Heisenberg Evolution WKB and Symplectic Area Phases

33 Pages·0.36 MB·English

by  T. A. Osborn

Heisenberg Evolution WKB and Symplectic Area Phases
T. A. Osborn
Department of Physics and Astronomy
University of Manitoba
2 Winnipeg, MB, Canada, R3T 2N2
0
0
2
M. F. Kondratieva
n
a
J Department of Mathematics and Statistics

8 University of Minnesota

Duluth, MN, USA 55812
1
v
9
2
0
1 Abstract
0
2 The Schro¨dinger and Heisenberg evolution operators are represented in phase
0 space T∗Rn by their Weyl symbols. Their semiclassical approximations are con-
/
h
structed in the short and long time regimes. For both evolution problems, the
p
WKB representation is purely geometrical: the amplitudes are functions of a Pois-
-
nt son bracket and the phase is the symplectic area of a region in T∗Rn bounded
a by trajectories and chords. A unified approach to the Schro¨dinger and Heisen-
u
berg semiclassical evolutions is developed by introducing an extended phase space
q
: χ T∗(T∗Rn). In this setting Maslov’s pseudodifferential operator version of
v 2 ≡
i WKB analysis applies and represents these two problems via a common higher
X
dimensional Schro¨dinger evolution, but with different extended Hamiltonians. The
r
a evolution of aLagrangian manifold inχ2,definedbyinitial data, controls thephase,
amplitude and caustic behavior. The symplectic area phases arise as a solution of
a boundary condition problem in χ . Various applications and examples are con-
2
sidered. Physically important observables generally have symbols that are free of
rapidly oscillating phases. The semiclassical Heisenberg evolution in this context
has been traditionally realized as an ~ power series expansion whose leading term
is classical transport. The extended Heisenberg Hamiltonian has a reflection sym-
metry that ensures this behavior. If the WKB initial phase is zero, it remains so
for all time, and semiclassical dynamics reduces to classical flow plus finite order ~
corrections.
1 Introduction
The version of quantum mechanics most suitable for semiclassical dynamics is Moyal’s
[1] representation. This classical like statement of quantum mechanics is achieved by
the Wigner–Weyl [2, 3] correspondence, which uniquely associates Hilbert space opera-
tors with Weyl symbols (functions) on phase space. This correspondence transforms the
non-commutative product of operators into a non-commutative product of symbols.
∗
Expectation values are realized as integrals over phase space.
For systems whose coordinate manifold is flat, the Wigner–Weyl isomorphism is most
compactly described [4, 5, 6] in terms of the quantizer. Let T(x) = exp[i(p qˆ q pˆ)/~] be
· − ·
the Heisenberg translation operator. Here x = (q,p) are the position, momentum coordi-
nates of the linear phase space χ T∗Rn, and xˆ = (qˆ,pˆ) are the corresponding quantum
1 ≡ q
counterparts. The parity operator on Hilbert space is Pψ(q) = ψ( q),ψ L2(Rn). The
− ∈ q
product of these two unitary operators defines the quantizer, ∆(x) 2nT(2x)P. In terms
≡
of ∆(x), the one-to-one Weyl symbol-operator pairing is
A = h−n dxA(x)∆(x), A(x) = [A] (x) = TrA∆(x),
w
Zχ1
where h = (2πb~). The first statement above is Weylbquantizationb; while the second is
de-quantization. Whenever the operator is a density matrix ρˆ, the quantity h−nρ(x) is
the Wigner distribution. The above map, from A to A(x), gives the same result as the
Wigner transform (A.3) of the kernel q A q′ .
h | | i
The star product is defined by the requiremenbt that [AB] = A B cf. Appendix A.
w
∗
For small ~, the multiplication is approbximately the commutative product of symbols,
∗
bb
A B(x) = A(x)B(x)+i~/2 A,B (x)+O(~2).
∗ { }
The Poisson bracket term measures, to leading ~ order, the non-commutative character
of the star product.
The two basic statements of quantum dynamics are the Schr¨odinger and Heisenberg
pictures. On the space, L2(Rn), each self-adjoint Hamiltonian H generates a unitary
q
Schr¨odinger evolution, U(t) = exp[ itH/~]. An initial density matrix ρˆ has Heisenberg
0
−
evolution, ρˆ(t) = U(t)ρˆ U(t)†. Let H, U(t) and ρ(t) be the Weyl sybmbols of H, U(t) and
0
ρˆ(t), respectively. In thibs setting the eqbuations of motion are
b b b b
i~∂U(t,x)/∂t = H(x) U(t,x), (1.1)
∗
∂ρ(t,x)/∂t = H,ρ(t) (x), (1.2)
M
{ }
with initial conditions U(0,x) = 1, and ρ(0,x) = ρ (x). In (1.2), the bracket operation is
0
defined by
1 1
A,B [A,B] = (A B B A).
{ }M ≡ i~ w i~ ∗ − ∗
This is the Moyal bracket. Like the qbuanbtum commutator, it is bilinear, skew and obeys
the Jacobi identity.
1
It is evident that the Weyl–Heisenberg symbol evolution problem (1.2) is formally as
close as it can be to classical dynamics. Since A,B = A,B + O(~2), Bohr’s cor-
M
{ } { }
respondence principle is realized in a transparent form. The time dependent expectation
value of an observable A is conveniently given by the phase space integral
bA TrAρˆ(t) = h−n dxA(x)ρ(t,x). (1.3)
t
h i ≡
Zχ1
The main goal of thisbpaper is bto obtain both short and long time WKB approximate
solutions for the Weyl symbol evolution problems (1.1) and (1.2). In Theorems 1 and
2, we prove that the phases entering these semi-classical representations are invariant
geometrical quantities — namely the symplectic areas defined by certain closed loops in
phase space. Furthermore, theamplitude functions aregiven by determinants ofa Poisson
bracket.
We employ a common method to solve both problems (1.1) and (1.2). Standard
Weyl symbol calculus identities show that H and H, are pseudodifferential opera-
M
∗ { ·}
tors, in other words they are functions of and multiplication by x. If one introduces
x
∇
an extended phase space χ T∗(T∗Rn), then Maslov’s [7] WKB analysis applies. In
2 ≡ q
particular, the semiclassical approximations for the evolutions U(t,x) and ρ(t,x) arechar-
acterized by a Lagrangian manifold, and the time evolution of this manifold is determined
by an extended Hamiltonian in χ . In order to distinguish these two phase spaces we call
2
χ the primary phase space (PPS), and χ the secondary phase space (SPS). Projecting
1 2
the χ analysis back onto the primary phase space provides explicit WKB representations
2
of U(t,x) and ρ(t,x).
The results in the literature nearest to this paper are found in three seminal works: by
Karasev and Nazaikinskii [8], Berry [9], and Marinov [10]. In the first of these, an analyt-
ical approach to a generalized semiclassical expansion based on a SPS with its associated
left-right representation of -product and Hamilton-Jacobi equation was developed. In
∗
the second pair of works, the geometrical (symplectic phase) interpretation of the Wigner
eigenfunction and of the dynamical WKB approximation was achieved. Specifically, in
the small time sector, where the time evolving χ Lagrangian manifold remains single
2
sheeted, Marinov found the WKB amplitude and phase for U(t,x). In addition to pro-
viding an entirely different proof, our treatment extends these known results in several
ways. First, we determine the long time version of the WKB approximation where the χ
2
Lagrangian manifold may be multi-sheeted with several points corresponding to a given
x χ . Second, by incorporating Maslov’s results one maintains control of the error
1
∈
estimates. Third, we solve a generalized version of (1.1) with rapidly oscillating initial
data. This latter generalization arises for the evolution U(t+ t ), t > 0. At t = 0 the
0 0
initial data generically will have the rapidly oscillating form, N (x)eiΦ0(x)/~.
0
In spite of the importance of problem (1.2), it has rbeceived insufficient study. The
version of the Heisenberg problem that has been investigated in detail [11, 12, 13, 14]
occurs for the case where the observable A is semiclassically admissible, namely its Weyl
symbol is ~ dependent and allows a finite order small ~ asymptotic expansion cf. (3.6).
In [15] a connected graph method was cbonstructed to obtain the coefficients of the ~
asymptotic expansion of [U(t)†AU(t)] (x). Based on this latter formalism, numerical
w
calculationshaveshown[16]thatthe~2-ordersemiclassical expansionaccuratelydescribes
b bb
2
noble gas atom-atom scattering. However, semiclassically admissible symbols are not
suitable for representing [17] density matrices. The requirement that one has a pure
state, ρ ρ = ρ , means that ρ (x) is an ~ 0 rapidly oscillating function cf. (2.12).
0 0 0 0
∗ →
The paper has thefollowing organization. Section 2 presents the extended Schr¨odinger
equation in the SPS setting and shows how the problems (1.1) and (1.2) are particular
examples of this extended evolution. Also, the geometric, commutative and coordinate
relationships between the PPS and SPS are reviewed. The trajectories entering the WKB
expansion are determined by the solutions of a boundary condition problem. In Section 3,
a detailed analysis of the BC problem and the manner in which it constructs phase space
loops is carried out. The semiclassical expansions for U(t,x) and ρ(t,x) are consolidated
in Theorems 1 and 2. Finally, in Section 4 the group generated phase addition rules are
obtained; the mutual compatibility of the Schr¨odinger and Heisenberg semiclassical evo-
lution is established; the exact solutions for quadratic Hamiltonian systems are compared;
and, the ~ 0 asymptotics of [U(t)AU(t)†] (x) for semiclassically admissible operators
w
→
A is realized as a special case of Theorem 2. There are three appendices containing spe-
cialized aspects of Weyl symbol bcalcublubs, pseudodifferential operators, Jacobi fields, and
abPoincar´e–Cartan identity.
2 Secondary Phase Space Dynamics
In the first part of this section we show how to interpret the H structure in the equations
∗
of motion as the action of a pseudodifferential operator (ΨDO). The normal ordered
symbols of these operators are then functions on the secondary phase space. For the
Schr¨odinger and Heisenberg problem we show how the classical flow in χ is composed in
2
terms of the standard Hamiltonian mechanics on χ . The necessity of representing the
1
initial value form of the density matrix as a rapidly oscillating ~ function is confirmed by
constructing the semiclassical Weyl symbol for a pure state.
Introduce the SPS canonical operators X,Y acting on functions f(x): Xf(x) = xf(x)
2 1
and Yf(x) = i~ f(x). Let (X,Y) be a normal ordered, symmetric ΨDO defined by
x
− ∇ H
the symbol (x,y). On the coordinate domain R2n, the extended Schr¨odinger evolution
H x
problem is
2 1
i~∂Ψ(t,x)/∂t = (X,Y)Ψ(t,x). (2.1)
H
The two problems (1.1) and(1.2) are particular cases of (2.1). The appropriaterealization
of ineach caseisdetermined by employing theleft-right product representation [8,18]
H ∗
(see Appendix A). Setting (x,y) = H(x 1Jy) selects the Schr¨odinger problem
H H1 ≡ − 2
(1.1), whereas the choice H(x 1Jy) H(x+ 1Jy) gives the Heisenberg problem
H2 ≡ − 2 − 2
(1.2). The solution Ψ(t,x) is either U(t,x) or ρ(t,x) depending on the choice of . In
H
SPS, the scalar function is a Hamiltonian which is the generator of χ classical flow.
2
H
Consistency of this extended Schr¨odinger problem requires that the normal ordered
symbol H(x 1Jy) define a symmetric operator on L2(R2n). That is does so is a con-
± 2 x
sequence of the argument structure (x 1Jy) which implies that cf. (A.16) the normal
± 2
ordered and Weyl ordered symbols are the same function. Since the Weyl ordered symbol
is real, one has that the ΨDO Hamiltonians in (2.1) are formally self-adjoint.
3
Consider the coordinate systems of χ and χ and their interconnections. Let y be
1 2
the momentum variable in SPS so that a general point m χ has Cartesian coordinates
2
∈
z(m) = (x,y). The symplectic structure of χ and χ is determined by their respective
1 2
canonical 2-forms
0 I
ω(1) dp dq = 1J dx dx , J = n ,
≡ ∧ 2 αβ β ∧ α I 0
n
(cid:20) − (cid:21)
ω(2) dy dx = 1J dz dz ,
≡ ∧ 2 jk k ∧ j
where I is the n n identity matrix, and J isethe symplectic matrix having the block
n
×
form of J with I substituted for I . Given the 2-forms, ω(1) and ω(2), the corresponding
2n n
Poisson brackets are e
f,g (x) = f(x) J g(x), F,G (z) = F(z) J G(z).
1 2
{ } ∇ · ∇ { } ∇ · ∇
The coordinate functions, x = (q,p) and z = (x,y), are canonical in χ and χ , namely
e 1 2
x ,x = J and z ,z = J . In order that (2.1) be a standard Schr¨odinger
α β 1 αβ j k 2 jk
{ } { }
equation the coordinates x = (q,p) must commute. This is the case since x ,x = 0.
α β 2
{ }
As one sees from functional formes of , a second natural coordinate system for χ is
2
H
defined by the (l,r) variables
l = x 1Jy, r = x+ 1Jy, l,r R2n,
− 2 2 ∈
(2.2)
x = 1(l+r), y = J(l r).
2 −
The link between the coordinate systems (x,y) and (l,r), and their commutative struc-
tures is the following.
Lemma 1. (i) The SPS commutative properties of the projections l,r : R4n R2n are
z → x
l ,l = J , r ,r = J , l ,r = 0.
α β 2 αβ α β 2 αβ α β 2
{ } { } − { }
(ii) The l variables are Poisson; the r variables are anti-Poisson, namely
f l,g l = f,g l, f r,g r = f,g r.
2 1 2 1
{ ◦ ◦ } { } ◦ { ◦ ◦ } −{ } ◦
(iii) The χ Poisson bracket and symplectic form have the l r decomposition
2
×
F,G = ( F) J ( G) ( F) J ( G) ,
2 l α αβ l β r α αβ r β
{ } ∇ ∇ − ∇ ∇
J dz dz = J dl dl J dr dr .
jk k j αβ β α αβ β α
∧ ∧ − ∧
The space structure for the left-right variables is χ χ′ (l,r). Here χ′ denotes the
e 1⊗ 1 ∋ 1
phase space defined by the 2-form 1J dr dr . Often the space χ χ′ is labelled [19]
−2 αβ β∧ α 1⊗ 1
the double phase space. We use the name secondary phase space for cotangent bundle
χ = T∗(T∗Rn) in order to distinguish it from χ χ′. The left-right map, (2.2) is
2 q 1 ⊗ 1
diffeomorphic.
The basic ingredients in any semiclassical approximation are the classical action phase
and the amplitude functions determined by classical transport. Because the extended
4
Hamiltonians are the simple composite functions H l and H l H r, it is possible
◦ ◦ − ◦
to describe the χ phase space motion as appropriately weighted sums of χ trajectories.
2 1
Let g(τ x ) = (q(τ x );p(τ x )) denote the solution of the PPS Hamiltonian system
0 0 0
| | |
g˙(τ x ) = J H(g(τ x )), g(0 x ) = x . (2.3)
0 0 0 0
| ∇ | |
The SPS motion is labelled G(τ z ) = (x(τ z );y(τ z )), where
0 0 0
| | |
G˙(τ z ) = J (G(τ z )), G(0 z ) = z . (2.4)
0 0 0 0
| ∇H | |
The initial data z for system (2.4) also has the left-right representation l = l(z ),r =
0 e 0 0 0
r(z ).
0
Lemma 2. The solution of the SPS Hamiltonian system, in terms of χ flows, takes the
1
form: (i) For the Schr¨odinger problem with = H l
1
H ◦
x(τ z ) = 1 g(τ l )+r , y(τ z ) = J g(τ l ) r . (2.5)
| 0 2 | 0 0 | 0 | 0 − 0
(cid:0) (cid:1) (cid:0) (cid:1)
(ii) For the Heisenberg problem with = H l H r
2
H ◦ − ◦
x(τ z ) = 1 g(τ l )+g(τ r ) , y(τ z ) = J g(τ l ) g(τ r ) . (2.6)
| 0 2 | 0 | 0 | 0 | 0 − | 0
(cid:0) (cid:1) (cid:0) (cid:1)
Proof. Consider part (i). Using the Poisson equations of motion, it is convenient to state
the SPS flow in the l,r variables. Lemma 1(iii) implies
˙
l (τ) = l ,H l = J H(l(τ)), r˙ (τ) = r ,H l = 0. (2.7)
α α 2 αβ β α α 2
{ ◦ } ∇ { ◦ }
So l(τ l ) = g(τ l ) and the r-motion is a constant, r(τ r ) = r . Using (2.2) recovers
0 0 0 0
| | (cid:3) |
(2.5). A similar argument verifies part (ii).
The identities (2.7) show that the left-right motions are decoupled. This decoupling
occurs because l (τ),r (τ) = 0. Further note that the system = H l has 2n
α β 2 1
{ } H ◦
constants of motion; r (τ) = const for α = 1...2n. Nevertheless, is not completely
α 1
H
integrable since the functions r (τ) are not in involution, i.e. r (τ),r (τ) = J = 0.
α α β 2 αβ
{ } − 6
Lagrangian manifolds in SPS create canonical transformations in PPS. Let us clarify
how this is realized in terms the left right projections. Suppose Λ is a χ manifold which
2
is locally defined by the smooth phase S : U R2n R, i.e. y = S(x),x U. This
⊆ x → ∇ ∈
identity relates l to r by
r l = J S(1(l+r)). (2.8)
− ∇ 2
If det(I 1JS′′) = 0, x U then the implicit function theorem defines l = l(r) and
± 2 6 ∈
r = r(l). These transformations are canonical. This follows from
dl (I 1JS′′)
= − 2 .
dr (I + 1JS′′)
2
The right side above is the Cayley transform of the symmetric S′′ and so defines a sym-
plectic matrix. A systematic study of generating functions of this type is found in [20]
5
An important aspect of WKB analysis is the ~ singularity structure of the Cauchy
initial data. As noted in the Introduction, a pure state density matrix must be a rapidly
oscillating function. Let us clarify this situation with an example. Suppose ψ (q) =
0
n(q)exp[is(q)/~] is a unit normalized L2(Rn) wave function, which defines a rank one
q
density matrix, ρˆ = ψ ψ . The functions n,s are real. The Weyl symbol of ρˆ is given
0 0 0 0
| ih |
by the Wigner transform (A.3)
i
ρ (x;~) = dvn(q + 1v)n(q 1v)exp φ(v,x), (2.9)
0 2 − 2 ~
Rn
Z
φ(v,x) = s(q + 1v) s(q 1v) p v. (2.10)
2 − − 2 − ·
The ~ 0 asymptotic form of ρ may be calculated by a stationary phase approxi-
0
→
mation [7]. Let φ′(v,x) = φ(v,x) and φ′′(v,x) = φ(v,x). The critical points of φ
v v v
∇ ∇ ∇
satisfy
2p = s(q 1v)+ s(q + 1v). (2.11)
∇ − 2 ∇ 2
Suppose (v ,x ) is a solution set for this equation where detφ′′(v ,x ) = 0, then the
0 0 0 0
6
implicit function theorem defines a function v = v(q,p) = v(x) obeying (2.11). Since
φ′(v,x)iseveninv,therootsoccurinpairs. Ifv = v(x)isasolution, thensoisv = v(x).
−
Incorporate this pairing behavior via the notation: q = q v(x)/2, p = s(q ),
± ± ±
± ∇
x = (q ,p ). Inthenon-singular regionwhere detφ′′(v(x),x) = 0,theWigner transform
± ± ±
6
above has asymptotic form
1 π
ρ0(x;~) ≈ hn2|det φ′′(v(x),x)|−1/2n(q+)n(q−)2cos ~S0(x)+ 4sgnφ′′(v(x),x) . (2.12)
n o
The phase S (x) = φ(v(x),x) and sgn denotes the signature of a symmetric matrix. If x
0
is in the neighborhood where (2.11) has no solutions, then ρ (x) is O(~∞) small.
0
As Berry first noticed [9] (in the n = 1 version of this problem), the phase S is a
0
symplectic area. To see this in the present context, define the χ Lagrangian manifold
1
λ x χ p = s(q) . Construct two pathsbetween theend points x andx . Let the
1 − +
≡ { ∈ | ∇ }
first path, C(x ,x ), be a straight line from x to x . As the second path, γ (x ,x ),
− + − + λ + −
choose any curve from x to x lying on the surface λ. Then one finds
+ −
S (x) = p dq. (2.13)
0
·
IC(x−,x+)+γλ(x+,x−)
The geometry of the loop in (2.13) is the well known chord mid-point construct [10, 9,
21, 22, 23]. The end points x of the directed chord C(x ,x ) lie on λ, and x is the
± − +
mid-point of this chord. By Stokes theorem, S (x) is also the area of any membrane
0
having boundary C(x ,x )+γ (x ,x ).
− + λ + −
3 Symplectic Areas and WKB Phases
This section shows how the secondary phase space WKB approximations for U(t,x) or
ρ(t,x) can be reformulated in terms of primary phase space flows. This reduction process
6
transforms all the χ WKB phases into symplectic areas defined by closed loops in χ .
2 1
A key element of this analysis is the solvability of the appropriate two point boundary
condition (BC) problem. We pose the BC problem in terms of Lagrangian manifolds and
construct both short and long time solutions.
To begin, we summarize Maslov’s WKB expansion for the extended Schr¨odinger
problem (2.1). The t = 0 state for system (2.1) is assumed to have the generic form,
Ψ (x) = α (x)eiβ0(x)/~. The real functions α ,β have support on the domain D R2n.
0 0 0 0 0 ⊆ x
The phase β defines a χ Lagrangian manifold Λ , via y(x) = β (x),x D . Let
0 2 0 0 0
∇ ∈
Π (x,y) = x be the projection onto R2n. It is assumed that projection Π : Λ D is a
1 x 1 0 → 0
diffeomorphism.
To proceed three assumptions are required: (a) The function Tm(R4n),m 2.
H ∈ + z ≥
cf. Appendix A. (b) For arbitrary finite time intervals [ T,T], the SPS Hamiltonian
−
system (2.4) has unique smooth solutions, i.e. G(t x,y) C∞([ T,T],R4n). (c) Let x
| ∈ − z
be a non-focal point (cf. Definition 3) with respect to the Π projection of the manifold
1
Λ = G(t)Λ . Assume there are a finite number of points xj D , j = 1,...,N such that
t 0 0 ∈ 0
Π G(t xj, β (xj)) = x.
1 | 0 ∇ 0 0
2 1
Each of these assumptions is required for an evident reason: (a) ensures that (X,Y)
H
is a well defined ΨDO, (b) prohibits finite time runaway trajectories, and (c) assumes the
existence of one or more solutions to the BC problem: which trajectories starting from
Λ have x as their final coordinate position? Locally varying (t,x) in (c) determines the
0
initial x as a function of (t,x), namely xj = xj(t,x).
0 0 0
Under the hypotheses (a–c) the ~ 0 asymptotic solution of the Cauchy problem
→
(2.1), with initial data Ψ , is
0
Ψ(t,x) = Ψsc(t,x)[1+O(~)], (3.1)
N
φ (t,x) i iπ
Ψsc(t,x) = j exp β (t,x) m (t,x) , t [ T,T]. (3.2)
√ J (t,x) ~ j − 2 j ∈ −
j
Xj=1 | | n o
Here m (t,x) is the Maslov index for trajectory G(τ zj) : τ [0,t] , where zj =
j { | 0 ∈ } 0
(xj(t,x), β (xj(t,x))) Λ . The phase and amplitude functions are
0 ∇ 0 0 ∈ 0
t
β (t,x) = β (xj)+ y(τ zj) x˙(τ zj) (G(τ zj)) dτ , (3.3)
j 0 0 | 0 · | 0 −H | 0
Z0
dx(t xj(cid:2), β (xj)) (cid:3)
J (t,x) = det | 0 ∇ 0 0 , (3.4)
j (cid:18) dxj0 (cid:19)(cid:12)xj0=xj0(t,x)
(cid:12)
t
(cid:12)
φ (t,x) = exp tr (G(τ z(cid:12)j))dτ α (xj). (3.5)
j ∇x∇yH | 0 0 0
(cid:18) Z0 (cid:19)
2 1
The family of operators (X,Y) consistent with the class Tm(R4n) is large. The
H + z
growth index, m, must be 2 or greater so as to include kinetic energy in H. The operators
2 1
(X,Y) need not have a polynomial dependence in momentum Y, nor do they need to be
H
partial differential operators. This generality is essential to the SPS method we employ:
normally, the ΨDO defined by and will not be a finite order partial differential
1 2
H H
7
operator. The phases β (t,x) have a geometric meaning. Define the flow transported
j
H
Lagrangian manifold, Λ G(t)Λ . Then the surfaces y (t,x) = β (t,x) describe the
t 0 j j
≡ ∇
different sheets of Λ . Representation (3.1–3.5) is Theorem 10.5 of [7].
t
In phase space quantum mechanics, the range of physical systems one may describe is
controlled by the allowed functional form of the Weyl symbol Hamiltonian. For reasons
of notational simplicity we presume H is static. Including time dependent Hamiltonians,
as a subsequent generalization, is a straightforward matter. The WKB summary above
indicates that two restrictions on H are necessary.
Assumption A1. The Hamiltonian operator H is semiclassically admissible, specifi-
cally H Tm(R2n),m 2. A2. On finite time intervals [ T,T], H generates unique
∈ + x ≥ −
classical flow, g(t x) C∞([ T,T],R2n). b
| ∈ − x
Assumption A is common to all the Propositions and Theorems of this section and
will not be explicitly cited in them.
Since l and r are linear functions of z, Assumption A1 implies that the composite
functions = H l and = H l H r are in Tm(R4n),m 2. Likewise, Lemma 2
H1 ◦ H2 ◦ − ◦ + z ≥
shows that Assumption A2 guarantees that G(t x,y) C∞([ T,T],R4n) for both and
| ∈ − z H1
flow. The requirement that H Tm(R2n) (cf. Appendix A) means that H = H(x;~)
H2 ∈ + x
has the ~ 0 uniform asymptotic expansion
→
J
H(x;~) = H (x)+ ~jH (x)+O(~J+1). (3.6)
0 j
j=1
X
For many physical systems H(x;~) = H (x). For example, this occurs for a mass m
0
charged particle moving in an external electromagnetic field. The higher order terms
H (x) have no effect on the leading order WKB approximation. So from here on we will
j
identify H with H and omit the 0 subscript.
0
3.1 The Boundary Condition Problem
The assumption A says nothing about the solvability of the BC problem. Rather than
dealing with this by assumption, as was done in the representation (3.1–3.5), we will find
existence proofs and explicit formulas for the BC solutions.
We discuss the BC conditions for the evolutions U(t,x) and ρ(t,x) in tandem. For the
U(t,x) evolution, take the initial state of the system to be
U (x) = N (x)eiΦ0(x)/~, x D R2n (3.7)
0 0 0
∈ ⊆
where the domain, D , is a simply connected open set. The phase Φ defines the single
0 0
sheeted Lagrangian manifold Λ = z χ y(x) = Φ (x),x D . The initial state for
0 2 0 0
{ ∈ | ∇ ∈ }
density matrix evolution ρ(t,x) is presumed to be
ρ (x) = α (x)eiS0(x)/~, x D R2n. (3.8)
0 0 0
∈ ⊆
Again, Λ will denote the simply connected manifold generated by S . The phases Φ and
0 0 0
S are both assumed to be C∞(R2n).
0
8
BC Problem. Determine the trajectories G(t x ,y ) which begin on Λ and have the
0 0 0
|
final position x D = Π Λ .
t 1 t
∈
(i) The Schr¨odinger case: Find the functions x = x (t,x) and y = y (t,x) from
0 0 0 0
1 g(t x 1Jy )+(x + 1Jy ) = x, (3.9)
2 | 0 − 2 0 0 2 0
y = Φ (x ), x D . (3.10)
(cid:2) 0 0 0 0 (cid:3)0
∇ ∈
(ii) The Heisenberg case: Find the functions x = x (t,x) and y = y (t,x) from
0 0 0 0
1 g(t x 1Jy )+g(t x + 1Jy ) = x, (3.11)
2 | 0 − 2 0 | 0 2 0
y = S (x ), x D . (3.12)
(cid:2) 0 0 0 0 0 (cid:3)
∇ ∈
Lemma 2 justifies writing the left sides of (3.9) and (3.11) as the weighted sum of
the left and right flows. It is natural to interpret each of these equations as a midpoint
condition. Conditions (3.10) and (3.12) ensure that (x ,y ) lies on Λ . Observe that
0 0 0
the initial density matrix (3.8) is not real valued like approximation (2.12). However, by
superposition with the complex conjugate of (3.8), the initial state can be made real.
We treat the small time and big time solutions to the BC problem with different
methods and assumptions. The small time problem may be formulated as follows. Notice
that the midpoint and Λ boundary conditions may be combined into a single statement,
0
(x′) = x.
t
M
Definition 1. (i) In the Schr¨odinger problem set
(x′) = M (x′) 1 g(t x′ 1J Φ (x′))+(x′ + 1J Φ (x′)) = x. (3.13)
Mt t ≡ 2 | − 2 ∇ 0 2 ∇ 0
(ii) In the Heisenberg probl(cid:2)em set (cid:3)
f
(x′) = M (x′) 1 g(t x′ 1J S (x′))+g(t x′ + 1J S (x′)) = x. (3.14)
Mt t ≡ 2 | − 2 ∇ 0 | 2 ∇ 0
The initialphase β (x) i(cid:2)n theextended Schr¨odinger problem is equal(cid:3)to either Φ (x) or
0 0
S (x) depending on whether one has case (i) or (ii). Here it is assumed that the support
0
of β (x) is all of R2n, and that x is any point in R2n. If the non-linear function is
0 t
M
invertible, then the BC problem has a single trajectory specified by the initial values,
x (t,x) = −1(x) and y (t,x) = β (x (t,x)). The arguments of the flows g(t) are the
0 t 0 0 0
M ∇
left-right projections of Λ .
0
Proposition 1. Let the Hessians H′′(x) and β′′(x) have x-uniform bounds
0
H′′(x) c < , β′′(x′) 2, (3.15)
k k ≤ 1 ∞ k 0 k ≤
then there exists a t > 0 such that the midpoint map : R2n R2n is a diffeomorphism
1 t
M →
for t [ t ,t ]. Specifically, x (t,x) = −1(x) is a unique C1([ t ,t ],R2n) solution of
∈ − 1 1 0 Mt − 1 1
the BC problems (i) and (ii), respectively.
Proof. Consider the Heisenberg problem. For parameters (t,x) ([ T,T],R2n), define
∈ −
the family of maps T : R2n R2n,
t,x
→
T (x′) x+x′ M (x′).
t,x t
≡ −
9