Operator formalism of quantum mechanics.pdf

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Operator formalism of quantum mechanics

a r X i v : m a t h - p h / 0 0 1 2 0 5 1 v 1 2 9 D e c 2 0 0 0 Operator formalism of quantum mechanics Jan Naudts Departement Natuurkunde, Universiteit Antwerpen UIA, Universiteitsplein 1, 2610 Antwerpen, Belgium E-mail: Jan.Naudts@ua.ac.be December 25, 2000 Abstract This is the first chapter of a new and unconventional textbook on quantum mechanics and quantum field theory. The chapter introduces standard quantum mechanics by means of a symmetry principle, with- out reference to classical mechanics. The mathematical foundation of this approach comes from a recent paper of Naudts and Kuna on co- variance systems. The standard representation of quantum mechanics is derived. Next, spin and mass of a quantum particle are explained as labels of projective representations of the Galilei group. Contents 1 Classical algebra of functions of position 3 1.1 A no-go statement . . . . . . . . . . . . . . . . . . . . 3 1.2 Pure and mixed states . . . . . . . . . . . . . . . . . . 3 1.3 Fourier decomposition and convolution algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Symmetries of physical state space 6 2.1 Covariant representations . . . . . . . . . . . . . . . . 6 2.2 The postulate of states . . . . . . . . . . . . . . . . . . 9 2.3 Irreducibility . . . . . . . . . . . . . . . . . . . . . . . 12 1 3 Standard representation of quantum mechanics 13 3.1 Construction of a covariant representation . . . . . . . 13 3.2 Physical states . . . . . . . . . . . . . . . . . . . . . . 15 3.3 Characteristic function of a physical state . . . . . . . 16 3.4 Generators of the group of shifts . . . . . . . . . . . . 17 3.5 Von Neumann uniqueness theorem . . . . . . . . . . . 19 4 Rotation symmetry 22 4.1 Group of shifts and rotations . . . . . . . . . . . . . . 22 4.2 Path depending representations . . . . . . . . . . . . . 24 4.3 The universal covering group . . . . . . . . . . . . . . 24 4.4 A projective representation of SO(3) . . . . . . . . . . 26 4.5 A spinor r

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