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Mechanics. Thus, we propose that the momentum-like operators Pb<sup>b</sup> and Pb<sup>f</sup> are the "adjoint" of

¼ Pbfψjf � �

<sup>i</sup>θψ1, <sup>f</sup><sup>N</sup> <sup>¼</sup> <sup>e</sup>

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

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

V xð Þ¼ <sup>∞</sup>, x <sup>≤</sup> <sup>0</sup>,

�

c x, x > 0,

ffiffiffiffiffiffiffiffi 2mc ℏ2 3 r

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

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

" # � �

<sup>x</sup> � <sup>E</sup> c

f

, (64)

<sup>i</sup><sup>θ</sup>f1, (65)

, (67)

α<sup>n</sup>þ1, n ¼ 0, 1, ⋯ (68)

�itEnψnð Þ<sup>x</sup> , (69)

(66)

b

ψjPbbf � �

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

ψ<sup>N</sup> ¼ e

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

finding of a time operator in Quantum Mechanics [10–13].

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

ψEð Þ¼ x d Ai

ψEð Þ¼ x ¼ 0 0 provides an expression for the energy eigenvalues E, which is

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

h i <sup>x</sup>j<sup>t</sup> <sup>¼</sup> <sup>X</sup> M

n¼0 e

ffiffiffiffiffiffiffiffiffi ℏ<sup>2</sup>c<sup>2</sup> 2m 3 s

En ¼ �

eigenfunctions of this time-type operator are calculated as in Eq. (38)

6. The particle in a linear potential

influence of the linear potential

each other, on a finite interval ½ � a; b , when

176 Numerical Simulations in Engineering and Science

Armando Martínez-Pérez and Gabino Torres-Vega\*

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

Physics Department, Cinvestav Mexico City, México
