Open access peer-reviewed chapter

Exact Finite Differences for Quantum Mechanics

By Armando Martínez-Pérez and Gabino Torres-Vega

Submitted: August 2nd 2017Reviewed: October 25th 2017Published: December 20th 2017

DOI: 10.5772/intechopen.71956

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. ); 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 gx(this class of functions is the domain of the momentum operator in Quantum Mechanics), on a partition P=x1x2xNof Nnon-uniformly spaced points xj1N, is defined through the requirement that

Dbgxjgxjgxjj1χ2j1=gxj,E1

where j=xj+1xjand the spacing function χ2j, which is a replacement for the usual spacing function j, is obtained by solving the above equality for χ2j,

χ2j1gxjgxjj1gxj=1gxjk=11k1k!gkxjj1k.E2

This is an expression which is valid for points xjdifferent from the zeroes of gx.

A definition for forward finite differences at xjis

Dfgxjgxj+jgxjχ1j=gxj,E3

where

χ1jgxj+jgxjgxj=1gxjk=11k!gkxjjk,E4

valid for points different from the zeroes of gx.

These definitions coincide with the usual finite differences derivative when the function to which they act on is the linear function gx=a0+a1x, a0,a1C. 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).

Example. For the quadratic function gx=a0+a1x+a2x2, a0,a1,a2C, the spacing function χ2xjbecomes

χ2xj=jj2a1a2+2x,E5

where xa1/2a2. A plot of this function is shown in Figure 1 for j=1. Figure 1.Three-dimensional plot of χ2xj for the quadratic function gx=a0+a1x+a2x2 with ∆j=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 χ1jand χ2j.

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

Let us consider the exact backward and forward finite differences derivatives of evx, at xj, given by

Dbevxj evxjevxj1χ2vj1=vevxjandDfevxj evxj+1evxjχ1vj=vevxj,E6

where vCcan be a pure real or pure imaginary constant, and the spacing functions χ1vjand χ2vjare defined as

χ1vjevj1vj+v2j2+Oj3,E7

and

χ2vj 1evjvjv2j2+Oj3.E8

Note that we recover the usual definitions of a finite differences derivative in the limit j0(N) in which case χ1vj=χ2vjj. Hereafter, the exact finite differences derivatives that we will consider are

Dbgj gjgj1χ2vj1andDfgj gj+1gjχ1vj,E9

with χ1vjand χ2vjgiven in Eqs. (7) and (8), and some properties of these definitions follow. There is a plot of χ1vjin Figure 2. The spacing function χ1vjis defined for finite values of vand j. Figure 2.Three-dimensional plot of χ1vj for the exponential function evx.

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

j=nmχ1vjDfgj=j=nmgj+1gj=gm+1gn,E10

where 1n<m<N, and

j=nmχ2vj1Dbgj=gmgn1,E11

where 1<n<mN.

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

j=nmχ1vjvevxj=j=nmχ1vjDfevxj=evxm+1evxn,E12

where 1n<m<N, and

j=nmχ2vj1vevxj=evxmevxn1,E13

where 1<n<mN.

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

Dfghxj=ghxj+1ghxjχ1vj=vghxj+1ghxjevhxj+1hxj1evhxj+1hxj1vχ1vj=Dfgjχ1vh,jχ1vj,E14

where

Dfgj vghxj+1ghxjevhxj+1hxj1,E15

χ1vh,j=evhxj+1hxj1vand h,j=hxj+1hxj, and

Dbghxj=Dbgjχ2vh,j1χ2vj1,E16

where χ2vh,j1=1evhxjhxj1/v,

Dbgj vghxjghxj11evhxjhxj1.E17

and hxis any absolutely continuous complex function on ab.

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

Dfghj=gj+1hj+1gjhjχ1vj=gj+1hj+1hjχ1vj+gj+1gjχ1vjhj=gj+1Dfhj+hjDfgj=evjgj+1Dbhj+1+hjDfgj,E18

where 1j<N. Also, for the backwards derivative, we have

Dbghj=gjDbhj+hj1Dbgj=gjDbhj+evj1hj1Dfgj1,E19

where 1<jN.

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

Dbghj=1χ2vj1gjhjgj1hj1=1χ2vj1gjhjhj1hjhj1+gjgj1hjhjhj1=Dbgjhj1gjDbhjhjhj1,E20
Dbghj=Dbgjhjgj1Dbhjhjhj1,E21
Dfghj=Dfgjhj+1gjDfhjhjhj+1,E22
Dfghj=Dfgjhjgj+1Dfhjhjhj+1.E23

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

1χ2vj1χ1vj=v,andχ1vjχ2vj=evj.E24

For instance, these equalities imply that

Dfgj=evjDbgj+1.E25

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,

j=nmχ2vjgj+1Dbhj+1+j=nmχ1vjhjDfgj=gm+1hm+1gnhn,E26

where 1j<N, and

j=nmχ2vj1gjDbhj+j=nmχ1vj1hj1Dfgj1=gmhmgn1hn1,E27

where 1<jN.

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 Db,fgiven by

E28

and

E29

We have used the definition for the backward derivative Dbgjfor all the rows of the backward derivative matrix Dbbut not for the first line in which we have instead used the forward derivative Dfg1. A similar thing was done for the forward derivative matrix Df. These matrices act on bounded vectors g=g1g2gNTCN.

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

Db2g1=vg2g1χ1v1=vDfg1,E30
Db2g2=vg2g1χ2v1=vDbg2,E31
Db2gj=1χ2vj1gjgj1χ2vj1gj1gj2χ2vj2=DfgjDfgj1χ2vj1,j=3,,N.E32

These expressions have the exponential function evxas 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,f

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

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

DbλI=λλ+1χ1v11χ2v11χ2v2λ1χ2v3λ1χ2vN1λ=0,E33

whose roots are λ0=0, λv=1/χ1v1+1/χ2v1=vand λj=1/χ2vj, 2≤ j ≤ N −1. Let us denote by eλ=eλ,1eλ,2eλ,NTto the eigenvector corresponding to the eigenvalue λ. The system of equations for the components of the eigenvectors is

eλ,kχ2vk+1χ2vkλeλ,k+1=0,E34

with k=1,,N1. Then, the eigenvectors are

e0=C111,ev=Cevx1evx2evxN,ej=C000yj+1yN,E35

where Cis the normalization constant, and

ym=k=mN11χ2vkχ2vj,E36

where 2j<mN. The quantities ymusually are very small.

On the other hand, for the forward finite difference matrix, Df, the characteristic polynomial is

DfλI=1N1λvλ1χ1v1+λ1χ1v2+λ1χ1vN2+λ=0,E37

Thus, the eigenvalues for the forward derivative are λ0=0, λv=vand λj=1/χ1vj, 1≤ j ≤ N −2, and the corresponding eigenvectors are

e0=C111,ev=Cevx1evx2evxN,ej=Cw1w2wj00,E38

where

wm=k=1m11χ1vkχ1vjE39

and 1m<jN2. The quantities wmare also very small; in fact, they vanish for the equally spaced partition.

The matrices, Db,f, have the same eigenvalues λ0and λv, and eigenvectors which are the discretization of the function gvx=Cevxon the partition x1x2xN(the eigenvector 111Tcorrespond to the eigenvalue 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 Db,fgjhave the same eigenfunctions as the matrices Db,fwhich 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:

Dbxj=xjxj1χ2vj1=j1χ2vj1=1+v2j1+Oj12,E40

from a local point of view, we have

Dbxgj=xjDbgj+gj1Dbxj=xjDbgj+gj1j1χ2vj1,E41

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

Dbxgj=gj11+v2j1+Oj12.

Thus, the commutator between Dband xbecomes one in the limit of small j, or large N, because it also happens that gj1gj.

We now consider the commutator between the coordinate matrix Qdiagx1x2xN, and the forward derivative matrix Df. That commutator is

E42

The small j(N) approximation of this commutator is just

E43

There is coincidence with the local calculation; as expected, this matrix approaches the identity matrix in the small jlimit. Note that this matrix is composed of backward translations of the first N1points and a forward translation of the point N1without 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 . Therefore, here we investigate briefly how translations are carried out by means of the derivative matrices Db,fused as their generators. We will focus on the translation of the common eigenvector ev=evx1evx2evxNTof 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

esDb,f=k=0sDb,fkk!,E44

where sR. Since evis an eigenvector of Db,fwith eigenvalue v(see Eqs. (35) and (38)), it follows that Db,fkev=vkev, k=1,2,, and then,

esDb,fev=k=0svkk!ev=esvev=Cevs+x1evs+x2evs+xN,E45

that is, esvis an eigenvalue of esDb,fwith corresponding eigenvector ev, but the right-hand side of this equality is also a translation by the amount sof the domain of the derivative operators. We point out that sis arbitrary and then the vector ev=esDb,fevis the function evxevaluated at the points of the translated partition P=x1+sx2+sxN+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 , a constant) and with periodic boundary conditions ev,1=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 ipshould be understood, and we will omit it from the formulae below for the sake of simplicity of notation.

Given a function gpin the L1-space and a non-uniform partition P=xNxN+1xN, with xN=xN, the function.

Fgxj12πeixjpgpdp,E46

is the continuous Fourier transform of gpat xj. Having introduced the summation with weights χ1vjof Eq. (10), here we define two discrete Fourier transforms at pas

Fbgpp2sinpxNj=NNχ2Δjeixj+1pgj+1.E47
Ffgpp2sinpxNj=NNχ1ΔjeixjpgjE48

Now, the discrete derivative of the product eixpgat xj, with derivative eigenvalue ip, is readily computed to give (see Eq. (18)).

Dfeixpgj=ipgj+1eixpj+eixjpDfgj.E49

The summation of this equality, with weights χ1ipj, results in

j=NN1χ1jDfeixpgj=ipj=NN1χ1jgj+1eixjp+j=NN1χ1jeixjpDfgj,E50

or

j=NN1χ1jeixjpiDfgj=pj=NN1χ2jeixj+1pgj+1+j=NN1χ1jiDfeixpgj.E51

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

FfiDfgp=pFbgpipeixNpgNeixNpgN2sinpxN.E52

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 g=eixp)

Dfeixpgj=eijpeixj+1pDbgj+1+gjDfeixpj.E53

The summation of this equality, with weights χ1j, results in

j=NN1χ2jeixj+1piDbgj+1=pj=NN1χ1jeixjpgj+j=NN1χ1jiDfeixpgj,E54

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

FbiDbgp=pFfgpipeixNpgNeixNpgN2sinpxN.E55

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

deixjpdp=ixjeixjpE56

results in

ixjdpeixjphp=dpeixjpdhpdp+eixjphpp=E57

or in terms of continuous Fourier transforms,

xjFhj=Fidhdpjih2πeixjphpp=.E58

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,ϕ2Pabdefined on the partition P=x1x2xNof ab. We obtain

j=1N1χ2vjψj+1iℏDbϕj+1j=1N1χ1vjϕjiℏDfψj=iℏψNϕNψ1ϕ1.E59

This equality is rewritten as

ψP̂bϕbP̂fψϕf=iℏψNϕNψ1ϕ1,E60

where the momentum-like operators P̂band P̂fare defined as

P̂b iℏDb,P̂f iℏDf,E61

and the bilinear forms ψϕband ψϕfare defined as

ψϕb j=1N1χ2vjψj+1ϕj+1,E62
ψϕf j=1N1χ1vjψjϕj.E63

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 P̂band P̂fare the “adjoint” of each other, on a finite interval ab, when

ψP̂bϕb=P̂fψϕf,E64

together with the boundary condition on the wave functions ψand ϕ,

ψN=eψ1,ϕN=eϕ1,E65

where θ02πis 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, 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

Vx=,x0,cx,x>0,E66

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

ψEx=dAi2mc23xEc,E67

where Ai denotes the Airy function and dis the normalization factor, mis the mass of the quantum particle and is Planck’s constant divided by 2π. The boundary condition ψEx=0=0provides an expression for the energy eigenvalues E, which is

En=2c22m3αn+1,n=0,1,E68

where αnare 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 T̂=iℏDb,f. The eigenfunctions of this time-type operator are calculated as in Eq. (38)

xt=n=0MeitEnψnx,E69

where ψnxis the eigenfunction of the Hamiltonian with energy En, Eq. (67). A plot of these time-type eigenstates, with M=600, is found in Figure 3. We can identify the classical trajectories with initial conditions x0p0=Enc0in that figure; they are the regions in which the probability is higher. We can also identify the interference pattern between them. Figure 3.Three-dimensional plot of the squared modulus qt2 of the time eigenstates for the wall-linear potential in coordinate representation, using 600 energy eigenfunctions. Dimensionless units.

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.

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Armando Martínez-Pérez and Gabino Torres-Vega (December 20th 2017). Exact Finite Differences for Quantum Mechanics, Numerical Simulations in Engineering and Science, Srinivas P. Rao, IntechOpen, DOI: 10.5772/intechopen.71956. Available from:

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