The Postulates of Quantum Mechanics: Postulate 1

The system referred to above could be a particle, an atom, or even a macroscopic superconductor (then, x stands for a very large set of coordinates), so the statement is quite strong and general. In any case, all the information that is available from all possible experiments is in \(\varPsi _{a}(x, t) \). The wave function must be taken to be normalized. In the case of a single degree of freedom \(\varPsi _{a}(x, t), \) the normalization condition reads as \(\int |\varPsi (x, t)|^{2} dx=1\), while in general, one must integrate the square modulus over all the variables. The function is complex, therefore \( \varPsi (x,t)=| \varPsi (x,t)| e^{i\phi (x, t)}\), where \(\phi (x, t)\) is the phase. One can change \(\phi (x, t)\) by a constant phase factor (for instance, multiplying \( \varPsi (x, t)\) by i) but the physical state remains the same; nevertheless, the phase difference between two wave functions does matter a lot, since the wave functions do interfere. We shall see that the phase can be changed in several ways (e.g. rotations, Galileo transformations, gauge changes).