS

A.Y. Shiekh

INTERNATIONAL ATOMIC ENERGY AGENCY

UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION

1987 MIRAMARE-TRIESTE

Quantum Canonical Transformations

IC/67/39

SI Path Integral flpproach to Quantum mechanics International Atomic Energy Agency and United Nations Educational Scientific and Cultural Organization

Classical mechanics is a description of nature using commuting variables that is well formulated by the Hamilton least action principle But

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

classical mechanics is not a complete model of nature; leaving several features unexplained. Some of the discrepancies in the classical theory have been overcome in the quantum generalization. Dirac 11925, 1958] proposed how a quantum theory might be induced

CANONICAL TRANSFORMATIONS II QUANTUM MECHANICS * (a canonically invariant path integral)

from the classical theory. He postulated that the classical commuting variables become non-commuting operators and suggested how the quantum dynamics be obtained from the classical Hamilton equations of

A.Y, Shiekh ** International Centre for Theoretical Physics, Trieste, Italy.

motion. Since the quantum theory is the more fundamental theory (the classical predictions following in the h—»O limit), it might be argued that this should be the starting point. Dirac's scheme, however, has the

ABSTRACT

advantage of starting from a well understood theory. In this scheme, to each classical system corresponds many quantum generalizations, (each

A particular form of the path integral is presented that allows the implementation of general canonical transformations in

yielding the same classical predictions forft—»0).Dirac's method however

quantum mechanics.

is ambiguous in not generating one unique member of the quantum generalizations. Some further specification (such as normal ordering) is required to completely specify the quantum theory. This might not seem a MIRAMARE - TRIESTE

disadvantage; but due to this ambiguity, classical techniques such as the

February 1987

use of canonical transformations cannot be used to directly induce a quantum counterpart. This results in the loss of powerful techniques such as the Hamilton-Jacobi approach so often employed In classical mechanics [Goldstein, 1980]. This work is a preliminary investigation of the alternative path integral

* **

To be submitted for publication. Permanent address: Blaekett Laboratory, Imperial College, London SW7 2BZ, England.

Page I

Quantum Canonical

Transformations

Quantum Canonical

quantization technique of Feynman, in an attempt to overcome some of

Transformations

§11 Deriving the Path Integral

these problems. The path integral technique, although equivalent, differs significantly from the usual operator formulation of quantum mechanics, in that it employs commuting (or badly called 'classical') variables. Although these variables commute, the theory being equivalent to the operator formalism must contain 'operator ordering1 within its structure. Understanding just how this occurs is crucial in the use of the path integral and is reviewed below.

Although no formalism of quantum mechanics is more fundamental than any other, each is supposed to be equivalent and so one should be derivable from another. Assuming a knowledge of traditional quantum mechanics one may deduce the path integral formalism. Starting from the position to position amplitude for Heisenberg eigenstates:

(This may be generalized to the transition between any two states.) Inserting position resolutions of unity: 1

leads to:

Recall that for Heisenberg eigenstates:

H being a Hermitean operator. The use of mid-point time is crucial for this evolution to be accurate to order At, as it must. So:

exp[-iH(q,p,k-i/2)At/-h] I k-n> where: At = (t h -t VN D

3

Proceeding by expanding the exponential leads to:

Page 2

Page 3



< q|p(k>> < poolqik-i) > (- i H(q(k-1 ),p(k),k-i/2)At/"h)2/2 This result generalizes, by induction, to finite m, leading to:

I l k , 1/g Further inserting momentum resolutions of unity: 1

' I l k E [ N exp[-iH(q(k-i),p(k),k-i/2)At/ti]

For the m=1 case this may be done in one of two ways, leading to alternative integrands, each therefore equivalent to order At, namely:

where: H(q,p,t) = /

! (-iH(q,p,k-i/2)At/ti) |p(k)Xpoolq(k-D> or ,... p) then f

- 1 1 , , Mdpi]>exp[F(p)] = exp[F0]/V(det(a/ic))

where F(p) = [-p'ap

+

p'p + y]

and Fo is the minimum of F(p) w.r.t. p i.e.

each and every path is then summed to yield the total end amplitude. A heuristic but very appealing argument for the classical limit is that paths around the classical contribute amplitudes that are in phase (since the

which may be seen from the fact that, since a can be chosen Hermitean, it

classical path is that with an extremum action); while those far from it

may be diagonalized by a unitary transformation (which has unit Jacobian)

contribute largely differing phases and so tend to cancel each other out. In

and the resulting Gaussian integrals trivially performed.

this way contributions from around the classical path are favoured and classical mechanics recovered in the ti-*0 limit. These arguments can be made more precise.

Identifying: a= ilAt/2mh where 1 is the NxN unit matrix, leads to:

In many cases the p integrations may be explicitly performed to yield a configuration space (Lagrange) path integral. For example if: ~H(q,p) = p2/2m +"V(q) (i.e. the non-relativistic case with a conservative potential). In this case

exp[iAt X k ., N {m/2 ((qao - qoc-n)/At)2 - V(q)/At

In performing the p integrals in: |

Transformations

Higher order differentials following from repeated application. = Lim Al ^ 0 (dfU)/dt - d2f(t)/dt2 At/2! + ... )

do dp exp[i(pAq -

from Taylor expansion (f being assumed analytic)

The 'strongest' terms in the action (such as that stemming from pq) are and obtaining the Lagrange formalism; p becomes mAq/At, so that in the 1 2

Lagrange formalism A q ~ (At) ' (c.f. p ~ ( A t r

U2

of order (At)0, stronger terms not occurring physically as they lead to

in the Hamiltonian

infinities in the At—'0 limit. Time derivatives appearing in such terms

formalism). It is in this way that the contributing class of paths are seen

should be represented accurately to order At, since we must work to this

to be stochastic (or Brownian) in nature. This behavior must be carefully

order. In this way the formula given above for the derivative is seen to be

taken into account when working to order At (as before); and is the

an inconsistent definition if used in a formalism, such as path integration,

manifestation of the path integrals sensitivity to the finite difference

that is sensitive to first order in At. Alternatively using a symmetric

4

scheme adopted in discretization. Terms, such as (Aq) /At in the

(mid-point) definition:

Lagrangian, give (in the At-»O limit), for paths smooth in p and q, no contribution. This is because finite 4

LimAl_0(f(t+At) - fu-At))/2At

p implies A q ~ A t and so

= L(m At ^ 0 (df(t)/dt + d3f(t)/dt3 (At) 2 /3! * ... )

3

( A q ) / A t ~ (At) , But such terms are finitely contributing for the dominating unsmooth paths (Aq-(At) 1 ' 2 ) and w i l l be referred to as stochastic. This dependence on stochastic paths is where operator ordering is concealed. It is here that it is see that a Hamiltonian can contribute like (Atr 1 , since it behaves like pV2m for small p, and that this is the strongest behaviour that is not divergent when the At-*o limit

This has all the error terms disappearing for the path integral in the At-*O limit, and so is a correct definition for use within the path integral. Other schemes might, in context, give no contribution, but the mid-point scheme is guaranteed not to. It might now be argued that the formal expression: =

is taken. The mid-point rule (being accurate to second order) generates no

Page 10

q < V = qa

Page 11



SIV Canonical transformations in Quantum mechanics

S(q,p)=(utBdt(pq-H(q,p,t)) can be made unambiguous if it is noted that all consistent discretization schemes lead to no stochastic terms and so the same answer in the At-»O limit. As was seen earlier, this use of mid-point expansion was not always necessary in Cartesian coordinates; but becomes essential when using a general coordinate system and is a matter approached in more detail later,

In this preliminary work canonical transformations for the path integral [Fanelli, 1976] are identified to be those of classical mechanics, and a time discretization scheme found that allows the transformation to a trivial Hamiltonian, as well as a consistent quantization of a classical system. Traditional quantization through operators [Dirac, 1958; Chernoff, 1981] does not generate a unique quantum theory. Equivalent classical systems (related by a canonical transformation) in general yield differing quantum systems [Kapoor, 1984] Also, in the operator formalism, it is not clear how to implement a canonical transformation due to the use of non-commuting variables. The ambiguities in quantizing the classical theory means that one cannot fall back on the classical theory to perform the canonical transformation. The path integral description of quantum mechanics [Feynman, 1948; Feynman and Hibbs,

! 965]

offers an

alternative method of quantization and poses a possible way out of this dilemma, since it uses commuting variables in its structure. Starting from the classical action, one can form the path integral expression:

JJ This formal expression is deceptive in that it employs commuting variables, but is supposed to be equivalent to the traditional operator formalism. The above formal expression is in fact ill defined. In order to evaluate it one can discretize it in time; but the answer is in fact dependent upon the finite difference scheme adopted. Factor ordering is Page 12

Page 13



carried within the prescription [Schulman, 1981; Hayes and Dowker, \972,

§V Canonical Transformations in the Path Integral

19731, in general, however, the prescription w i l l change under a canonical transformation [Klauder, 1980], so a quantization scheme based on a particular prescription w i l l in general generate inequivalent quantum systems from equivalent classical ones. These features have been dealt with in more detail in the introduction. However, a particular path integral has been found that is invariant under general canonical transformations, and opens the way to canonicai transformations in quantum mechanics, and so a quantum mechanical application of the Hamilton-Jacobi theory of classical mechanics [Goldstein, 1980; Schulman, 198!].

In classical mechanics a canonical transformation is one that preserves the least action principle [Goldstein,

1980}, For the path integral one

might analogously require that there be a path integral representation in the new variables (Q,P,t), if one existed in the old ones (q,p,t) Such a t r a n s f o r m a t i o n should be system independent, that

is to say, the

transformation should be canonical not only for some specific system, but for ail problems with the same degrees of freedom. The amplitude may alter under such a transformation by at most a phase factor. i.e. formally, w i t h end points (a,b) in phase space held fixed: \\ e

1

^ " H ) t u / h Dq Dp = e i(F B " F a " h \\ e" ( p ° " K)at/h

DQ DP

V H(q,Pjt)

F being an arbitrary smooth function. Assuming that a canonically invariant discretization prescription exists (just such a scheme being sought), that is to say this forma] statement becomes true for that scheme. Any other expression should be manipulated into this form w i t h the resulting OCh2) term additions to the Hamiltonian. These terms may be replaced by 'potential like 1 terms of the same effect [McLaughlin

and

Schulman, 19761, the technique for achieving this being illustrated later. It is being claimed that the quantum canonical transformation is a cleaner object when used w i t h an invariant path integral schemeSince the above equation is to be true f o r a l l Hamiltonians, the integrands must be equal. This is perhaps most easily seen by choosing Hamiltonians that are highly localized in phase space. The integrands must then be equal at the 'localization point'. By choosing Hamiltonians localized at each point, it rollows that the integrands must be equal

Page 14

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the generator of which is determined by the Hamiiton-Jacobi equation. For

everywhere. This implies that;

this purpose it is convenient to work with an alternative generating

pq - H = PQ - K • dF/dt if the end points in phase space are fixed, i.e. we should work with a

function given by:

coherent state type path integral [Klauder, 1980]. This is the same requirement as in classical mechanics [Goldstein, 1980], as welt as the condition that the Jacobian of the transformation be unity (which follows from the above [Goldstein, 1980] ).

pq - H= PQ - K+ dF/dt

from which follows, by the independence of q and P:

pq - H = PQ - K + OF/3q) Q t q + OF/aQ) qt Q + OF/dt) q Q

= OF/dP)

and by the independence of q and Q:

P = -OF/3Q)

SO:

= -QP - K * OF /dq) pt q + OF /3P) qt P + OF /dt) q p

Suppose F = F(g,Q,t); then because:

p = OF/dq).

F(q,p,t) = F(q,Q,t) + QP

qt

= OF/aq) p t

K = H + OF/dt) q p

leading to the Hamiiton-Jacobi equation: K = H + OF/dtK

H(q,(df /dq)pt,t)

+

OF / 3 t )

p

=0

F being now seen to be the generating function of the canonical

Classically one has transformed into a frame that 'tracks' the system so

transformation. One concludes from this that the quantum canonical

that it then has trivial motion (constant phase space position). The

transformations are the same as those for classical mechanics, excepting

transformation then carries the motion.

that scaling transformations are excluded. It is possible that the momenta (defined by p = OLcq,q>/dq)q) are not all independent of the coordinates. Independence, and so a Hamiltonian description, can be achieved by employing the constraint analysis of Dirac, where the constraints are moved into the Action using Lagrange multipliers [Dirac, 1964], If one could perform general canonical transformations in quantum mechanics, then one might consider emulating the Hamiiton-Jacobi philosophy of classical mechanics. In this approach [Goldstein, 1980; Schulman, 1981], rather than directly solve the equations of motion following from a given Hamiltonian H(q,p,t), a canonical transformation is implemented that renders the transformed Hamiltonian (Kamiltonian K

1/8 / MJ

At At At

+ i/e FM,QQP

AQAQAP

+

AQAQAt

AQ AQ +

AP AP

+

At At AQAP

1/8 FM,QQt 1/8 F n -

Pp

i/a FM,Q

t PP

APAP At AQ APAP

* we F M , Q t t

AQAtAt

p

AQAP At

+ 1/4 F M , Q

AQ A t

APAPAP

t/24FM,tu

t

APAt

The generating function derivatives may be converted to p,q derivatives by starting from: AQ i

P

5F = OF/dQ) nf 5Q + (dF/dq) Qt 5q * O F / 3 t ) n n 5t qt 'qo'

AP At

= -P5Q + p5q * OF/dt)

5t

AQ AQ AQ Page 22

Page 23



leading to: F, = -p i- P

opposed to q m and p j .

P

Jacobian:

= q,tP

J =
(

again applying the canonical transformation at the M mid-point: K

M = HM * ^

AP AP AP

* 3qr,JtPt1,tt-FM,t

At At At AQ AQ AP

1/4 QQPM'QP p n PP P P 0J p M'Q

p

- q r i , Q t PM. t)

AQ AQ AP AP At At

Flk-i N( '

+ J a c o b l a n

stochastic terms)

Jacob fan

Page 2 6

Page 27



exp[i(PMik)(Q(k)-Qik-i)) - K(QM(k),PM(k),k-w2) At + Action stochastic terms)/"h)

SIX Hamilton-Jacobi Transformations

Now work to replace the stochastic terms by a potential like term [NcLaughlin and Schulman, 1976],

The Hamilton-Jacob) transformation is the special case where the canonical transformed Hamiltonian CKamiltonian) given by:

V(Q M )At of

t h e same e f f e c t

( t o o r d e r At)- T h i s can only

be a c h i e v e d i f

the

stochastic terms are of order At,

K = H+

qo

is null. i.e. H(q,p,t)

So further define:

+

=0

but recall: F

F

h

j ^ e" i)- a " Lim N ^ (1 J..Jn)- Ii N-i

(dQ{ )dp

J

P=

( )/2 tti)

MJ '

Qt

so that the generator (D of the required transformation is given by: exp[i ^ - . k = |

N(

P M (k)(Q(k)-Q(k-l)) - KCQM(k),Pn(k),k-l/2) At - V

H(q,or/dq)Qt,t) Working to order At:

+

Or/at) qQ = 0

the Hamilton-Jacobi equation. It should be recalled that this Hamiltonian will in general look 'quantum mechanical' due to the adoption of h terms in

j~~[k=| NCI + Jacobian stochastic terms

+

i Action stochastic terms/1=0

weyl ordering the Hamiltonian. In this special case of a Hamilton-Jacobi transformation:

exp[i ( Pn(k)(Q(k)-Q(k-n) -

n(k),k-!«)

At }/Ts]

and

flk-i N t!

+

Jacobian stochastic terms

+

i Action stochastic terms/th)

J = exp[i [ PMM(0(k)-Q(k-

to be compared with: exp[i [ PH(k)(Q(k)-Q(lj..Jn j . ) l N-! ( d Q ( J> d p M ( J> / 2 l t t l )

AP At At In this special case of a Hamilton-Jacobi transformation K=0, so:

expti ( PM(k)(Qck)-Q(k-0) - P M 2 (k)At/2A )/f\]

H = -(dr/3t) qQ

where the stochastic terms are

= -r. t + P q . t as was shown previously.

AO AQ

This leads to the Hamilton stochastic terms canceling many of their pq

AP AP

At At pp

Jacobian

counterparts, since the Hamilton stochastic contribution:

AQ AP contribution

'/e(q M , P M , 0 Q

AQ At pp _P • n q P M' M'Qt

q

M'Q PM' t

q

M'Qt

P

M'

'

AP At

2q + q M 'QQt Pfi '* M'Qt PM' Q PP PP - 1/8 / To evaluate this object one should commute factors in the Hamiltonian

for Heisenberg eigenstates:

operator (using [q,p] = i l t i which implies [a,a'] = 1), such that a operators

lp,q,t*At> = exp[(H(p,q,t+At/2)At/-h]|p,q,t>

are shifted to the right and can be applied to the coherent state, while the

(accurate to order At), H being a Hermitean operator, so:

a 1 operators (now on the l e f t ) are also suitably applied This s i f t i n g

-

induces additional terms that carry the operator ordering information for the path integral.

P| | ( | N

The m=2 case has three such equivalent alternatives; namely

exp[-tH(p,q,k-i/2)At/ti] |p,q(k-

,q(k-nlp(k-2),q>/2

where the end points in phase space are not integrated over. Proceeding by expanding the exponential, leads to: Page 42

Page 43



p

the representation:

and the fact [q.p] =TJ"h

as weli as: eA*B = e A e B e -iA,Bl

exp[i

if [A,B] commutes with both A and B, it then follows that

pmck)(q -oo

[ ooJ

dphdqh/27ih f -oo

oo

r

D

n

D

J -oo

f ooJ

exp[-( ( x h -

dp dq -oo

oo

r

a

^

exp[i(r b -r a )/ti] exp[-(

but from the definition for the coherent state:

a"(qa-ipa))

2+

Pa2)/2t1]

where the Hamilton-Jacobi analysis yields:

- ( i c h r ^ e x p H (x - (q+ip))2 + which may be derived by developing and solving for partial differential equations in p and q and using the fact that:

qa(p,Q,ta)

Pa(P,Q,ta)

r a (P,o,t a )

qb(p,Q,tb)

pb(P,o,tb)

rb(P,Q,tb)

this is illustrated for the case of the simple Harmonic Oscillator.

=) the contribution is I-oo «•••!-«>