A variation principle, similar to the Delves principle, is given for the (real) wave function itself. The principle does not give a bound. The relationship between both principles is discussed. — After proper generalization a variation principle for arbitrary probability amplitudes (or generalized Fourier coefficients) is obtained.

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General Principles

Yu. I. Ostrovsky, Mikhail M. Butusov, G. V. Ostrovskaya

Published Jan 1, 1980• 0 citations

Publisher

DOI: 10.1007/978-3-540-39008-4_1

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\int_{-\infty}^\infty |f(x)|^2w_1(x)dx

, (

w_1(x)

is a non-negative function). 2) The weighted frequency spreading

\int_{-\infty}^\infty |\hat{f}(ω)|^2w_2(ω)dω

. 3) The time weight scale

a

{w_1}_a(x)=w_1(xa^{-1})

and 4) The frequency weight scale

b

{w_2}_b(ω)=w_2(ωb^{-1})

. "Generalized Uncertainty Principle" is an inequality that summarizes the constraints on the relations between the two spreading quantities and two scaling parameters. For any two reasonable weights

w_1(x)

and

w_2(ω)

, we introduced a three dimensional set in

R^3

that is in the essence of many uncertainty principles. The set is called "possibility body". We showed that classical uncertainty principles (such as the Heiseneberg-Pauli-Weyl uncertainty principle) stem from lower bounds for different functions defined on the possibility body. We investigated qualitative properties of general uncertainty principles and possibility bodies. Using this approach we derived new (quantitative) uncertainty principles for Landau-Pollak-Slepian weights. We found the general uncertainty principles related to homogeneous weights,

w_1(x)=w_2(x)=x^k

k\in N

, up to a constant.