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Birkhäuser Basel eBooks

Spectrum and Quantum Dynamics

Published Jan 1, 2009• 0 citations
Publisher

Abstract

Different spectral subspaces of a self-adjoint operator T in general entail different behaviors of the unitary evolution group e −itT (particularly as | t |→∞). In this chapter many such dynamical issues are discussed; the main motivation is when T corresponds to the Schrödinger operator of a quantum system. Some related physical concepts, such as quantum return probability and test operators, are used to probe the large-time behaviors. The cornerstones of such results are the concepts of precompactness, almost periodicity and the Wiener and Riemann-Lebesgue lemmas. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-3-7643-8795-2_14

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Different spectral subspaces of a self-adjoint operator T in general entail different behaviors of the unitary evolution group e −itT (particularly as | t |→∞). In this chapter many such dynamical issues are discussed; the main motivation is when T corresponds to the Schrödinger operator of a quantum system. Some related physical concepts, such as quantum return probability and test operators, are used to probe the large-time behaviors. The cornerstones of such results are the concepts of precompactness, almost periodicity and the Wiener and Riemann-Lebesgue lemmas. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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The authors present briefly the basic notions of the quantum formalism: pure and mixed states; quantum observables; quantum probability; Born’s rule, superposition, and ‘state collapse’; the projection postulate (von Neumann–Lüders postulate); Dirac’s ket and bra-vector notations; elements of quantum information theory and quantum logic; Schrödinger’s and von Neumann’s equations; unitary dynamics; and positive operator valued measures. They also define the basic mathematical notions related to the quantum formalism: Hilbert space; scalar product; norm; Hermitian operator (matrix); projector; and unitary operator and adjoint operator. This chapter may be useful for newcomers to the field, but for those readers who have preliminary knowledge about quantum mechanics (QM) they can proceed directly to advanced chapters. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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