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Back to the Schrödinger equation on a lattice

Efstratios Manousakis

Published Nov 1, 2015• 0 citations
Publisher

Abstract

Abstract This chapter goes back to the discrete lattice in order to transform the basis to momentum eigenstates. In addition, the chapter takes the continuum limit of space and we recover known relations and the delta function.
DOI
10.1093/acprof:oso/9780198749349.003.0003

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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.
Abstract This paper presents a numerical method for solving Schrödinger equations with arbitrary potentials with the adoption of the qunatum diagonalization scheme and basis functions in the discrete wavelet transform. The quantum diagonalization scheme, developed to solve the Hubbard model and so on, is known to allow the diagonalization of a Hamiltonian matrix of a much large size because it does not require to maintain the entire matrix in memory. And, the adoption of wavelet basis functions enables representing eigenstates efficiently using a number of bases much smaller than the number of grid points to integral and reducing the computation time to perform the diagonalization process. With the help of another additional techniques, the method allows to find solutions of a Schrödinger equation much exactly and efficiently. The validity of the method and its efficiency depending on several conditions are proven and surveyed by applying it to several problems for which the exact solutions are known.
This paper provides a rigorous mathematical derivation demonstrating how discrete multiway state updates on a hypergraph with long-range spatial or branchial shortcuts converge to the continuous Fractional Schrödinger Equation utilized in the Franquero Entropy Framework (FEF). By analyzing a discrete master equation on a multiway graph, I show that when non-local causal updates are introduced, the transition probability distribution develops heavy algebraic tails characterized by the power law W(r) ~ [C_(α,d) * a^α] / |r|^(d + α). Taking the spatial/branchial Fourier transform and applying the continuous time and space limit (a → 0, τ → 0), the discrete hypergraph parameters must satisfy the fractional scaling relation D_α = lim(a→0, τ→0) [A_α * a^α] / τ to obtain a finite continuum limit. Applying the inverse Fourier transform converts the spectral power -|k|^α back to the continuous spatial domain via Riesz fractional derivatives, (-∇²)^(α/2), recovering the non-unitary FEF Time-Dependent Schrödinger Equation: iħ ∂ψ(x,t)/∂t = [ D_α(-ħ²∇²)^(α/2) + V(x) - iħ Γ_ent(ψ) ] ψ(x,t) This mapping establishes a direct physical interpretation between discrete rulial systems and the FEF, where long-range hypergraph rewrites generate fractional dispersion (driving a +3.27° angular deflection shift at α = 1.6), multiway causal bottlenecking emerges as entropic damping, and branchial path divergence physically manifests as fringe visibility loss.