Abstract
We define a finite-differences derivative operation, on a non uniformly spaced partition, which has the exponential function as an exact eigenvector. We discuss some properties of this operator and we propose a definition for the components of a finite-differences momentum operator. This allows us to perform exact discrete calculations.
Highlights
Works on discrete quantum mechanics [1,2,3,4,5,6] only consider discrete, uniform jumps along some variable and along its conjugate direction
We focus on realizing continuous translations on discrete systems with non uniform separation between the discrete values of a given variable, which we will call the coordinate variable
Which is a translation of the eigenvector ev by an amount u along the coordinate direction. This can be seen as having the same function but the domain has changed, it was translated by the amount u and the function is evaluated at the new points of the domain, a continuous translation [10]
Summary
Works on discrete quantum mechanics [1,2,3,4,5,6] only consider discrete, uniform jumps along some variable and along its conjugate direction. We begin by introducing a first-order finite-differences definition of a derivative operator acting on vectors on a non-uniform mesh. Our definition is based on a simple equality and is characterized by having the exponential function as an exact eigenfunction. We will consider a mesh M( N ) := {qi }1N of N non spaced points qi on the finite interval [ a, b] ∈ R, a < b < ∞, with separations ∆ j = q j+1 − q j between them. The nice property of these finite-differences operators is that the exponential function is an exact eigenvector of them. We will discuss some additional properties of these operators
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