Abstract
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).
The axiomatic formulation of Quantum Mechanics is generally presented as a set of four postulates, introduced by John von Neumann. The physical meaning of each of them requires a nontrivial, careful enquiry.
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Cini, M. (2018). The Postulates of Quantum Mechanics: Postulate 1. In: Elements of Classical and Quantum Physics. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-71330-4_12
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DOI: https://doi.org/10.1007/978-3-319-71330-4_12
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