Axiom 1: The wave function
for a particle with mass
moving in a potential
obeys the
Schrödinger equation
1. In formulating this axiom, we already made an abstract assumption of one and only one particle that can be somewhere in space. Only the interaction with the potential is included, which is assumed to be a given function of . This potential can be created by, e.g., electric fields and therewith indirectly by the interactions with other particles which are, however, are not included explicitly.
2. There are no relativistic effects included here.
3. The normalization condition (1.24) is necessary for an interpretation of as a probability density. must be square integrable. Functions that are square integrable belong to an infinite dimensional vector space of functions, the Hilbert space . The Hilbert space is a central object in the mathematical theory of quantum mechanics. Basically, it replaces the phase space of points from classical mechanics.