1.3. OVERVIEW 5
Quantum evolution. We can quantize the foregoing by putting
P =
pw(x,
hD), A =
aw(x,
hD)
and defining
(1.2.4) A(t) := F
−1(t)AF
(t)
for F (t) :=
e−
itP
h
. The operator A(t) represents, according to the so-called
Heisenberg picture of quantum mechanics, the evolution of the quantum
observable A under the flow (1.2.1). Then we have the evolution equation
(1.2.5) ∂tA(t) =
i
h
[P, A(t)],
an obvious analogue of (1.2.3). Here then is a basic principle we will later
work out in some detail: an assertion about Hamiltonian dynamics, and so
the Poisson bracket {·, ·}, will involve at the quantum level the commutator
[·, ·].
REMARK: h and ¯. h In this book h denotes a dimensionless parameter
and is consequently not immediately to be identified with the dimensional
physical quantity
= Planck’s constant/2π = 1.05457 ×
10−34joule-sec.
As the example of the damped wave equation (1.1.1) shows, the use of h 0
asymptotics is not restricted to problems motivated by quantum mechanics.
1.3. OVERVIEW
Chapters 2–4 develop the basic machinery, followed by applications to partial
differential equations in Chapters 5–7. We develop more advanced theory
and applications in Chapters 8–13, and in Chapters 14 and 15 we discuss
semiclassical analysis on manifolds.
The following diagram indicates the dependencies of the chapters and
may help in selective reading of the book:
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