Uncertainty Relation

Abstract

In the chapter about Fourier and wavelet transforms we saw that all functions, including windows, obey the uncertainty principle, which states that sharp localizations in time and in frequency are mutually exclusive. Roughly speaking, if a nonzero function g of time is small outside a time-interval of length T and its Fourier transform is small outside a frequency band of width Ω, then an inequality of the type ΩT ≥ c must hold for some positive constant c≈1. The precise value of c depends on how the widths T and Ω of the signal in time and frequency are measured. In this chapter we discuss the uncertainty relation (Prugovecki [47], Con-stantinescu and Magyari [15]) in quantum mechanics. In particular we study the uncertainty relation for the momentum and coordinate operators. First we discuss the commutation relation of unbounded operators (Collatz [14]).