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 g xð Þ (this class of functions is the domain of the momentum operator in Quantum Mechanics), on a partition P ¼ f g x1; ; x2; ⋯; xN of N non-uniformly spaced points xj � �<sup>N</sup> <sup>1</sup> , is defined through the requirement that

$$g(D\_b g)(\mathbf{x}\_j) \coloneqq \frac{g(\mathbf{x}\_j) - g(\mathbf{x}\_j - \Delta\_{j-1})}{\chi\_2(j-1)} = g'(\mathbf{x}\_j),\tag{1}$$

Example. For the quadratic function g xð Þ¼ <sup>a</sup><sup>0</sup> <sup>þ</sup> <sup>a</sup>1<sup>x</sup> <sup>þ</sup> <sup>a</sup>2x2, <sup>a</sup>0, a1, a<sup>2</sup> <sup>∈</sup> <sup>C</sup>, the spacing function

In the remaining part of this chapter, we only consider the derivative of the exponential

Let us consider the exact backward and forward finite differences derivatives of ev x, at xj,

where v∈ C can be a pure real or pure imaginary constant, and the spacing functions χ1ð Þ v; j

ffi Δ<sup>j</sup> þ v 2 Δ2

v x

<sup>j</sup> <sup>≔</sup> <sup>e</sup>v xjþ<sup>1</sup> � <sup>e</sup>v xj

<sup>j</sup> <sup>þ</sup> <sup>O</sup> <sup>Δ</sup><sup>3</sup> j

<sup>χ</sup>1ð Þ <sup>v</sup>; <sup>j</sup> <sup>¼</sup> v ev xj

, (7)

, (6)

2

, (5)

Exact Finite Differences for Quantum Mechanics http://dx.doi.org/10.5772/intechopen.71956 165

a1 <sup>a</sup><sup>2</sup> þ 2x

<sup>χ</sup>2ð Þ¼ <sup>x</sup>; <sup>j</sup> <sup>Δ</sup><sup>j</sup> � <sup>Δ</sup><sup>j</sup>

Figure 1. Three-dimensional plot of <sup>χ</sup>2ð Þ <sup>x</sup>; <sup>j</sup> for the quadratic function g xð Þ¼ <sup>a</sup><sup>0</sup> <sup>þ</sup> <sup>a</sup>1<sup>x</sup> <sup>þ</sup> <sup>a</sup>2x<sup>2</sup> with <sup>Δ</sup><sup>j</sup> <sup>¼</sup> 1.

where x 6¼ �a1=2a2. A plot of this function is shown in Figure 1 for Δ<sup>j</sup> ¼ 1.

function; this choice fixes the form of the spacing functions χ1ð Þj and χ2ð Þj .

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

<sup>χ</sup>2ð Þ <sup>v</sup>; <sup>j</sup> � <sup>1</sup> <sup>¼</sup> v ev xj and Dfe

<sup>χ</sup>1ð Þ <sup>v</sup>; <sup>j</sup> <sup>≔</sup> <sup>e</sup><sup>v</sup> <sup>Δ</sup><sup>j</sup> � <sup>1</sup>

v

χ2ð Þ x; j becomes

function

given by

and

Dbe

and χ2ð Þ v; j are defined as

v x ð Þ<sup>j</sup> <sup>≔</sup> <sup>e</sup>v xj � <sup>e</sup>v xj�<sup>1</sup>

where Δ<sup>j</sup> ¼ xjþ<sup>1</sup> � xj and the spacing function χ2ð Þj , which is a replacement for the usual spacing function Δj, is obtained by solving the above equality for χ2ð Þj ,

$$\chi\_2(j-1) \coloneqq \frac{g(\mathbf{x}\_j) - g(\mathbf{x}\_j - \Delta\_{j-1})}{g'(\mathbf{x}\_j)} = \frac{1}{g'(\mathbf{x}\_j)} \sum\_{k=1}^n \frac{(-1)^{k-1}}{k!} g^{(k)}(\mathbf{x}\_j) \Delta\_{j-1}^k. \tag{2}$$

This is an expression which is valid for points xj different from the zeroes of g<sup>0</sup> ð Þx .

A definition for forward finite differences at xj is

$$\mathbf{g}\left(\mathbf{D}\_{\hat{f}}\mathbf{g}\right)\left(\mathbf{x}\_{\hat{i}}\right) \coloneqq \frac{\mathbf{g}\left(\mathbf{x}\_{\hat{i}} + \boldsymbol{\Delta}\_{\hat{i}}\right) - \mathbf{g}\left(\mathbf{x}\_{\hat{i}}\right)}{\chi\_{1}(\hat{j})} = \mathbf{g}'(\mathbf{x}\_{\hat{i}}) \,. \tag{3}$$

where

$$\chi\_1(j) \coloneqq \frac{g\left(\mathbf{x}\_j + \boldsymbol{\Delta}\_j\right) - g\left(\mathbf{x}\_j\right)}{g'\left(\mathbf{x}\_j\right)} = \frac{1}{g'\left(\mathbf{x}\_j\right)} \sum\_{k=1}^{\bullet} \frac{1}{k!} g^{(k)}\left(\mathbf{x}\_j\right) \left(\boldsymbol{\Delta}\_j\right)^k. \tag{4}$$

valid for points different from the zeroes of g<sup>0</sup> ð Þx .

These definitions coincide with the usual finite differences derivative when the function to which they act on is the linear function g xð Þ¼ a<sup>0</sup> þ a1x, a0, a<sup>1</sup> ∈ C. An exact finite differences derivative of other functions need of more terms than the one found in the usual definition of a finite differences derivative, as can be seen in Eqs. (2) and (4).

Figure 1. Three-dimensional plot of <sup>χ</sup>2ð Þ <sup>x</sup>; <sup>j</sup> for the quadratic function g xð Þ¼ <sup>a</sup><sup>0</sup> <sup>þ</sup> <sup>a</sup>1<sup>x</sup> <sup>þ</sup> <sup>a</sup>2x<sup>2</sup> with <sup>Δ</sup><sup>j</sup> <sup>¼</sup> 1.

Example. For the quadratic function g xð Þ¼ <sup>a</sup><sup>0</sup> <sup>þ</sup> <sup>a</sup>1<sup>x</sup> <sup>þ</sup> <sup>a</sup>2x2, <sup>a</sup>0, a1, a<sup>2</sup> <sup>∈</sup> <sup>C</sup>, the spacing function χ2ð Þ x; j becomes

$$
\lambda \chi\_2(\mathbf{x}, \mathbf{j}) = \Delta\_{\dot{\mathbf{j}}} - \frac{\left\Delta\_{\dot{\mathbf{j}}}^2}{\frac{a\_1}{a\_2} + \mathbf{2x}},\tag{5}
$$

where x 6¼ �a1=2a2. A plot of this function is shown in Figure 1 for Δ<sup>j</sup> ¼ 1.

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 χ1ð Þj and χ2ð Þj .
