Quantum Mechanics: Two-state Systems

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 $L^2$ functions on ${\bf R}^3$, 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 ${\bf C}^2$, and the operators can all be represented as $2\times 2$ 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

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