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arXiv:quant-ph/9811048

Schroedinger equation as the universal continuum limit of nonrelativistic coherent hopping on a cubic spatial lattice

L. Polley

Published Nov 18, 1998• 0 citations
arXivHTMLPDF

Abstract

The Schroedinger equation with scalar and vector potentials is the continuum limit of any coherent hopping process (where position eigenstates superpose with neighbouring eigenstates after a time step) whose hopping amplitudes are homogeneous in quadratic order of the inverse lattice spacing, inhomogeneous in first order, and satisfying a summability condition with respect to higher-than-next neighbours.

DOI
10.48550/arxiv.quant-ph/9811048

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The Schroedinger equation with scalar and vector potentials is the continuum limit of any coherent hopping process (where position eigenstates superpose with neighbouring eigenstates after a time step) whose hopping amplitudes are homogeneous in quadratic order of the inverse lattice spacing, inhomogeneous in first order, and satisfying a summability condition with respect to higher-than-next neighbours.
Abstract This chapter begins by studying the one-dimensional time-dependent Schrödinger equation. In order to do this the chapter discretizes the spatial coordinates, and thus the differential equation becomes a set of coupled difference equations first order in time. Dealing with a finite number or a numerable set of states makes it easier to understand the concepts of Hilbert space, its measure, and the overlap between states. Furthermore, the chapter only has to deal with the Kronecker symbol as opposed to the Dirac delta function. Later, in Chapter 3, it will consider the continuum limit where we will naturally recover the Schrödinger differential equation, the state normalization condition as an integration condition, and the delta function.
The quantum state of a particle can be completely specified by a position at one instant of time. This implies a lack of information, hence a symmetry, as to where the particle will move. We here study the consequences for free particles of spin 0 and spin 1/2. On a cubic spatial lattice a hopping equation is derived, and the continuum limit taken. Spin 0 leads to the Schroedinger equation, and spin 1/2 to the Weyl equation. Both Hamiltonians are hermitian automatically, if time-reversal symmetry is assumed. Hopping amplitudes with a "slight" inhomogeneity lead to the Weyl equation in a metric-affine space-time.
The Schrödinger equation is derived classically assuming that particles present local \nrandom spatial fluctuations compatible with the presence of the zero-point field. With- \nout specifying the forces arising from this permanent matter-field interaction but ex- \nploring its fundamental properties (homogeneity, isotropy and random aspect) to justify \nthe emergence of the continuity equation in one-particle context, these fluctuations are \ndescribed in terms of the probability density. Specifically, the starting point is the as- \nsumption that the local activities, which turn the path followed by the particle totally \nunpredictable, must be associated with an energy proportional to \n@ \nP \n=@ \nt \n. The polar form \nof the wave function, which connects the obtained classical equations with the corre- \nsponding quantum equation, emerges as a by-product of the approach.
Schroedinger's equation with scalar and vector potentials is shown to describe "nothing but" hopping of a quantum particle on a lattice; any spatial variation of the hopping amplitudes acts like an external electric and/or magnetic field. The main point of the argument is the superposition principle for state vectors; Lagrangians, path integrals, or classical Hamiltonians are not (!) required. Analogously, the Hamiltonian of the free electromagnetic field is obtained as a twofold continuum limit of unitary hopping in Z(N) link configuration space, if gauge invariance and C and P symmetries are imposed.