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 Db,f given by

$$\mathbf{D}\_{h} := \begin{pmatrix} \frac{-1}{\chi(v\_{1})} & \frac{1}{\chi(v\_{1})} & 0 & 0 & \dots & 0 & 0 & 0\\ \frac{-1}{\chi(v\_{1})} & \frac{1}{\chi(v\_{1})} & 0 & 0 & \ddots & 0 & 0 & 0\\ 0 & \frac{-1}{(\chi(v\_{2}))} & \frac{1}{\chi(v\_{2})} & 0 & \ddots & 0 & 0 & 0\\ 0 & 0 & \frac{-1}{(\chi(v\_{3}))} & \frac{1}{\chi(v\_{3})} & \ddots & 0 & 0 & 0\\ \vdots & & & & &\\ 0 & 0 & 0 & 0 & \dots & \frac{-1}{(\chi(v\_{n}))} & \frac{1}{\chi(v\_{1}) - 1} & 0\\ 0 & 0 & 0 & 0 & \dots & 0 & \frac{-1}{(\chi(v\_{n}) - 1)} & \frac{1}{\chi(v\_{2}) - 1} \end{pmatrix} \tag{28}$$

and

ð Þ Dbgh <sup>j</sup> ¼ gj

168 Numerical Simulations in Engineering and Science

where 1 < j ≤ N.

Db g h � �

ratio of two functions we have

<sup>j</sup> <sup>¼</sup> <sup>1</sup>

χ2ð Þ v; j � 1

For instance, these equalities imply that

Xm j¼n

χ2ð Þ v; j � 1 gj

by parts results,

where 1 ≤ j < N, and

where 1 < j ≤ N.

Xm j¼n

gj hj

� gj�<sup>1</sup> hj�<sup>1</sup> � �

> Db g h � �

> > Df g h � �

Df g h � �

1 <sup>χ</sup>2ð Þ <sup>v</sup>; <sup>j</sup> � <sup>1</sup>

<sup>χ</sup>2ð Þ <sup>v</sup>; <sup>j</sup> gjþ<sup>1</sup>ð Þ Dbh <sup>j</sup>þ<sup>1</sup> <sup>þ</sup>X<sup>m</sup>

ð Þ Dbh <sup>j</sup> <sup>þ</sup>X<sup>m</sup>

j¼n

ð Þ Dbh <sup>j</sup> þ hj�<sup>1</sup>ð Þ Dbg <sup>j</sup> ¼ gj

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

� gj

<sup>¼</sup> ð Þ Dbg <sup>j</sup> hj

<sup>¼</sup> Df <sup>g</sup> � � j hjþ<sup>1</sup>

<sup>¼</sup> Df <sup>g</sup> � � j hj

<sup>¼</sup> <sup>1</sup>

<sup>¼</sup> ð Þ Dbg <sup>j</sup> hj�<sup>1</sup>

j

j

j

Df g � �

<sup>j</sup> ¼ e �v Δ<sup>j</sup>

j¼n

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

<sup>χ</sup>1ð Þ <sup>v</sup>; <sup>j</sup> hj Df <sup>g</sup> � �

<sup>χ</sup>1ð Þ <sup>v</sup>; <sup>j</sup> � <sup>1</sup> hj�<sup>1</sup> Df <sup>g</sup> � �

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

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

ð Þ Dbh <sup>j</sup> þ e

<sup>χ</sup>2ð Þ <sup>v</sup>; <sup>j</sup> � <sup>1</sup> � gj hj � hj�<sup>1</sup>

0 @

ð Þ Dbh <sup>j</sup> hjhj�<sup>1</sup> ,

� gj�<sup>1</sup>

� gj

� gjþ<sup>1</sup>

<sup>χ</sup>1ð Þ <sup>v</sup>; <sup>j</sup> <sup>¼</sup> v, and <sup>χ</sup>1ð Þ <sup>v</sup>; <sup>j</sup>

ð Þ Dbh <sup>j</sup> hjhj�<sup>1</sup>

Df h � � j hjhjþ<sup>1</sup>

> Df h � � j hjhjþ<sup>1</sup>

<sup>χ</sup>2ð Þ <sup>v</sup>; <sup>j</sup> <sup>¼</sup> <sup>e</sup>

v Δ<sup>j</sup>

ð Þ Dbg <sup>j</sup>þ<sup>1</sup>: (25)

<sup>j</sup> <sup>¼</sup> gmþ<sup>1</sup>hmþ<sup>1</sup> � gnhn, (26)

<sup>j</sup>�<sup>1</sup> <sup>¼</sup> gmhm � gn�<sup>1</sup>hn�<sup>1</sup>, (27)

<sup>v</sup> <sup>Δ</sup>j�<sup>1</sup> hj�<sup>1</sup> Df <sup>g</sup> � �

þ

gj � gj�<sup>1</sup> � �

hjhj�<sup>1</sup>

, (21)

, (22)

: (23)

: (24)

� � hjhj�<sup>1</sup>

<sup>j</sup>�<sup>1</sup>, (19)

hj

1 A

(20)

$$\mathbf{D}\_{f} := \begin{pmatrix} \frac{-1}{\chi\_{1}(v,1)} & \frac{1}{\chi\_{1}(v,1)} & 0 & 0 & \chi\_{1}(v,1) & 0 & 0 & 0\\ 0 & \frac{-1}{\chi\_{1}(v,2)} & \frac{1}{\chi\_{1}(v,2)} & 0 & \cdots & 0 & 0 & 0\\ 0 & 0 & \frac{-1}{\chi\_{1}(v,3)} & \frac{1}{\chi\_{1}(v,3)} & \cdots & 0 & 0 & 0\\ \vdots & & & & & &\\ 0 & 0 & 0 & 0 & \frac{-1}{\chi\_{1}(v,N-2)} & \frac{1}{\chi\_{1}(v,N-2)} & 0\\ 0 & 0 & 0 & 0 & \cdots & 0 & \frac{-1}{\chi\_{1}(v,N-1)} & \frac{1}{\chi\_{1}(v,N-1)}\\ 0 & 0 & 0 & 0 & \cdots & 0 & \frac{-1}{\chi\_{1}(v,N-1)} & \frac{1}{\chi\_{1}(v,N-1)}\\ \end{pmatrix} \tag{29}$$

We have used the definition for the backward derivative ð Þ Dbg <sup>j</sup> for all the rows of the backward derivative matrix Db but not for the first line in which we have instead used the forward derivative Df g <sup>1</sup>. A similar thing was done for the forward derivative matrix Df . These matrices act on bounded vectors g ¼ g1; g2; ⋯; gN <sup>T</sup> ∈ C<sup>N</sup>.

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 Db,f . Expressions for the exact second finite differences derivative associated to the exponential function are obtained through the square of the derivative matrices Db,f . These expressions are

$$\left(D\_b^2 \mathbf{g}\right)\_1 = \upsilon \frac{\mathbf{g}\_2 - \mathbf{g}\_1}{\chi\_1(\upsilon, 1)} = \upsilon \left(D\_f \mathbf{g}\right)\_{1'} \tag{30}$$

y<sup>m</sup> ¼ N Y�1 k¼m

<sup>χ</sup>1ð Þ <sup>v</sup>; <sup>1</sup> <sup>þ</sup> <sup>λ</sup> � � 1

where 2 ≤ j < m ≤ N. The quantities y<sup>m</sup> usually are very small.

<sup>λ</sup>ð Þ <sup>v</sup> � <sup>λ</sup> <sup>1</sup>

1

CCCA, <sup>e</sup><sup>v</sup> <sup>¼</sup> <sup>C</sup>

wm ¼

continuous derivative. We note that the local derivatives Db,f g � �

the null vector, which is the trivial eigenvector of the derivative.

<sup>χ</sup>2ð Þ <sup>v</sup>; <sup>j</sup> � <sup>1</sup> <sup>¼</sup> <sup>Δ</sup><sup>j</sup>�<sup>1</sup>

4.3. The commutator between coordinate and derivative

ð Þ Db <sup>x</sup> <sup>j</sup> <sup>¼</sup> xj � xj�<sup>1</sup>

Since the following equality holds:

from a local point of view, we have

m Y�1 k¼1

0

BBB@

<sup>D</sup><sup>f</sup> � <sup>λ</sup> <sup>I</sup> � � �

where

spaced partition.

� ¼ �ð Þ<sup>1</sup> <sup>N</sup>�<sup>1</sup>

N �2, and the corresponding eigenvectors are

e<sup>0</sup> ¼ C

<sup>1</sup> � <sup>χ</sup>2ð Þ <sup>v</sup>; <sup>k</sup> χ2ð Þ v; j

<sup>χ</sup>1ð Þ <sup>v</sup>; <sup>2</sup> <sup>þ</sup> <sup>λ</sup>

� �<sup>⋯</sup> <sup>1</sup>

w1 w2 ⋮ wj 0 ⋮ 0

1

CCCCCCCCCCCA

� � (39)

0

BBBBBBBBBBB@

On the other hand, for the forward finite difference matrix, D<sup>f</sup> , the characteristic polynomial is

Thus, the eigenvalues for the forward derivative are λ<sup>0</sup> ¼ 0, λ<sup>v</sup> ¼ v and λ<sup>j</sup> ¼ �1=χ1ð Þ v; j , 1≤ j ≤

ev x<sup>1</sup> ev x<sup>2</sup> ⋮ ev xN 1

<sup>1</sup> � <sup>χ</sup>1ð Þ <sup>v</sup>; <sup>k</sup> χ1ð Þ v; j

and 1 ≤ m < j ≤ N � 2. The quantities wm are also very small; in fact, they vanish for the equally

The matrices, Db,f , have the same eigenvalues λ<sup>0</sup> and λv, and eigenvectors which are the discretization of the function gvð Þ¼ <sup>x</sup> C ev x on the partition f g <sup>x</sup>1; <sup>x</sup>2; <sup>⋯</sup>; xN (the eigenvector ð Þ <sup>1</sup>; <sup>1</sup>; <sup>⋯</sup>; <sup>1</sup> <sup>T</sup> correspond to the eigenvalue <sup>v</sup>=0). This is the same eigenfunction that is found in the continuous variable case because the exponential function is indeed an eigenfunction of the

as the matrices Db,f which are global objects. The other eigenvectors are fluctuations around

<sup>χ</sup>2ð Þ <sup>v</sup>; <sup>j</sup> � <sup>1</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup>

v 2

<sup>Δ</sup><sup>j</sup>�<sup>1</sup> <sup>þ</sup> <sup>O</sup> <sup>Δ</sup><sup>2</sup>

j�1

CCCA, <sup>e</sup><sup>j</sup> <sup>¼</sup> <sup>C</sup>

0

BBB@

� �, (36)

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

> <sup>χ</sup>1ð Þ <sup>v</sup>; <sup>N</sup> � <sup>2</sup> <sup>þ</sup> <sup>λ</sup> � �

¼ 0, (37)

171

, (38)

<sup>j</sup> have the same eigenfunctions

� �, (40)

$$\pi(D\_b^2 \mathbf{g})\_2 = \upsilon \frac{\mathbf{g}\_2 - \mathbf{g}\_1}{\chi\_2(\upsilon, 1)} = \upsilon \ (D\_b \mathbf{g})\_{2\prime} \tag{31}$$

$$\left(D\_{b}^{2}\mathbf{g}\right)\_{j} = \frac{1}{\chi\_{2}(\upsilon,j-1)} \left(\frac{\mathbf{g}\_{j} - \mathbf{g}\_{j-1}}{\chi\_{2}(\upsilon,j-1)} - \frac{\mathbf{g}\_{j-1} - \mathbf{g}\_{j-2}}{\chi\_{2}(\upsilon,j-2)}\right) = \frac{\left(D\_{\mathbf{j}}\mathbf{g}\right)\_{j} - \left(D\_{\mathbf{j}}\mathbf{g}\right)\_{j-1}}{\chi\_{2}(\upsilon,j-1)}, \quad j = \mathbf{3}, \cdots, \text{N}.\tag{32}$$

These expressions have the exponential function ev x as one of their eigenfunctions with eigenvalue v2, as is also the case of the continuous variable derivative. Higher order derivatives can be obtained in an analogous way.

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 Db,<sup>f</sup>

Now that we have the matrices Db,f representing the backward and forward derivatives, we are interested in finding their eigenvalues λ∈ C and its corresponding eigenvectors eλ. Therefore, we begin by finding the values of λ for which the matrices Db,f � λ I are not invertible, that is, when they are singular.

On one hand, for the backward finite difference matrix Db, the characteristic polynomial is

$$|\mathcal{D}\_{\mathbb{D}} - \lambda \, I| = \lambda \left(\lambda + \frac{1}{\chi\_1(v, 1)} - \frac{1}{\chi\_2(v, 1)}\right) \left(\frac{1}{\chi\_2(v, 2)} - \lambda\right) \left(\frac{1}{\chi\_2(v, 3)} - \lambda\right) \cdots \left(\frac{1}{\chi\_2(v, N - 1)} - \lambda\right) = 0,\tag{33}$$

whose roots are λ<sup>0</sup> ¼ 0, λ<sup>v</sup> ¼ �1=χ1ð Þþ v; 1 1=χ2ð Þ¼ v; 1 v and λ<sup>j</sup> ¼ 1=χ2ð Þ v; j , 2≤ j ≤ N �1. Let us denote by e<sup>λ</sup> ¼ ð Þ eλ,1;eλ,2; ⋯;eλ,N <sup>T</sup> to the eigenvector corresponding to the eigenvalue λ. The system of equations for the components of the eigenvectors is

$$-\frac{e\_{\lambda,k}}{\chi\_2(v,k)} + \left(\frac{1}{\chi\_2(v,k)} - \lambda\right) e\_{\lambda,k+1} = 0,\tag{34}$$

with k ¼ 1, ⋯, N � 1. Then, the eigenvectors are

$$\mathbf{e}\_0 = \mathbb{C}\begin{pmatrix} 1\\1\\\vdots\\1 \end{pmatrix}, \quad \mathbf{e}\_v = \mathbb{C}\begin{pmatrix} e^{\mathcal{P}\cdot\mathbf{x}\_1} \\ e^{\mathcal{P}\cdot\mathbf{x}\_2} \\\vdots \\ e^{\mathcal{P}\cdot\mathbf{x}\_N} \end{pmatrix}, \quad \mathbf{e}\_j = \mathbb{C}\begin{pmatrix} 0\\0\\\vdots\\0\\y\_{j+1} \\\vdots\\y\_N \end{pmatrix},\tag{35}$$

where C is the normalization constant, and

Exact Finite Differences for Quantum Mechanics http://dx.doi.org/10.5772/intechopen.71956 171

$$\mathbf{y}\_m = \prod\_{k=m}^{N-1} \left( 1 - \frac{\chi\_2(\mathbf{v}, k)}{\chi\_2(\mathbf{v}, j)} \right) \tag{36}$$

where 2 ≤ j < m ≤ N. The quantities y<sup>m</sup> usually are very small.

On the other hand, for the forward finite difference matrix, D<sup>f</sup> , the characteristic polynomial is

$$\left|\mathbf{D}\_{\boldsymbol{\theta}} - \lambda \, I\right| = (-1)^{N-1} \lambda (\boldsymbol{v} - \lambda) \left(\frac{1}{\chi\_1(\boldsymbol{v}, 1)} + \lambda\right) \left(\frac{1}{\chi\_1(\boldsymbol{v}, 2)} + \lambda\right) \cdots \left(\frac{1}{\chi\_1(\boldsymbol{v}, N - 2)} + \lambda\right) = 0,\tag{37}$$

Thus, the eigenvalues for the forward derivative are λ<sup>0</sup> ¼ 0, λ<sup>v</sup> ¼ v and λ<sup>j</sup> ¼ �1=χ1ð Þ v; j , 1≤ j ≤ N �2, and the corresponding eigenvectors are

$$\mathbf{e}\_0 = \mathbf{C} \begin{pmatrix} 1 \\ 1 \\ \vdots \\ 1 \end{pmatrix}, \quad \mathbf{e}\_v = \mathbf{C} \begin{pmatrix} e^{p \cdot x\_1} \\ e^{p \cdot x\_2} \\ \vdots \\ e^{p \cdot x\_N} \end{pmatrix}, \quad \mathbf{e}\_j = \mathbf{C} \begin{pmatrix} w\_1 \\ w\_2 \\ \vdots \\ w\_j \\ 0 \\ \vdots \\ 0 \end{pmatrix}, \tag{38}$$

where

D2 <sup>b</sup><sup>g</sup> � �

D2 <sup>b</sup><sup>g</sup> � �

<sup>χ</sup>2ð Þ <sup>v</sup>; <sup>j</sup> � <sup>1</sup> � gj�<sup>1</sup> � gj�<sup>2</sup>

� �

gj � gj�<sup>1</sup>

D2 <sup>b</sup><sup>g</sup> � �

<sup>j</sup> <sup>¼</sup> <sup>1</sup>

shown in a previous section.

that is, when they are singular.

j j D<sup>b</sup> � λ I ¼ λ λ þ

χ2ð Þ v; j � 1

170 Numerical Simulations in Engineering and Science

can be obtained in an analogous way.

4.2. Eigenfunctions and eigenvectors of Db,<sup>f</sup>

1 <sup>χ</sup>1ð Þ <sup>v</sup>; <sup>1</sup> � <sup>1</sup>

with k ¼ 1, ⋯, N � 1. Then, the eigenvectors are

e<sup>0</sup> ¼ C

where C is the normalization constant, and

us denote by e<sup>λ</sup> ¼ ð Þ eλ,1;eλ,2; ⋯;eλ,N

χ2ð Þ v; 1

The system of equations for the components of the eigenvectors is

� <sup>e</sup>λ, <sup>k</sup> <sup>χ</sup>2ð Þ <sup>v</sup>; <sup>k</sup> <sup>þ</sup>

1

CCCA, <sup>e</sup><sup>v</sup> <sup>¼</sup> <sup>C</sup>

0

BBB@

� � 1

<sup>1</sup> <sup>¼</sup> <sup>v</sup> <sup>g</sup><sup>2</sup> � <sup>g</sup><sup>1</sup>

<sup>2</sup> <sup>¼</sup> <sup>v</sup> <sup>g</sup><sup>2</sup> � <sup>g</sup><sup>1</sup>

χ2ð Þ v; j � 2

These expressions have the exponential function ev x as one of their eigenfunctions with eigenvalue v2, as is also the case of the continuous variable derivative. Higher order derivatives

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

Now that we have the matrices Db,f representing the backward and forward derivatives, we are interested in finding their eigenvalues λ∈ C and its corresponding eigenvectors eλ. Therefore, we begin by finding the values of λ for which the matrices Db,f � λ I are not invertible,

On one hand, for the backward finite difference matrix Db, the characteristic polynomial is

<sup>χ</sup>2ð Þ <sup>v</sup>; <sup>2</sup> � <sup>λ</sup> � � 1

whose roots are λ<sup>0</sup> ¼ 0, λ<sup>v</sup> ¼ �1=χ1ð Þþ v; 1 1=χ2ð Þ¼ v; 1 v and λ<sup>j</sup> ¼ 1=χ2ð Þ v; j , 2≤ j ≤ N �1. Let

1 <sup>χ</sup>2ð Þ <sup>v</sup>; <sup>k</sup> � <sup>λ</sup> � �

> ev x<sup>1</sup> ev x<sup>2</sup> ⋮ ev xN

1

CCCA, <sup>e</sup><sup>j</sup> <sup>¼</sup> <sup>C</sup>

0

BBB@

<sup>χ</sup>1ð Þ <sup>v</sup>; <sup>1</sup> <sup>¼</sup> v Df <sup>g</sup> � �

<sup>¼</sup> Df <sup>g</sup> � �

<sup>χ</sup>2ð Þ <sup>v</sup>; <sup>3</sup> � <sup>λ</sup> � �

<sup>⋯</sup> <sup>1</sup>

<sup>T</sup> to the eigenvector corresponding to the eigenvalue λ.

0 0 ⋮ 0 yjþ<sup>1</sup> ⋮ yN

1

CCCCCCCCCCCA

0

BBBBBBBBBBB@

<sup>χ</sup>2ð Þ <sup>v</sup>; <sup>N</sup> � <sup>1</sup> � <sup>λ</sup> � �

eλ, <sup>k</sup>þ<sup>1</sup> ¼ 0, (34)

, (35)

¼ 0, (33)

<sup>1</sup>, (30)

<sup>χ</sup>2ð Þ <sup>v</sup>; <sup>j</sup> � <sup>1</sup> , j <sup>¼</sup> <sup>3</sup>, <sup>⋯</sup>, N: (32)

<sup>χ</sup>2ð Þ <sup>v</sup>; <sup>1</sup> <sup>¼</sup> v Dð Þ bg <sup>2</sup>, (31)

j�1

<sup>j</sup> � Df <sup>g</sup> � �

$$w\_m = \prod\_{k=1}^{m-1} \left( 1 - \frac{\chi\_1(v,k)}{\chi\_1(v,j)} \right) \tag{39}$$

and 1 ≤ m < j ≤ N � 2. The quantities wm are also very small; in fact, they vanish for the equally spaced partition.

The matrices, Db,f , have the same eigenvalues λ<sup>0</sup> and λv, and eigenvectors which are the discretization of the function gvð Þ¼ <sup>x</sup> C ev x on the partition f g <sup>x</sup>1; <sup>x</sup>2; <sup>⋯</sup>; xN (the eigenvector ð Þ <sup>1</sup>; <sup>1</sup>; <sup>⋯</sup>; <sup>1</sup> <sup>T</sup> correspond to the eigenvalue <sup>v</sup>=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 Db,f g � � <sup>j</sup> have the same eigenfunctions as the matrices Db,f which are global objects. The other eigenvectors are fluctuations around the null vector, which is the trivial eigenvector of the derivative.

### 4.3. The commutator between coordinate and derivative

Since the following equality holds:

$$(D\_b \ge \mathbf{x})\_j = \frac{\mathbf{x}\_j - \mathbf{x}\_{j-1}}{\chi\_2(v, j-1)} = \frac{\Delta\_{j-1}}{\chi\_2(v, j-1)} = 1 + \frac{v}{2}\Delta\_{j-1} + \mathcal{O}\left(\Delta\_{j-1}^2\right),\tag{40}$$

from a local point of view, we have

$$(D\_b \text{ xg})\_j = \mathbf{x}\_j (D\_b \text{ g})\_j + \mathbf{g}\_{j-1} (D\_b \text{ x})\_j = \mathbf{x}\_j (D\_b \text{ g})\_j + \mathbf{g}\_{j-1} \frac{\Delta\_{j-1}}{\chi\_2(v, j-1)} \tag{41}$$

Let the linear transformation represented by the matrix formed by means of the standard

s Db,f � �<sup>k</sup>

s ve<sup>v</sup> <sup>¼</sup> <sup>C</sup>

ev sð Þ <sup>þ</sup>x<sup>1</sup> ev sð Þ <sup>þ</sup>x<sup>2</sup> ⋮ ev sð Þ <sup>þ</sup>xN


�ixjþ1p

1

0

BBB@

<sup>k</sup>! , (44)

Exact Finite Differences for Quantum Mechanics http://dx.doi.org/10.5772/intechopen.71956 173

<sup>v</sup> <sup>¼</sup> <sup>e</sup><sup>s</sup> <sup>D</sup>b,f <sup>e</sup><sup>v</sup> is the function <sup>e</sup>v x evaluated at

g pð Þdp, (46)

gjþ<sup>1</sup>: (47)

gj (48)

CCCA, (45)

k¼0

where s ∈ R. Since e<sup>v</sup> is an eigenvector of Db,f with eigenvalue v (see Eqs. (35) and (38)), it

that is, es v is an eigenvalue of e<sup>s</sup> <sup>D</sup>b,f with corresponding eigenvector ev, but the right-hand side of this equality is also a translation by the amount s of the domain of the derivative operators.

the points of the translated partition P<sup>0</sup> ¼ f g x<sup>1</sup> þ s; x<sup>2</sup> þ s; ⋯; xN þ s . Thus, we can perform not

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 Δ, a constant) and

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 �ip should be understood, and we will omit it from the formulae below

definition of a translation operator and of the exponential operator, given by

<sup>s</sup> <sup>D</sup>b,f <sup>¼</sup> <sup>X</sup><sup>∞</sup>

e

b,f <sup>e</sup><sup>v</sup> <sup>¼</sup> vk <sup>e</sup>v, <sup>k</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>⋯</sup>, and then,

<sup>s</sup> <sup>D</sup>b,f <sup>e</sup><sup>v</sup> <sup>¼</sup> <sup>X</sup><sup>∞</sup>

only discrete translations but continuous translations as well.

k¼0

ð Þ s v <sup>k</sup>

4.5. Fourier transforms between coordinate and derivative representations

ð Þ Fg xj

ð Þ <sup>F</sup>bg ð Þ<sup>p</sup> <sup>≔</sup> <sup>p</sup>

<sup>F</sup><sup>f</sup> <sup>g</sup> � �ð Þ<sup>p</sup> <sup>≔</sup> <sup>p</sup>

� �<sup>≔</sup> <sup>1</sup>

weights χ1ð Þ v; j of Eq. (10), here we define two discrete Fourier transforms at p as

2 sin ð Þ pxN

2 sin ð Þ pxN

ffiffiffiffiffiffi <sup>2</sup><sup>π</sup> <sup>p</sup> ð ∞

�∞ e �ixjp

X N

χ<sup>2</sup> Δ<sup>j</sup> � �e

> χ<sup>1</sup> Δ<sup>j</sup> � �e �ixjp

j¼�N

X N

j¼�N

is the continuous Fourier transform of g pð Þ at xj. Having introduced the summation with

<sup>k</sup>! <sup>e</sup><sup>v</sup> <sup>¼</sup> <sup>e</sup>

e

We point out that s is arbitrary and then the vector e<sup>0</sup>

with periodic boundary conditions ev,<sup>1</sup> ¼ ev,N.

for the sake of simplicity of notation.

Given a function g pð Þ in the <sup>L</sup><sup>1</sup>

with x�<sup>N</sup> ¼ �xN, the function.

follows that D<sup>k</sup>

and then, the commutator between x and Db, acting on g, is given by

$$([D\_{\mathbb{D}}, \mathfrak{x}]g)\_{\flat} = g\_{\flat - 1} \left( 1 + \frac{\upsilon}{2} \Delta\_{\flat - 1} + \mathcal{O} \left( \Delta\_{\flat - 1}^2 \right) \right).$$

Thus, the commutator between Db and x becomes one in the limit of small Δj, or large N, because it also happens that gj�<sup>1</sup> ! gj .

We now consider the commutator between the coordinate matrix Q≔diagð Þ x1; x2; ⋯; xN , and the forward derivative matrix D<sup>f</sup> . That commutator is

$$[\mathbf{D}\_f, \mathbf{Q}] = \begin{pmatrix} 0 & \frac{\Delta\_1}{\chi\_1(v, 1)} & 0 & \dots & 0 & 0 & 0\\ 0 & 0 & \frac{\Delta\_2}{\chi\_1(v, 2)} & \dots & 0 & 0 & 0\\ \vdots\\ 0 & 0 & 0 & \dots & 0 & \frac{\Delta\_{N-2}}{\chi\_1(v, N-2)} & 0\\ 0 & 0 & 0 & \dots & 0 & 0 & \frac{\Delta\_{N-1}}{\chi\_1(v, N-1)}\\ 0 & 0 & 0 & \dots & 0 & \frac{\Delta\_{N-1}}{\chi\_2(v, N-1)} & 0 \end{pmatrix} \tag{42}$$

The small Δ<sup>j</sup> (N ! ∞) approximation of this commutator is just

$$\begin{aligned} \begin{bmatrix} \mathbf{D}\_{f}, \mathbf{Q} \end{bmatrix} \underset{\Delta\_{f} \to 0}{\longrightarrow} \begin{pmatrix} 0 & 1 & 0 & \dots & 0 & 0 & 0 \\ 0 & 0 & 1 & \dots & 0 & 0 & 0 \\ \vdots & & & & & \\ 0 & 0 & 0 & \dots & 0 & 1 & 0 \\ 0 & 0 & 0 & \dots & 0 & 0 & 1 \\ 0 & 0 & 0 & \dots & 0 & 1 & 0 \end{pmatrix} \end{aligned} \tag{43}$$

There is coincidence with the local calculation; as expected, this matrix approaches the identity matrix in the small Δ<sup>j</sup> limit. Note that this matrix is composed of backward translations of the first N � 1 points and a forward translation of the point N � 1 without periodicity; the value of the first point is lost.

### 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 Db,f used as their generators. We will focus on the translation of the common eigenvector <sup>e</sup><sup>v</sup> <sup>¼</sup> <sup>e</sup>v x<sup>1</sup> ;ev x<sup>2</sup> ; <sup>⋯</sup>;ev xN ð Þ<sup>T</sup> of both 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

ð Þ Db xg <sup>j</sup> <sup>¼</sup> xjð Þ Db <sup>g</sup> <sup>j</sup> <sup>þ</sup> gj�<sup>1</sup>ð Þ Db <sup>x</sup> <sup>j</sup> <sup>¼</sup> xjð Þ Db <sup>g</sup> <sup>j</sup> <sup>þ</sup> gj�<sup>1</sup>

v 2

Thus, the commutator between Db and x becomes one in the limit of small Δj, or large N,

We now consider the commutator between the coordinate matrix Q≔diagð Þ x1; x2; ⋯; xN , and

There is coincidence with the local calculation; as expected, this matrix approaches the identity matrix in the small Δ<sup>j</sup> limit. Note that this matrix is composed of backward translations of the first N � 1 points and a forward translation of the point N � 1 without periodicity; the value of

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 Db,f used as their generators. We will focus on the translation of the common

<sup>Δ</sup><sup>j</sup>�<sup>1</sup> <sup>þ</sup> <sup>O</sup> <sup>Δ</sup><sup>2</sup>

j�1

:

and then, the commutator between x and Db, acting on g, is given by

because it also happens that gj�<sup>1</sup> ! gj

172 Numerical Simulations in Engineering and Science

the first point is lost.

4.4. Translations

the forward derivative matrix D<sup>f</sup> . That commutator is

The small Δ<sup>j</sup> (N ! ∞) approximation of this commutator is just

eigenvector <sup>e</sup><sup>v</sup> <sup>¼</sup> <sup>e</sup>v x<sup>1</sup> ;ev x<sup>2</sup> ; <sup>⋯</sup>;ev xN ð Þ<sup>T</sup> of both matrices.

ð Þ ½ � Db; <sup>x</sup> <sup>g</sup> <sup>j</sup> <sup>¼</sup> gj�<sup>1</sup> <sup>1</sup> <sup>þ</sup>

.

Δ<sup>j</sup>�<sup>1</sup>

<sup>χ</sup>2ð Þ <sup>v</sup>; <sup>j</sup> � <sup>1</sup> , (41)

ð42Þ

ð43Þ

$$\varepsilon^{s\ D\_{b,f}} = \sum\_{k=0}^{\infty} \frac{\left(s\ D\_{b,f}\right)^k}{k!},\tag{44}$$

where s ∈ R. Since e<sup>v</sup> is an eigenvector of Db,f with eigenvalue v (see Eqs. (35) and (38)), it follows that D<sup>k</sup> b,f <sup>e</sup><sup>v</sup> <sup>¼</sup> vk <sup>e</sup>v, <sup>k</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>⋯</sup>, and then,

$$\mathbf{e}^{s^\*D\_{b,f}}\mathbf{e}\_v = \sum\_{k=0}^{\bullet} \frac{(s\ v)^k}{k!} \mathbf{e}\_v = \mathbf{e}^{s^\*} \mathbf{e}\_v = \mathbf{C} \begin{pmatrix} \mathbf{e}^{v\ (s+x\_1)}\\ \mathbf{e}^{v\ (s+x\_2)}\\ \vdots\\ \mathbf{e}^{v\ (s+x\_N)} \end{pmatrix} \tag{45}$$

that is, es v is an eigenvalue of e<sup>s</sup> <sup>D</sup>b,f with corresponding eigenvector ev, but the right-hand side of this equality is also a translation by the amount s of the domain of the derivative operators. We point out that s is arbitrary and then the vector e<sup>0</sup> <sup>v</sup> <sup>¼</sup> <sup>e</sup><sup>s</sup> <sup>D</sup>b,f <sup>e</sup><sup>v</sup> is the function <sup>e</sup>v x evaluated at the points of the translated partition P<sup>0</sup> ¼ f g x<sup>1</sup> þ s; x<sup>2</sup> þ s; ⋯; xN þ s . Thus, we can perform not only discrete translations but continuous translations as well.

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 Δ, a constant) and with periodic boundary conditions ev,<sup>1</sup> ¼ ev,N.

### 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 �ip should be understood, and we will omit it from the formulae below for the sake of simplicity of notation.

Given a function g pð Þ in the <sup>L</sup><sup>1</sup> -space and a non-uniform partition P ¼ f g x�<sup>N</sup>; x�Nþ<sup>1</sup>; ⋯; xN , with x�<sup>N</sup> ¼ �xN, the function.

$$\mathbf{u}(F\mathbf{g})\left(\mathbf{x}\_{\rangle}\right) \coloneqq \frac{1}{\sqrt{2\pi}} \int\_{-\infty}^{\infty} e^{-i\mathbf{x}\cdot\mathbf{p}} \mathbf{g}(\mathbf{p}) d\mathbf{p} \,\tag{46}$$

is the continuous Fourier transform of g pð Þ at xj. Having introduced the summation with weights χ1ð Þ v; j of Eq. (10), here we define two discrete Fourier transforms at p as

$$\chi(\mathcal{F}\_{\mathbb{B}}\mathbf{g})(p) \coloneqq \frac{p}{2} \frac{p}{\sin\left(p\mathbf{x}\_{\mathcal{N}}\right)} \sum\_{j=-N}^{N} \chi\_2\left(\boldsymbol{\Delta}\_{\boldsymbol{\beta}}\right) e^{-i\boldsymbol{x}\_{j+1}p} g\_{j+1}.\tag{47}$$

$$\chi(\mathcal{F}\_{\boldsymbol{f}}\boldsymbol{\mathcal{g}})(\boldsymbol{p}) \coloneqq \frac{\boldsymbol{p}}{2} \frac{\boldsymbol{p}}{\sin\left(p\boldsymbol{\chi}\_{\mathcal{N}}\right)} \sum\_{j=-N}^{N} \chi\_{1}(\boldsymbol{\Delta}\_{j}) e^{-i\boldsymbol{\chi}\_{\mathcal{I}}\boldsymbol{p}} \mathbf{g}\_{j} \tag{48}$$

Now, the discrete derivative of the product <sup>e</sup>�ixpg at xj, with derivative eigenvalue �ip, is readily computed to give (see Eq. (18)).

$$\left(\mathcal{D}\_{\rangle}e^{-ixp}\mathcal{g}\right)\_{\rangle} = -i\text{p }\mathcal{g}\_{\rangle+1}\left(e^{-ixp}\right)\_{\rangle} + e^{-ixp}\left(\mathcal{D}\_{\hbar}\mathcal{g}\right)\_{\rangle}.\tag{49}$$

d eixjp

dp eixjph pð Þ¼� <sup>ð</sup><sup>∞</sup>

xjð Þ Fh <sup>j</sup> <sup>¼</sup> F i d h

5. Quantum mechanical momentum and time operators

N X�1 j¼1 χ∗

� Pbfψjf � �

ð Þ ψjf <sup>b</sup> ≔

ð Þ ψjf <sup>f</sup> ≔

N X�1 j¼1

> N X�1 j¼1 χ∗ <sup>1</sup>ð Þ <sup>v</sup>; <sup>j</sup> <sup>ψ</sup><sup>∗</sup>

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

f

rewrite Eq. (26) in terms of complex wave functions ψ, f∈ ℓ

<sup>j</sup>þ<sup>1</sup>ð Þ �iℏDb<sup>f</sup> <sup>j</sup>þ<sup>1</sup> �

ψjPbbf � �

and the bilinear forms ð Þ ψjf <sup>b</sup> and ð Þ ψjf <sup>f</sup> are defined as

where the momentum-like operators Pb<sup>b</sup> and Pb<sup>f</sup> are defined as

b

P ¼ f g x1; x2; ⋯; xN of ½ � a; b . We obtain

<sup>χ</sup>2ð Þ <sup>v</sup>; <sup>j</sup> <sup>ψ</sup><sup>∗</sup>

This equality is rewritten as

N X�1 j¼1

dp � �

results in

form.

i xj ð∞ �∞

or in terms of continuous Fourier transforms,

dp <sup>¼</sup> i xje

dp eixjp dh pð Þ

dp <sup>þ</sup> <sup>e</sup>

ixjp h pð Þ�<sup>∞</sup>

p¼�∞

2

<sup>N</sup>f<sup>N</sup> � <sup>ψ</sup><sup>∗</sup>

<sup>j</sup> ¼ �i<sup>ℏ</sup> <sup>ψ</sup><sup>∗</sup>

<sup>1</sup>f<sup>1</sup>

Pb<sup>b</sup> ≔ � iℏDb, Pb<sup>f</sup> ≔ � iℏDf , (61)

�∞

j � ih ffiffiffiffiffiffi <sup>2</sup><sup>π</sup> <sup>p</sup> <sup>e</sup> ixjp h pð Þ � � � � ∞

These equalities are like the usual properties between the spaces related by the Fourier trans-

We can apply the results of previous sections to discrete Quantum Mechanics theory. Let us

<sup>1</sup>ð Þ <sup>v</sup>; <sup>j</sup> <sup>f</sup><sup>j</sup> �iℏDf<sup>ψ</sup> � �<sup>∗</sup>

¼ �i<sup>ℏ</sup> <sup>ψ</sup><sup>∗</sup>

<sup>χ</sup>2ð Þ <sup>v</sup>; <sup>j</sup> <sup>ψ</sup><sup>∗</sup>

ixjp (56)

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

<sup>p</sup>¼�<sup>∞</sup> (57)

175

: (58)

ð Þ P; ½ � a; b defined on the partition

<sup>N</sup>f<sup>N</sup> � <sup>ψ</sup><sup>∗</sup>

� �, (60)

<sup>j</sup>þ<sup>1</sup>f<sup>j</sup>þ<sup>1</sup>, (62)

<sup>j</sup> fj: (63)

<sup>1</sup>f<sup>1</sup> � �: (59)

The summation of this equality, with weights χ1ð Þ �ip; j , results in

$$\sum\_{j=-N}^{N-1} \chi\_1(j) \left( D\_f e^{-i\mathbf{p}\cdot\mathbf{y}} g \right)\_j = -i\mathbf{p} \sum\_{j=-N}^{N-1} \chi\_1(j) g\_{j+1} e^{-i\mathbf{x}\cdot\mathbf{y}} + \sum\_{j=-N}^{N-1} \chi\_1(j) e^{-i\mathbf{x}\cdot\mathbf{y}} \left( D\_f g \right)\_{j'} \tag{50}$$

or

$$\sum\_{j=-N}^{N-1} \chi\_1(j) e^{-i\mathbf{j}\cdot\mathbf{p}} \left(-i\mathbf{D}\_\dagger \mathbf{g}\right)\_j = p \sum\_{j=-N}^{N-1} \chi\_2(j) e^{-i\mathbf{j}\cdot\mathbf{p}} \mathbf{g}\_{j+1} + \sum\_{j=-N}^{N-1} \chi\_1(j) \left(-i\mathbf{D}\_\dagger e^{-i\mathbf{p}\cdot\mathbf{p}}\right)\_j. \tag{51}$$

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

$$\left(\mathcal{F}\_f(-i\mathcal{D}\_\delta g)\right)(p) = p\left(\mathcal{F}\_b g\right)(p) - i\left.p\frac{e^{-i\mathbf{x}\cdot\mathbf{y}}g\_N - e^{-i\mathbf{x}\cdot\mathbf{x}\cdot\mathbf{y}}g\_{-N}}{2\sin\left(p\mathbf{x}\_N\right)}.\tag{52}$$

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 <sup>g</sup> <sup>¼</sup> <sup>e</sup>�ixp)

$$\left(D\_f e^{-i\mathbf{x}p}\mathbf{g}\right)\_j = e^{i\Lambda\_f p} e^{-i\mathbf{x}\_{j+1}p} (D\_b \mathbf{g})\_{j+1} + \mathbf{g}\_j \left(D\_f e^{-i\mathbf{x}p}\right)\_j. \tag{53}$$

The summation of this equality, with weights χ1ð Þj , results in

$$\sum\_{j=-N}^{N-1} \chi\_2(j) e^{-i\chi\_{j+1}p} (-i\mathcal{D}\_b \mathbf{g})\_{j+1} = p \sum\_{j=-N}^{N-1} \chi\_1(j) e^{-i\mathbf{x}\cdot p} \mathbf{g}\_j + \sum\_{j=-N}^{N-1} \chi\_1(j) \left(-i\mathcal{D}\_f e^{-i\mathbf{x}\cdot p} \mathbf{g}\right)\_{j'} \tag{54}$$

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

$$(\mathcal{F}\_b(-i\mathcal{D}\_b\mathbf{g}))(p) = p \ (\mathcal{F}\_f\mathbf{g})(p) - i \ p \ \frac{e^{-i\mathbf{x}\cdot\mathbf{y}}\mathcal{G}\_N - e^{-i\mathbf{x}\cdot\mathbf{x}\cdot\mathbf{y}}\mathcal{G}\_{-N}}{2\sin\left(p\mathbf{x}\_N\right)}.\tag{55}$$

These are the equivalent to the well-known identities found in continuous variables theory. Thus, the multiplication by p in forward p-space corresponds to the backward finite differences derivative in coordinate space. Additionally, the multiplication by p in backward p-space corresponds to the forward finite differences derivative in coordinate space, when choosing vanishing or periodic boundary conditions.

The integration by parts of the simple relationship

Exact Finite Differences for Quantum Mechanics http://dx.doi.org/10.5772/intechopen.71956 175

$$\frac{d}{dp}\frac{e^{i\mathbf{x}\cdot p}}{dp} = i\,\mathbf{x}\_{\rangle}e^{i\mathbf{x}\_{\rangle}p} \tag{56}$$

results in

Now, the discrete derivative of the product <sup>e</sup>�ixpg at xj, with derivative eigenvalue �ip, is

�ixp � �

<sup>χ</sup>1ð Þ<sup>j</sup> gjþ<sup>1</sup><sup>e</sup>

�ixjþ1p

χ2ð Þj e

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

<sup>F</sup><sup>f</sup> �iDf <sup>g</sup> � � � � ð Þ¼ <sup>p</sup> <sup>p</sup> ð Þ <sup>F</sup>bg ð Þ� <sup>p</sup> i p <sup>e</sup>�ixNpgN � <sup>e</sup>�ix�Npg�<sup>N</sup>

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 <sup>g</sup> <sup>¼</sup> <sup>e</sup>�ixp)

> χ1ð Þj e �ixjp gj þ N X�1 j¼�N

<sup>j</sup> þ e

�ixjp <sup>þ</sup>

gjþ<sup>1</sup> <sup>þ</sup>

ð Þ Dbg <sup>j</sup>þ<sup>1</sup> <sup>þ</sup> gj Dfe

�ixjp Df g � �

N X�1 j¼�N

N X�1 j¼�N

2 sin ð Þ pxN

�ixp � �

2 sin ð Þ pxN

j

χ1ðÞ � j iDfe

�ixpg � �

j

χ1ð Þj e

χ1ðÞ � j iDfe

�ixjp Df g � �

�ixpg � �

: (49)

j

j

: (52)

: (53)

j

: (55)

, (54)

, (50)

: (51)

<sup>j</sup> ¼ �ip gjþ<sup>1</sup> <sup>e</sup>

N X�1 j¼�N

readily computed to give (see Eq. (18)).

174 Numerical Simulations in Engineering and Science

χ1ð Þj Dfe

χ1ð Þj e

N X�1 j¼�N

N X�1 j¼�N

Fourier transforms.

N X�1 j¼�N

χ2ð Þj e

�ixjþ1p

vanishing or periodic boundary conditions.

The integration by parts of the simple relationship

or

Dfe �ixpg � �

�ixpg � �

�ixjp �iDf <sup>g</sup> � �

Dfe �ixpg � �

The summation of this equality, with weights χ1ð Þj , results in

ð Þ �iDbg <sup>j</sup>þ<sup>1</sup> <sup>¼</sup> <sup>p</sup>

The summation of this equality, with weights χ1ð Þ �ip; j , results in

<sup>j</sup> ¼ �ip

<sup>j</sup> ¼ p N X�1 j¼�N

<sup>j</sup> ¼ e iΔjp e �ixjþ1p

> N X�1 j¼�N

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

ð Þ <sup>F</sup>bð Þ �iDbg ð Þ¼ <sup>p</sup> <sup>p</sup> <sup>F</sup><sup>f</sup> <sup>g</sup> � �ð Þ� <sup>p</sup> i p <sup>e</sup>�ixNpgN � <sup>e</sup>�ix�Npg�<sup>N</sup>

These are the equivalent to the well-known identities found in continuous variables theory. Thus, the multiplication by p in forward p-space corresponds to the backward finite differences derivative in coordinate space. Additionally, the multiplication by p in backward p-space corresponds to the forward finite differences derivative in coordinate space, when choosing

$$\text{si } \mathbf{x}\_{\rangle} \int\_{-\infty}^{\infty} dp \, e^{i\mathbf{x}\cdot p} h(p) = -\int\_{-\infty}^{\infty} dp \, e^{i\mathbf{x}\cdot p} \frac{d}{dp} \left. \frac{h(p)}{dp} + e^{i\mathbf{x}\cdot p} h(p) \right|\_{p=-\infty}^{\infty} \tag{57}$$

or in terms of continuous Fourier transforms,

$$\mathbf{x}\_{\circ}(Fh)\_{\circ} = \left(F \text{ i} \frac{d}{dp} \right)\_{\circ} - \left. \frac{\text{i}h}{\sqrt{2\pi}} \right. \left. e^{i\mathbf{x}\_{\circ}p} h(p) \right|\_{p=-\circ}^{\circ}. \tag{58}$$

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 ψ, f∈ ℓ 2 ð Þ P; ½ � a; b defined on the partition P ¼ f g x1; x2; ⋯; xN of ½ � a; b . We obtain

$$\sum\_{j=1}^{N-1} \chi\_2(\boldsymbol{\upsilon}, \boldsymbol{j}) \psi\_{j+1}^\* (-i\hbar \mathcal{D}\_\boldsymbol{\vartheta} \boldsymbol{\phi})\_{j+1} - \sum\_{j=1}^{N-1} \chi\_1^\*(\boldsymbol{\upsilon}, \boldsymbol{j}) \boldsymbol{\phi}\_{\boldsymbol{j}} (-i\hbar \mathcal{D}\_\boldsymbol{\ell} \boldsymbol{\psi})\_{\boldsymbol{j}}^\* = -i\hbar \left(\psi\_N^\* \boldsymbol{\phi}\_N - \psi\_1^\* \boldsymbol{\phi}\_1\right). \tag{59}$$

This equality is rewritten as

$$\left(\psi|\widehat{P}\_b\phi\right)\_b - \left(\widehat{P}\_f\psi|\phi\right)\_f = -i\hbar \left(\psi\_N^\*\phi\_N - \psi\_1^\*\phi\_1\right),\tag{60}$$

where the momentum-like operators Pb<sup>b</sup> and Pb<sup>f</sup> are defined as

$$
\widehat{P}\_b := -\
i\hbar D\_{b\prime} \quad \widehat{P}\_f := -\
i\hbar D\_{f\prime} \tag{61}
$$

and the bilinear forms ð Þ ψjf <sup>b</sup> and ð Þ ψjf <sup>f</sup> are defined as

$$(\psi|\phi)\_{\boldsymbol{b}} \coloneqq \sum\_{j=1}^{N-1} \chi\_2(\boldsymbol{v}, j) \psi\_{j+1}^\* \phi\_{j+1\prime} \tag{62}$$

$$\chi(\psi|\phi)\_{\boldsymbol{f}} \coloneqq \sum\_{j=1}^{N-1} \chi\_1^\*(\boldsymbol{v}, \boldsymbol{j}) \psi\_{\boldsymbol{f}}^\* \phi\_{\boldsymbol{f}}.\tag{63}$$

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 Pb<sup>b</sup> and Pb<sup>f</sup> are the "adjoint" of each other, on a finite interval ½ � a; b , when

$$\left(\psi|\widehat{P}\_b\phi\right)\_b = \left(\widehat{P}\_f\psi|\phi\right)\_f. \tag{64}$$

together with the boundary condition on the wave functions ψ and f,

$$
\psi\_N = e^{i\theta} \psi\_{1'} \quad \phi\_N = e^{i\theta} \phi\_{1'} \tag{65}
$$

where θ∈½ Þ 0; 2π is an arbitrary phase. This gets rid of boundary terms.

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 [10–13].

### 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

$$V(\mathbf{x}) = \begin{cases} \; \; \approx \; \; \; \; \; \; \mathbf{x} \le \mathbf{0}, \\\; \; c \; \; \; \; \; \; \; \; x > \mathbf{0}, \end{cases} \tag{66}$$

where ψnð Þx is the eigenfunction of the Hamiltonian with energy En, Eq. (67). A plot of these time-type eigenstates, with M ¼ 600, is found in Figure 3. We can identify the classical trajec-

Figure 3. Three-dimensional plot of the squared modulus j j h i <sup>q</sup>j<sup>t</sup> <sup>2</sup> of the time eigenstates for the wall-linear potential in

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.

[1] Boole G. A Treatise on the Calculus of Finite Differences. New York: Cambridge Univer-

[2] Harmuth HF, Meert B. Dogma of the Continuum and the Calculus Fo Finite Differences

[3] Jordan C. Calculus of Finite Differences. 2nd ed. New York: Chelsea Publishing Com-

[4] Richardson CH. An Introduction to the Calculus Offinite Differences. Toronto: D. Van

in Quantum Physics. San Diega: Elsevier Academic Press; 2005

<sup>c</sup> ; <sup>0</sup> in that figure; they are the regions in which the

Exact Finite Differences for Quantum Mechanics http://dx.doi.org/10.5772/intechopen.71956 177

<sup>¼</sup> En

coordinate representation, using 600 energy eigenfunctions. Dimensionless units.

probability is higher. We can also identify the interference pattern between them.

tories with initial conditions x0; p<sup>0</sup>

Armando Martínez-Pérez and Gabino Torres-Vega\*

Physics Department, Cinvestav Mexico City, México

\*Address all correspondence to: gabino@fis.cinvestav.mx

Author details

References

sity Press; 2009 (1860)

pany; 1950

Nostrand; 1954

where c > 0. The eigenfunction corresponding to this potential is

$$\psi\_E(\mathbf{x}) = d \operatorname{Ai} \left[ \sqrt[3]{\frac{2mc}{\hbar^2}} \left( \mathbf{x} - \frac{E}{c} \right) \right],\tag{67}$$

where Ai denotes the Airy function and d is the normalization factor, m is the mass of the quantum particle and ℏ is Planck's constant divided by 2π. The boundary condition ψEð Þ¼ x ¼ 0 0 provides an expression for the energy eigenvalues E, which is

$$E\_n = -\sqrt[3]{\frac{\hbar^2 c^2}{2m}} \alpha\_{n+1} \quad \text{ } n = 0, 1, \dots \text{ } \tag{68}$$

where f g α<sup>n</sup> are the roots of the Airy function, which are negative quantities.

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 Tb ¼ �iℏDb,f . The eigenfunctions of this time-type operator are calculated as in Eq. (38)

$$\langle \mathbf{x} | t \rangle = \sum\_{n=0}^{M} e^{-i \, \, t \, E\_n} \psi\_n(\mathbf{x}),\tag{69}$$

Figure 3. Three-dimensional plot of the squared modulus j j h i <sup>q</sup>j<sup>t</sup> <sup>2</sup> of the time eigenstates for the wall-linear potential in coordinate representation, using 600 energy eigenfunctions. Dimensionless units.

where ψnð Þx is the eigenfunction of the Hamiltonian with energy En, Eq. (67). A plot of these time-type eigenstates, with M ¼ 600, is found in Figure 3. We can identify the classical trajectories with initial conditions x0; p<sup>0</sup> <sup>¼</sup> En <sup>c</sup> ; <sup>0</sup> in that figure; they are the regions in which the probability is higher. We can also identify the interference pattern between them.

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.
