First Axiom

Axiom 1: The wave function $\Psi({\bf x},t)$ for a particle with mass $m$ moving in a potential $V({\bf x})$ obeys the Schrödinger equation

$$i\hbar\frac{\partial}{\partial t} \Psi({\bf x},t) =\left[-\frac{\hbar^2\Delta}{2m} + V({\bf x}) \right]\Psi({\bf x},t).$$ (22)

$\vert\Psi({\bf x},t)\vert^2d^3x$ is the probability for the particle to be in the (infinitesimal small) volume $d^3x$ around ${\bf x}$ at time t. The probability $P(\Omega)$ for the particle to be in a finite volume $\Omega$ of space is given by the integral over this volume:

$$P(\Omega)=\int_{\Omega}d^3x \vert\Psi({\bf x},t)\vert^2.$$ (23)

The probability to find the particle somewhere in space must be one and hence

$$\int_{R^3}d^3x \vert\Psi({\bf x},t)\vert^2=1.$$ (24)

Remarks:

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 $V({\bf x})$ is included, which is assumed to be a given function of ${\bf x}$. 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 $\vert\Psi\vert^2$ as a probability density. $\Psi$ must be square integrable. Functions $\Psi$ that are square integrable belong to an infinite dimensional vector space of functions, the Hilbert space $L^2(R^3)$. The Hilbert space is a central object in the mathematical theory of quantum mechanics. Basically, it replaces the phase space of points $(x,p)$ from classical mechanics.