In this paper we prove a character formula expressing the classes of simple representations in principal block simply-connected semisimple algebraic group G terms baby Verma modules, under assumption that characteristic base field is bigger than 2h-1, where h Coxeter number G. This provides replacement for Lusztig’s conjecture, valid reasonable on characteristic.

New applications of the formula A|ψ〉 = 〈A〉|ψ〉 +∆A|ψ⊥〉 are discussed. Simple derivations of the Heisenberg uncertainty principle and of related inequalities are presented. In addition, the formula is used in an instructive paradox which clarifies a fundamental notion in quantum mechanics.

Abstract We employ the Markov approximation and well-known Fresnel-
integral to derive in "one-line" familiar expression for Landau-Zener
transition probability. Moreover, we provide numerical as well analytical
justifications our approach, identify three characteristic motions of the
probability amplitude complex plane.