Abstract
The Du Fort–Frankel scheme for the one-dimensional Schrödinger equation is shown to be equivalent, under a time-dependent unitary transformation, to the Ablowitz–Kruskal–Ladik scheme for the Klein–Gordon equation. The Schrödinger equation describes a non-relativistic quantum particle, while the Klein–Gordon equation describes a relativistic particle. The conditional convergence of the Du Fort–Frankel scheme to solutions of the Schrödinger equation arises because solutions of the Klein–Gordon equation only approximate solutions of the Schrödinger equation in the non-relativistic limit. The time-dependent unitary transformation is the discrete analog of the transformation that arises from seeking a non-relativistic limit using the interaction picture of quantum mechanics to decompose the Klein–Gordon Hamiltonian into the relativistic rest energy and a remainder. The Ablowitz–Kruskal–Ladik scheme is in turn decomposed into a quantum lattice gas automaton for the one-dimensional Dirac equation, which is also the one-dimensional discrete time quantum walk. This relativistic interpretation clarifies the origin of the known discrete invariant of the Du Fort–Frankel scheme as expressing conservation of probability for the 2-component wavefunction in the one-dimensional Dirac equation under discrete unitary evolution. It also leads to a second invariant, the matrix element of the evolution operator, whose imaginary part gives a discrete approximation to the expectation of the non-relativistic Schrödinger Hamiltonian.
Highlights
The Schrödinger equation for a particle of mass m in a potential V is ih ∂t ψ = − h 2 2m ∇2ψ + V (x)ψ, (1)where his the reduced Planck’s constant [1,2,3]
We can construct a second discrete invariant related to the quantum-mechanical energy, the expectation of the Hamiltonian operator [31]
The du Fort–Frankel scheme will converge to solutions of the Schrödinger equation if one rescales both time and the particle mass with increasing resolution N
Summary
Where his the reduced Planck’s constant [1,2,3]. This equation describes the evolution of a wavefunction ψ(x, t), with the interpretation that |ψ(x, t)|2 is the probability density for the particle to be located at position x at time t. To construct a unitary and time-reversible approximation to exp(−i tH/h ) for the evolution between two time levels, given concretely by ihψ n+1 j It is implicit, because each timestep requires the solution of a tridiagonal linear system involving the operator to determine the ψ n+1 j at all grid points j. The same asymmetry appears in the stencil for the Crank–Nicholson scheme, which contains three spatial points at each of two time levels This asymmetry between space and time makes the Schrödinger equation incompatible with special relativity, which requires an equation to be invariant under Lorentz transformations that mix the space and time coordinates. Space and time appear symmetrically in the four-point stencil for the Du Fort–Frankel scheme shown in
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