=
is taken. The mid-point rule (being accurate to second order) generates no
Page 10
q < V = qa
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Quantum Canonical Transformations
Quantum Canonical Transformations
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
Quantum Canonical Transformations
Quantum Canonical Transformations
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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Quantum Canonical Transformations
Quantum Canonical Transformations
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
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Quantum Canonical Transformations
Quantum Canonical Transformations
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
1/4QQPM'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
Quantum Canonical Transformations
Quantum Canonical Transformations
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
Quantum Canonical Transformations
Quantum Canonical Transformations
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 «•••!-«>