Generalized uncertainty relations and coherent and squeezed states

Abstract

Characteristic uncertainty relations and their related squeezed states are briefly reviewed and compared in accordance with the generalizations of three equivalent definitions of the canonical coherent states. The standard SU(1,1) coherent states are shown to be the unique states that minimize the Schrödinger uncertainty relation for every pair of the three generators and the Robertson relation for the three generators. The characteristic uncertainty inequalities are naturally extended to the case of several states. It is shown that these inequalities can be written in the equivalent complementary form.OSIC codes: 270.6570, 270.0270.

Generalizations of Heisenberg uncertainty relation

“…This family can be defined [8] as the unique set of states that minimize inequality (7). It was only recently realized [7], that the famous canonical coherent states (introduced in [5]) can be uniquely defined as states that minimize (4).…”

“…In section 3.3 a general scheme for construction of UR's for n observables and m states is provided. The relation of the conventional inequalities to the sets of the widely used canonical coherent states [5] and squeezed states [6,7] is also reminded.…”

Quantum mechanics and the generalized uncertainty principle

“…See also Refs. [27,28,29,30,31,32,33]. Our emphasis is different, however, in that we are primarily concerned with the properties of the uncertainty relations for a compactified phase space in the large radius limit.…”

On the position uncertainty measure on the circle

“…The counterpart of this inequality on the real line is ∆ 2 x + ∆ 2 p x ≥ 1, which is minimized in the canonical CS |α only [11]. There are no periodic wave functions on the circle, that precisely minimize (20).…”