Muga G. Time in Quantum Mechanics - Vol. 2
Подождите немного. Документ загружается.
3 Post-Pauli’s Theorem 31
a solution, the CCR may, too, not admit a solution for a given required set of
properties of the sought-after canonical pair. For example, we may require a solution
in L
2
(0, ∞) for the position and momentum operator pair that are both self-adjoint.
But, while the position operator is self-adjoint, the momentum operator cannot be
self-adjoint. Hence, no solution exists for the sought pair. It is clear though that
the reason for the non-existence of solution lies in the inhomogeneous property of
the underlying configuration space; and the non-self-adjointness of the momentum
operator, being the generator of translation in the configuration space, is a statement
of this fact in the system Hilbert space. It is important to bear in mind that, while
the set of properties we require of a canonical pair may be physically motivated,
the mathematical structure of Hilbert spaces is under no obligation to submit to our
wishes. If no solution exists, it may be because our required properties are incon-
sistent or not physically possible in the first place. And if we pay attention to why
no solution exists or why a certain class of solutions exists, we may have a better
understanding of the physical underpinnings of the system under consideration.
3.2.4 Example: Dense and Closed Category Solutions
to the Time–Energy Canonical Commutation Relation
Consider a particle in a force-free interval [−l, l]. The system Hilbert space is
H = L
2
[−l, l]. Let the Hamiltonian be (Hϕ)(q) =−
2
ϕ
(q)/2μ subject to the
boundary conditions ϕ(−l) = e
−2iγ
ϕ(l) and ϕ
(−l) = e
−2iγ
ϕ
(l), for some fixed
γ with 0 < |γ | <π/2. We demonstrate that there exist at least two self-adjoint
time operators conjugate to H, forming dense and closed category solutions to the
time–energy canonical commutation relation [T, H]ϕ = iϕ. Moreover, we find
these time operators to be both compact and hence non-covariant.
Now there exists a compact and self-adjoint operator T
1
such that T
1
and H form
a canonical pair of dense category [20]. This operator has the integral representation
(
T
1
ϕ
)
(q) =
l
−l
i
k,k
ϕ
(γ )
k
(q)ϕ
(γ )
k
(q
)
∗
E
k
− E
k
ϕ(q
)dq
, (3.1)
where the φ
(γ )
k
(q)’s are the eigenfunctions of H and the E
k
’s are the corresponding
eigenvalue (see Eq. 3.42), in which the primed sum indicates that k = k
is excluded.
That is, the pair T
1
and H satisfy the canonical commutation relation in some dense
subspace D
(1)
c
of H,
(
[T
1
, H]ϕ
)
(q) = iϕ(q), for all ϕ(q) ∈ D
(1)
c
, (3.2)
D
(1)
c
=
ϕ(q) =
k
a
k
ϕ
(γ )
k
(q),
k
|
a
k
|
2
< ∞,
k
a
k
= 0
. (3.3)
32 E.A. Galapon
Since the canonical domain is dense, i.e., orthogonal only to the zero vector, the
canonical pair C(T
1
, H; D
(1)
c
) is of dense category. T
1
is compact because its kernel
is square integrable.
There exists a compact and self-adjoint operator T
2
such that T
2
and H form a
canonical pair of closed category. This operator has the integral representation
(
T
2
ϕ
)
(q)=
l
−l
μ
4 sin γ
(q +q
)
e
i γ
H(q −q
) + e
−i γ
H(q
−q)
ϕ(q
)dq
,(3.4)
where H(q − q
) is the Heaviside function. That is, the pair H and T
2
satisfies the
canonical commutation relation in a non-dense subspace of H,
(
[T
2
, H]ϕ
)
(q) = i ϕ(q) for all ϕ(q) ∈ D
(2)
c
, (3.5)
D
(2)
c
=
l
−l
ϕ(q)dq = 0,ϕ(−l) = ϕ(l) = 0,ϕ
(−l) = ϕ
(l) = 0
. (3.6)
Since the canonical domain D
(2)
c
is not dense, i.e., it is orthogonal to any vector
φ(q) = constant = 0, the canonical pair C(T
2
, H; D
(2)
c
)isofclosedcategory.T
2
is
likewise compact because its kernel is square integrable.
This example shows that it is possible for a given Hamiltonian to have numer-
ous distinct associated time operators. So which time operator? Learning from our
example with the position and momentum operators, we cannot know which one
until we knew the origins of these operators or studied their properties. Remem-
ber that the quantum time problem has many aspects – quantum arrival, quantum
traversal, quantum tunneling – so it is possible that one time operator is more
appropriate than the other for a certain aspect of the quantum time problem. While
both operators are compact, it is not necessary that they will exhibit the same
dynamical behaviors, so that they do not necessarily represent the same aspect of
time.
The rest of the chapter is devoted to solving the time of arrival problem by finding
the appropriate time of arrival operator solution to the time–energy canonical com-
mutation relation. From this we will uncover the physical origin of the time operator
T
2
and find the operator −T
2
as a time of arrival operator for a spatially confined
particle under certain boundary condition. More importantly, we will find that the
solution set for such a system in the presence of a continuous interaction poten-
tial consists of self-adjoint, compact, and conjugate time operators – the confined
time of arrival (CTOA) operators. It will become clear why the CTOA operators
are appropriately referred to as time of arrival operators; and, in the process, it will
be made evident why compact and non-covariant time operators can be physically
meaningful.
3 Post-Pauli’s Theorem 33
3.3 Time of Arrival Operators
3.3.1 The Quantum Time of Arrival Problem
The quantum time of arrival problem seeks to find the time of arrival distribution
of a quantum particle at a given arrival point in the configuration space, for a given
initial state [50]. A solution to the problem within standard quantum mechanics
constitutes finding a time of arrival operator T conjugate to the system Hamilto-
nian H,[H, T]ϕ = iϕ, that admits a spectral resolution, not necessarily projection
valued, from which the time of arrival distribution can be computed in the standard
way.
However, the consensus is that no time of arrival operator can be constructed in
the most general case of arbitrary arrival point and of arbitrary interaction potential.
In the early days of quantum mechanics, the reason for this consensus is that no
self-adjoint time of arrival operator can be constructed, in accordance with Pauli’s
theorem. This belief has been bolstered by the observation that the quantization
of the free classical time of arrival expression is non-self-adjoint and admits no
self-adjoint extension. But even with the acceptance of POVMs as observables,
so that the non-self-adjoint free time of arrival operator can be interpreted as a
POVM observable [14], the belief is still there that no time of arrival operator can be
constructed.
In one dimension, the most quoted reason is that the classical time of arrival at
some point x, which is given by
T
x
(q, p) =−sgn(p)
μ
2
q
x
dq
√
H(q, p) − V (q
)
, (3.7)
where H is the Hamiltonian and V is the interaction potential and (q, p)arethe
position and momentum at t = 0, does not admit a sensible quantization because
Eq. (3.7) is generally not everywhere real and single valued in the entire phase space
[50, 57]. Moreover, the known existence of obstruction to quantization in Euclidean
space forbids the existence of quantization that satisfies the Dirac Poisson-bracket-
commutator correspondence; this implies that even if somehow we can quantize
Eq. (3.7), then in general the quantized time of arrival operator is not conjugate
with the Hamiltonian. For these reasons it is believed that if a theory of quan-
tum arrival existed it could not rest on the spectral resolution of a time of arrival
operator.
However, it is now clear that these objections to the construction of time of arrival
operators can be overcome. In this section, we describe how a time of arrival oper-
ator conjugate to a given Hamiltonian can be constructed. And in later sections we
will describe how self-adjoint, compact, and conjugate time of arrival operators can
be obtained from this time of arrival operator. We will see how these self-adjoint
operators can address not only the quantum time of arrival problem but also the
question of appearance of particles in quantum mechanics.
34 E.A. Galapon
3.3.2 The Idea of Supraquantization
Quantization seeks to derive the quantum counterpart of a classical observable
f (q, p) by some associative mapping Q of the real-valued function f (q, p)toa
maximally symmetric operator F in the system Hilbert space H, i.e., Q( f ) → F.
A paramount requirement of quantization is that the Poisson bracket of two (classi-
cal) observables quantizes into the commutator of the separately quantized observ-
ables, in particular, Q(
{
f, g
}
) = (i)
−1
[
Q( f ), Q(g)
]
. However, there is a well-
known obstruction to quantization in Euclidean space (and other spaces) which
says that no quantization exists that satisfies the Poisson-bracket-commutator corre-
spondence requirement for all observables [32, 33, 31, 34, 30, 35, 38, 61]. This
is unsatisfactory because the said correspondence is necessary, for example, in
ensuring that required evolution properties of a certain class of observables are
satisfied.
Thus in [18, 21] we addressed the problem of obstruction to quantization by
proposing the method of supraquantization – the construction of quantum observ-
ables without quantization and the subsequent quantum mechanical derivation of its
classical counterpart. The central idea of supraquantization is that quantum observ-
ables can be grouped meaningfully into distinct classes of observables, with each
class possessing a set of properties that distinguishes the observables of the class
from other observables not belonging to the class. It is the central problem of
supraquantization to determine this set of properties shared by the class of observ-
ables. Once these properties are known, the observables of the class are determined
by imposing the axioms of quantum mechanics in conjunction with other principles
of physics and by requiring that the quantum observables of the class reduce to their
classical counterparts in the classical limit.
For a specific class of classical observables, the required supraquantization may
be accomplished by referring to one of the members of the class and employing a
transfer principle to the rest. The transfer principle can be expressed as follows: Each
element of a class of observables shares a common set of properties with the rest of
its class such that when a particular property is identified for a specific element of
the class that property can be transferred to the rest of the class without discrimina-
tion. This, together with the axioms of quantum mechanics and the correspondence
principle, allows us to infer a general property of the observables of the class by
solving a particular observable of the class and then abstracting from that particular
observable the sought after property.
The idea of supraquantization is employed in [18, 21] in constructing time
of arrival operators without quantization and is described in the rest of this sec-
tion. We describe below how solving for the free quantum time of arrival opera-
tor without quantization leads to solving the time of arrival operator in the pres-
ence of interaction potential using the transfer principle. This consequently leads
to the derivation of the classical time of arrival from pure quantum mechanical
consideration.
3 Post-Pauli’s Theorem 35
3.3.3 Construction of Time of Arrival Operators Without
Quantization
Before we proceed, observe that by changing variables from (q, p)to(
˜
q = q −
x,
˜
p = p) in Eq. (3.7), we find that Eq. (3.7) becomes the time of arrival at the
origin for the potential
˜
V (
˜
q) = V (
˜
q + x). Hence it is sufficient for us to consider
the time of arrival at the origin in the development to follow, for when the arrival
point is different from the origin we only have to appropriately change variables.
Our problem now is to find the appropriate time of arrival operator T at the origin
for a given Hamiltonian H for a quantum particle in one dimension.
While our ultimate goal is to construct self-adjoint and conjugate Hilbert space
time of arrival operators for a given interaction potential, we pose the construc-
tion problem in the rigged Hilbert space Φ
×
⊃ H
∞
⊃ Φ as an intermediate step,
where Φ is the fundamental space of infinitely differentiable functions in the real
line with compact supports, H
∞
= L
2
(−∞,∞) is the Hilbert space closure of Φ
under the usual metric of quantum mechanics and Φ
×
is the space of functionals
on Φ. The operator T that we seek generally maps Φ into Φ
×
and has the integral
representation
(
Tϕ
)
(q) =
∞
−∞
q
|
T
q
ϕ(q
)dq
, (3.8)
for all ϕ(q)inΦ. In this form, solving for T is finding for the kernel
q
|
T
q
.But
how do we determine the kernel
q
|
T
q
without resorting to quantization?
We accomplish this in two steps. First, it is by identifying the set of properties of
the time of arrival operator T and the set of appropriate physical principles that will
uniquely identify T. Second, it is by implementing the transfer principle where we
solve the construction of the free time of arrival operator using the results of first
step to abstract out the general form of the time of arrival operator kernel.
Let us enumerate the properties of the time of arrival T that we require:
1. Being a time of arrival operator, it must at least evolve according to dT/dt =
−I
Φ
, where I
Φ
is the identity in Φ, at least in the neighborhood of t = 0. This
condition translates to the generalized canonical commutation relation
[H
×
, T]ϕ
(q) = iϕ(q) (3.9)
for all ϕ(q)inΦ, where H
×
is the rigged Hilbert space extension of H in Φ
×
.
This is a property arising from the axioms of quantum mechanics.
2. For the operator T to be identifiable as a time of arrival operator, it must reduce
to the classical time of arrival in the classical limit. That is, it must satisfy the
condition
T
0
(q, p) =lim
→0
T
(q, p) , (3.10)
36 E.A. Galapon
where T
0
(q, p) is the classical time of arrival at the origin and T
(q, p)isthe
Wigner transform of the kernel
T
(q, p) =
∞
−∞
q +
v
2
T
q −
v
2
exp
−i
v p
!
dv. (3.11)
This is a property arising from the correspondence principle.
3. The operator T must satisfy the time reversal symmetry ΘTΘ
−1
=−T, where
Θ is the time reversal operator; this translates to the condition
q
|
T
q
∗
=−
q
|
T
q
. (3.12)
4. In keeping with the requirement that quantum observables must yield real expec-
tation values, we require T to be Hermitian; this translates into the condition
"
q
T
|
q
∗
=
q
|
T
q
. (3.13)
The time reversal and hermicity conditions already restrict the functional form
of the kernel. Equation (3.12) implies that
q
|
T
q
is purely complex, so that
q
|
T
q
= iτ (q, q
), where τ (q, q
) is real valued. On the other hand, Eq.
(3.13) implies that τ (q, q
) =−τ (q
, q), which allows us to write τ (q, q
) =
T (q, q
)S(q, q
), where T (q, q
) = T (q
, q) and S(q, q
) =−S(q
, q). The time
reversal and hermicity conditions then dictate that the kernel is of the form
q
|
T
q
= iT (q, q
)S(q, q
), with T (q, q
) and S(q, q
) to be determined. The
conjugacy (3.9) and the correspondence principle (3.10) conditions will further
restrict the forms of T (q, q
) and S(q, q
), but we will find the above-enumerated
conditions are not sufficient to uniquely identify T. However, supplementing them
with the condition that only parameters of the system enter into the construction will
identify T uniquely.
Recognize that solving for the time of arrival operator T for a given Hamilto-
nian H is essentially solving for the generalized canonical commutation relation
(3.9) under certain conditions specified by Eqs. (3.10), (3.12), and (3.13). This is
an example of what we have discussed earlier that canonical pairs may arise as
solutions to specific problems. We will see below that there are in fact numerous
solutions to Eq. (3.9), but only a particular solution will be found acceptable.
3.3.4 Non-interacting Case
We now solve the free particle kernel without quantization.
3
The problem is to deter-
mine the unknown T (q, q
) and S(q, q
) for the free particle. Substituting the free
Hamiltonian and the time of arrival operator back into Eq. (3.9) and after performing
3
Here we give a more transparent solution than the one provided in [21].
3 Post-Pauli’s Theorem 37
two integrations by parts, we arrive at
(
[H, T]ϕ
)
(q) =
Σ
i
−
2
2μ
∂
2
T (q, q
)
∂q
2
+
2
2μ
∂
2
T (q, q
)
∂q
2
S(q, q
)ϕ(q
)dq
+
Σ
i
2
2μ
−
2
∂T
∂q
∂ S
∂q
+ T (q, q
)
∂
2
S
∂q
2
+
2
∂T
∂q
∂ S
∂q
+ T (q, q
)
∂
2
S
∂q
2
ϕ(q
)dq
, (3.14)
where Σ is the support of ϕ(q). We now have to choose T (q, q
) and S(q, q
) such
that T is a solution to the generalized CCR. The first term can be made to vanish
without specifying S(q, q
) by imposing that T (q, q
) is a continuous solution to the
partial differential equation
−
2
2μ
∂
2
T (q, q
)
∂q
2
+
2
2μ
∂
2
T (q, q
)
∂q
2
= 0 . (3.15)
This has the general solution T (q, q
) = f (q + q
) + g(q −q
), where f and g are
arbitrary; however, since T (q, q
) = T (q
, q), g must be an even function.
Moreover, functions T (q, q
) and S(q, q
) must be chosen such that the second
term equals the right-hand side of Eq. (3.9). This is only possible if the brack-
eted quantity in the second term involves the Dirac delta function and, perhaps, its
derivatives. Since T (q, q
) has already been chosen to satisfy a second-order partial
differential equation, so that it has continuous second-order partial derivatives, the
delta function must only come from S(q, q
). The general form of S(q, q
) that leads
to the required Dirac delta function is S(q, q
) = CH(q − q
) + BH(q
− q),
where C and B are constants and H(x) is the Heaviside step function. But since
S(q, q
) =−S(q
, q), we must have B =−C or S(q, q
) = C sgn(q − q
), where
sgn(x) = H(x) − H(−x) is the sign function.
Substituting S(q, q
) = C sgn(q −q
) back into the second term leads to
(
[H, T]ϕ
)
(q) =
−2C
μ
∂T
∂q
+
∂T
∂q
q
=q
iϕ(q) . (3.16)
Hence T is a solution to the CCR if the bracketed quantity is unity. If we choose
C =−μ/,wemusthave
∂T
∂q
+
∂T
∂q
q
=q
=
1
2
. (3.17)
This implies that T (q, q
) has finite first derivatives along the line q = q
.Fora
solution of the form T (q, q
) = f (q + q
), the only possible solution satisfying
(3.17) is given by T (q, q
) = (q + q
)/4; however, the general solution satisfying
38 E.A. Galapon
(3.17) is given by T
(q, q
) = (q + q
)/4 + h((q − q
)
2
), where h(x) continuous
second derivative everywhere.
We now require that the correspondence principle be satisfied. That is, the clas-
sical time of arrival at the origin, given by T =−μq/ p, must be derived from T.
We first consider the solution T (q, q
) = (q +q
)/4 so that we have the kernel
q
|
T
q
=
μ
i
(q +q
)
4
sgn(q −q
) . (3.18)
This is an acceptable solution if the corresponding operator T reduces to the classi-
cal free time of arrival. Indeed we have its Wigner transform
∞
−∞
q +
v
2
T
q −
v
2
exp
−i
v p
!
dv =
μ
2i
q
∞
−∞
sgn(v)e
−ivp/
dv =−μ
q
p
. (3.19)
Thus Eq. (3.18) solves the kernel problem. However, as we have noted above, the
function T (q, q
) is not, so far, uniquely determined because we can have, say, the
solution T
(q, q
) = (q + q
)/4 + a(q − q
)
2
for some real constant a. This gives
another solution T
whose Wigner transform is given by
∞
−∞
q +
v
2
T
q −
v
2
exp
−i
v p
!
dv =−μ
q
p
−4aμ
2
1
p
3
, (3.20)
which reduces to the classical free time of arrival as → 0. For a fixed we find
that the two operators T and T
satisfy the correspondence principle in the classical
limit.
Hence the enumerated properties of the time of arrival operator that we seek is not
sufficient to uniquely identify the solution. They must somehow be supplemented.
Notice that T
involves a constant a with unit of inverse length that cannot be gener-
ated from the available parameters of the free particle, which are and μ. Hence to
consider T
we have to introduce another constant of nature which is questionable
because quantum mechanics does well without the need for such another physical
constant. Thus restricting ourselves to solutions involving parameters of the system
alone, we are left with the solution T (q, q
) = (q +q
)/4, and the free time of arrival
operator kernel is given by Eq. (3.18).
3.3.5 Interacting Case
Having solved the free particle kernel, we proceed in implementing the transfer
principle. We hypothesize that all time of arrival operator kernels assume the same
form. Thus, from Eq. (3.18), the kernel takes on the form
q
|
T
q
=
μ
i
T (q, q
)sgn(q −q
) , (3.21)
3 Post-Pauli’s Theorem 39
where T(q, q
) depends on the interaction potential and which we refer to as the ker-
nel factor. From the time reversal symmetry and hermicity requirement, we require
that T (q, q
) be real valued and symmetric, T (q, q
) = T (q
, q). We determine
T (q, q
) by imposing condition (3.9) on T. Substituting Eq. (3.21) back into the
left-hand side of Eq. (3.9) and performing two successive integration by parts, we
arrive at
[H
×
, T]ϕ
(q) = i
dT(q, q)
dq
+
∂T (q
, q
)
∂q
+
∂T (q, q)
∂q
ϕ(q) dq
−i
μ
Σ
−
2
2μ
∂
2
∂q
2
+ V (q)
T (q, q
) −
−
2
2μ
∂
2
∂q
2
+ V (q
)
T (q, q
)
×sgn(q −q
)ϕ(q
)dq
, (3.22)
where Σ is the support of ϕ(q). We point out that our ability to arrive at this expres-
sion has been made possible by extending the formulation in a rigged Hilbert space.
If H
×
and T are to be conjugate in Φ, then the right-hand side of Eq. (3.22) must
reduce to Eq. (3.9). Following the lead of the free particle case, Eq. (3.22) reduces
to Eq. (3.9) if we impose that T (q, q
) solves the partial differential equation
−
2
2μ
∂
2
T (q, q
)
∂q
2
+
2
2μ
∂
2
T (q, q
)
∂q
2
+
V (q) − V (q
)
T (q, q
) = 0 (3.23)
and satisfies the condition
dT(q, q)
dq
+
∂T (q, q
)
∂q
q=q
+
∂T (q, q
)
∂q
q
=q
= 1 . (3.24)
Equation (3.24) defines a family of operators conjugate to the Hamiltonian in the
sense required by Eq. (3.9).
The conditions guaranteeing uniqueness of T (q, q
) is found by appealing to the
transfer principle. We determine the set of conditions satisfied by the free particle
kernel factor that ensures it uniqueness and then impose the same conditions in the
interacting case. By inspection, the free particle kernel factor T (q, q
) = (q +q
)/4
satisfies both (3.23) and (3.24), and it satisfies the conditions
T (q, q) =
q
2
, T (q, −q) = 0 . (3.25)
These two conditions uniquely identify the free particle solution. By virtue of
the transfer principle, we impose the same conditions (3.25) on the solution to
Eq. (3.23). It can be established that the boundary conditions (3.25) guarantee that
boundary condition (3.24) is satisfied; moreover, Eqs. (3.25) already imposes the
condition T (q, q
) = T (q
, q) [21]. Likewise, it can be shown that for continuous
potentials Eq. (3.23) has a unique solution in the entire qq
-plane [22, 47].
40 E.A. Galapon
In general, the solution to the time kernel equation falls into whether the system
under consideration is linear or non-linear. For linear systems, systems with the
interacting potential V (q) = aq
2
+ bq + c for some constants a, b, and c,the
solution is given by T
lin
(q, q
) = T
0
(q, q
), where
T
0
(q, q
) =
1
2
η
0
0
F
1
;1;
μ
2
2
[
V (η) − V(s)
]
!
ds
η=
q+q
2
, (3.26)
in which
0
F
1
is a specific hypergeometric function. On the other hand, for non-linear
systems with everywhere analytic potentials, the solution to the time kernel equation
is given by T
nli
(q, q
) = T
0
(q, q
) + T
1
(q, q
) + T
2
(q, q
) +···, where T
1
(q, q
),
T
2
(q, q
),... are determined by the potential and are found by solving explicitly the
time kernel equation for the given potential.
The correspondence principle can now be established via the Wigner transform
of the kernel of T. For linear systems, we have T
(q, p) = t
0
(q, p); while for non-
linear systems, T
(q, p) = t
0
(q, p) +
2
t
1
(q, p) +
4
t
2
(q, p) +···, where t
0
(q, p)
is given by
t
0
(q, p) =−
∞
k=0
(−1)
k
(2k − 1)!!
k!
μ
k+1
p
2k+1
q
0
[V (q) − V(q
)]
k
dq
, (3.27)
and the
2 j
t
j
(q, p)’s arise from the T
j
(q, q
) in the solution for T
nli
(q, q
). For
potentials continuous in the neighborhood of the origin, Eq. (3.27) can be summed
to yield the classical time of arrival T
0
(q, p) at the origin. That is, t
0
(q, p)isan
expansion of T
0
(q, p) in the neighborhood of the arrival point; for this reason, we
referred to t
0
(q, p) as the local time of arrival. Hence we have our operator T reduc-
ing to the classical time of arrival in the classical limit. And we have solved the
supraquantization of the time of arrival operator, at least, for continuous potentials.
3.3.6 Time of Arrival Operators in Representation-Free Form
In the construction of the confined time of arrival operators in the next section, we
will need to explicitly write our time of arrival operator T in terms of the position
and momentum operators, q and p, that is, in representation-free form. T must be
written in them such that, in coordinate representation, the kernel of T is given by
Eq. (3.21). Moreover, T must be in the form such that the generalized time–energy
CCR, ([H
×
, T]ϕ)(q) = iϕ(q), assumes the formal relation [H, T] = iI, where H
is the formal Hamiltonian H = p
2
/2μ + V(q)[28].
This can be accomplished by Weyl quantizing the transform T
(q, p)ofthe
kernel. For everywhere analytic potentials, T
(q, p) is an expansion in q
n
p
−m
for
positive integers n and m. The explicit operator form of T is then obtained with the
formal replacement scheme