ABSTRACT
In this chapter we take up systems whose state spaces form a continuum. These are, naturally, infinite dimensional vector spaces (often called Hilbert spaces). Unfortunately, we could not do any better than treating such systems by formal analogy with their finite dimensional counterparts. A rigorous treatment would require a background in functional analysis and is beyond the scope of the present work. Thus, we essentially lay out a dictionary for translating the mathematical ingredients and formulae, from discrete to continuum descriptions, motivated by plausibility arguments. In particular, we argue why states are now described by complex square integrable functions (called wave functions) and observables by Hermitian differential operators. We then move on to show how to quantize quantum systems with classical analogues, which comprise the bulk of real life physical systems. Here, we explicitly show how the most nontrivial component of the quantization, i.e., the description of the representation of the momentum operator (in position space) can be deduced from both, the de Broglie hypothesis, and the canonical commutation relation. The “particle in a box” problem is treated as an illustration. The parity operator, Ehrenfest's theorem and continuity conditions have been relegated to the problem section.
