The framework of quantum mechanics rests on three pillars: the Hilbert space of quantum states; the Hermitian operators, also called observables; and the unitary evolution operators. I start by trying to attach some pictures to these abstractions.

The simplest classical system consists of a single point particle coasting along in space (perhaps subject to a force field). To ``quantize'' this, you'll need the Hilbert space of complex-valued functions on , and you'll encounter unbounded operators on this space. So goes the tale of history: Heisenberg, Schrödinger, Dirac and company cut their milk teeth on this problem.

I will take an ahistorical but mathematically gentler approach. The
Hilbert space for a two-state quantum system is , and the
operators can all be represented as complex matrices. The spin
of an electron provides a physical example. That is, if we simply ignore
position and momentum (``mod them out'', so to speak), we have a physical
picture that can be modelled by this (relatively) simple framework. (As
noted before, Feynman's *Lectures*, volume III, starts off like this.)

© 2001 Michael Weiss