Elements of Classical and Quantum Physics by Michele Cini

Author:Michele Cini
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

This is commonly referred to as the expectation value, even if everybody understands that this is a misnomer. For instance, a probability distribution symmetric around could vanish right there, so would be unattainable but still the expectation value.

The average position at time t is

which implies

As in every statistical treatment, the width of the distribution is given by the standard deviation defined by

One finds that

where

The wave function is complex and can be taken to be real only in special cases. This is far from obvious since the measurements always yield real results. However we have seen that the De Broglie wave is complex, and is the eigenfunction of momentum, which is itself a complex operator. We shall see that the phase is crucial for angular momentum eigenstates, for the existence of a current, for enabling a change of gauge and a change of reference in a Galileo transformation; later on, we shall discover more subtle ways the phase is necessary (topological phase). On the other hand, the wave function, like the electromagnetic potentials, is not observable.

In simple systems like a particle in a quantum well, one measurement of energy can tell us which one is the state. In general, the task of labelling a system with a quantum state can be demanding. Complex systems with several degrees of freedom may require a series of measurements. How many quantities and which quantities are needed? This complicated issue is studied by Quantum Tomography.

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