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: