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Books > Science & Mathematics > Physics > Quantum physics (quantum mechanics)
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Introduction To The Mathematical Structure Of Quantum Mechanics, An: A Short Course For Mathematicians (2nd Edition) (Hardcover, 2nd ed.)
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Introduction To The Mathematical Structure Of Quantum Mechanics, An: A Short Course For Mathematicians (2nd Edition) (Hardcover, 2nd ed.)
Series: Advanced Series In Mathematical Physics, 28
Expected to ship within 12 - 17 working days
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The second printing contains a critical discussion of Dirac
derivation of canonical quantization, which is instead deduced from
general geometric structures. This book arises out of the need for
Quantum Mechanics (QM) to be part of the common education of
mathematics students. The mathematical structure of QM is
formulated in terms of the C*-algebra of observables, which is
argued on the basis of the operational definition of measurements
and the duality between states and observables, for a general
physical system.The Dirac-von Neumann axioms are then derived. The
description of states and observables as Hilbert space vectors and
operators follows from the GNS and Gelfand-Naimark Theorems. The
experimental existence of complementary observables for atomic
systems is shown to imply the noncommutativity of the observable
algebra, the distinctive feature of QM; for finite degrees of
freedom, the Weyl algebra codifies the experimental complementarity
of position and momentum (Heisenberg commutation relations) and
Schroedinger QM follows from the von Neumann uniqueness theorem.The
existence problem of the dynamics is related to the
self-adjointness of the Hamiltonian and solved by the Kato-Rellich
conditions on the potential, which also guarantee quantum stability
for classically unbounded-below Hamiltonians. Examples are
discussed which include the explanation of the discreteness of the
atomic spectra.Because of the increasing interest in the relation
between QM and stochastic processes, a final chapter is devoted to
the functional integral approach (Feynman-Kac formula), to the
formulation in terms of ground state correlations (the quantum
mechanical analog of the Wightman functions) and their analytic
continuation to imaginary time (Euclidean QM). The quantum particle
on a circle is discussed in detail, as an example of the interplay
between topology and functional integral, leading to the emergence
of superselection rules and sectors.
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