Mathematical Concepts of Quantum Mechanics

Abstract

In this chapter, we derive a convenient representation for the integral kernel of the Schrodinger evolution operator, e-itH / n . This representation, the "Feynman path integral" , will provide us with a heuristic but effective tool for investigating the connection between quantum and classical mechanics. This investigation will be undertaken in the next section. The Feynman Path IntegralConsider a particle in IRd described by a self-adjoint Schrodinger operatorRecall that the dynamics of such a particle is given by the Schrodinger equation . 8'IjJRecall also that the solution to this equation, with the initial condition is given in terms of the evolution operator U(t):= e-iHt / n asOur goal in this section is to understand the evolution operator U(t) = e-iHt / n by finding a convenient representation of its integral kernel. We denote the integral kernel of U (t) by Ut (y, x) (also called the propagator from x to y). A representation of the exponential of a sum of operators is provided by the Trotter product formula (Theorem 10.2) which is explained in Section 10.3 at the end of this chapter. The Trotter product formula says that S. J. Gustafson et al., Mathematical Concepts of Quantum Mechanics

On Threshold Eigenvalues and Resonances for the Linearized NLS Equation

“…[15], also [2]) for the Birman-Schwinger operator K α, ε is the operator F : RanP 0 → RanP 0 , such that…”

“…Since the Feshbach operator has the isolated eigenvalue −1 of multiplicity N as well as the BirmanSchwinger operator (see p.208 of [15], [2]) we have the spectral problem…”

Decoherence and Thermalization

“…In this paper we consider the decoherence phenomenon for quite general non-solvable models. Our analysis is based on the modern theory of resonances for quantum statistical systems as developed in [8][9][10][11][12][13][14][15] (see also the book [16]), which is related to resonance theory in non-relativistic quantum electrodynamics [9,17].…”

A general treatment of deformation effects in Hamiltonians for inhomogeneous crystalline materials

“…This is the largest space for which the deformed Hamiltonian is defined. 15 The deformation has, according to the inverse function theorem, a twice differentiable inverse. This ensures that H can be written with respect to coordinates on B, as it was done in Eq.…”

“…The Lebesgue space is needed here because of the definition of the Sobolev space. 15 It is also the appropriate space with respect to wave functions, because these give the probability that an electron is in a certain area, and, as a consequence, they only need to be defined almost everywhere, i.e., up to sets of zero measure. The spaces…”