ABSTRACT
Chapter 4
The Quantum Plane
4.1 Position and momentum
We now have the necessary machinery to treat what is probably the
most fundamental example in quantum mechanics: the one-dimensional
particle. Its classical analog was mentioned in Section 1.1, and there we
pointed out that the phase space of this classical system can be identied
with the real plane R
. The phase space of the corresponding quantum
system is modelled on the Hilbert space L
(R), and this space, together
with some associated structure, plays the role of a \quantum" plane. As
we will see in this and later chapters, the complex, topological, measure
theoretic, metric, and dierentiable structures of the ordinary plane all
have quantum analogs. In this section we introduce \coordinates" and
\translations" on the quantum plane.
