## Abstract

We introduce a finite difference derivative, on a non-uniform partition, with the characteristic that the derivative of the exponential function is the exponential function itself, times a constant, which is similar to what happens in the continuous variable case. Aside from its application to perform numerical computations, this is particularly useful in defining a quantum mechanical discrete momentum operator.

### Keywords

- exact finite differences derivative
- discrete quantum mechanical momentum operator
- time operator

## 1. Introduction

Even though the calculus of finite differences is an interesting subject on its own [1, 2, 3, 4] that scheme is mainly used to perform numerical computations with the help of a computer. Finite differences methods give approximate expressions for operators like the derivative or the integral of functions, and it is expected that we get a good approximation when the separation between the points of the partition is small; the smaller it becomes the better.

The momentum operator of Quantum Mechanics, when considering continuous variables, is related to the derivative of functions, but its form, when the variable takes discrete values, is not known yet (an approach is found in Ref. [5]); we need an exact expression for the momentum operator in discrete Quantum Mechanics. Thus, to have an expression for the quantum mechanical momentum operator on a mesh of points, we need an exact expression for the derivate on a mesh of points. In this chapter, we intend to modify the usual finite differences definition of the derivative on a partition to propose an operator that can be used as a momentum operator for discrete Quantum Mechanics.

## 2. Exact first-order finite differences derivatives of functions

In this section, we intend to introduce a finite differences derivative, which has the same eigenfunction as for the continuous variable case. We start with results valid for any function, but we will concentrate, later in the chapter, on the exponential function because that function is used to perform translations along several directions in the quantum realm. The resulting derivative operator will depend on the point at which it is evaluated as well as on the partition of the interval and on the function of interest. This is the trade-off for having exact finite differences derivatives.

### 2.1. Backward and forward finite differences derivatives

An exact, backward, finite differences derivative of an absolutely continuous function

where

This is an expression which is valid for points

A definition for forward finite differences at

where

valid for points different from the zeroes of

These definitions coincide with the usual finite differences derivative when the function to which they act on is the linear function

**Example.** For the quadratic function

where

In the remaining part of this chapter, we only consider the derivative of the exponential function; this choice fixes the form of the spacing functions

## 3. Exact first-order finite differences derivative for the exponential function

Let us consider the exact backward and forward finite differences derivatives of

where

and

Note that we recover the usual definitions of a finite differences derivative in the limit

with

*The summation of a derivative.* As is the case for continuous systems, the summation is the inverse operation to the derivative,

where

where

*The exponential function is also an eigenfunction of the summation operation.* The usual integral of the exponential function also has its equivalent expression in exact finite differences terms

where

where

*Chain rule.* The finite differences versions of the chain rule are

where

where

and

*The derivative of a product of functions.* The exact finite differences derivative of a product of functions is

where

where

*The derivative of the ratio of two functions.* For the finite differences, backward derivative of the ratio of two functions we have

*Additional properties.* A couple of equalities that will be needed below are

For instance, these equalities imply that

*Summation by parts.* An important result is the summation by parts. The sum of equalities (18) and (19) combined with equalities (10) and (11) provide the exact finite differences summation by parts results,

where

where

The integration by parts theorem of continuous functions is the basis that allows to define adjoint, symmetric and self-adjoint operations for continuous variables [8, 9]. Therefore, the summation by parts results can be used in the finding of an appropriate momentum operator for discrete quantum systems. The summation by parts relates two operators between themselves and with boundary conditions on the functions.

## 4. The matrix associated to the exact finite differences derivative

It is advantageous to use a matrix to represent the finite differences derivative on the whole interval so that we can consider the whole set of derivatives on the partition at once. Let us consider the backward and forward exact finite differences derivative matrices

and

We have used the definition for the backward derivative

The matrix formulation of the derivative operators allows the derivation of some useful results for the derivative itself.

### 4.1. Higher order derivatives

Many properties can be obtained with the help of the derivative matrices

These expressions have the exponential function

The derivative matrices are singular, which means that they do not have an inverse matrix, but, at a local level, the inverse operator to the derivative is the summation, as we have already shown in a previous section.

### 4.2. Eigenfunctions and eigenvectors of D b , f

Now that we have the matrices

On one hand, for the backward finite difference matrix

whose roots are

with

where

where

On the other hand, for the forward finite difference matrix,

Thus, the eigenvalues for the forward derivative are

where

and

The matrices, *v*=0). This is the same eigenfunction that is found in the continuous variable case because the exponential function is indeed an eigenfunction of the continuous derivative. We note that the local derivatives

### 4.3. The commutator between coordinate and derivative

Since the following equality holds:

from a local point of view, we have

and then, the commutator between

Thus, the commutator between

We now consider the commutator between the coordinate matrix

The small

There is coincidence with the local calculation; as expected, this matrix approaches the identity matrix in the small *backward translations* of the first *forward translation* of the point

### 4.4. Translations

It is well known that the derivative is the generator of the translations of its domain [8]. Therefore, here we investigate briefly how translations are carried out by means of the derivative matrices

Let the linear transformation represented by the matrix formed by means of the standard definition of a translation operator and of the exponential operator, given by

where

that is,

The usual periodic, discrete translation found in the papers of other authors [6, 7] is obtained when the separation between the partition points is the same (denoted by

### 4.5. Fourier transforms between coordinate and derivative representations

In this section, we define continuous and discrete Fourier transforms and establish some of their properties regarding the Fourier transform of continuous and discrete derivatives. The derivative eigenvalue

Given a function

is the continuous Fourier transform of

Now, the discrete derivative of the product

The summation of this equality, with weights

or

According to Eqs. (10), (24), (47) and (48), this equality can be rewritten in terms of discrete Fourier transforms.

Another expression for the finite differences of the derivative of a function is obtained as follows. Considering the relationship (see Eq. (18), the second expression with

The summation of this equality, with weights

and, according to Eq. (10), this equality can be rewritten as the discrete Fourier transform

These are the equivalent to the well-known identities found in continuous variables theory. Thus, the multiplication by

The integration by parts of the simple relationship

results in

or in terms of continuous Fourier transforms,

These equalities are like the usual properties between the spaces related by the Fourier transform.

## 5. Quantum mechanical momentum and time operators

We can apply the results of previous sections to discrete Quantum Mechanics theory. Let us rewrite Eq. (26) in terms of complex wave functions

This equality is rewritten as

where the momentum-like operators

and the bilinear forms

We recognize Eqs. (60) and (61) as the finite differences versions of the equation that is used to define the adjoint operator and the symmetry of an operator in continuous Quantum Mechanics. Thus, we propose that the momentum-like operators

together with the boundary condition on the wave functions

where

With these definitions, we are closer to have a finite differences version of a self-adjoint momentum operator on an interval [12, 13] for use in discrete Quantum Mechanics. We believe that our results will lead to a sound definition of a discrete momentum operator and to the finding of a time operator in Quantum Mechanics [10, 11, 12, 13].

## 6. The particle in a linear potential

As an application of the ideas presented in this chapter, we consider the particle under the influence of the linear potential

where

where Ai denotes the Airy function and

where

In this case, the energy values are discrete and non-uniformly spaced, and the operator conjugate to the Hamiltonian would be a time-type operator with a discrete derivative

where

In conclusion, we can have an exact derivative without the need of many terms, and this allows for the definition of adjoint operators related to the derivative on a mesh of points.