naa a22 4500 510767176 CHVBK 20180411083200.0 cr unu---uuuuu 180411e20130501xx s 000 0 eng 10.1007/s12043-013-0517-4 doi (NATIONALLICENCE)springer-10.1007/s12043-013-0517-4 DE VINCENZO SALVATORE Escuela de Física, Facultad de Ciencias, Universidad Central de Venezuela, A.P. 47145, 1041-A, Caracas, Venezuela aut Confinement, average forces, and the Ehrenfest theorem for a one-dimensional particle [Elektronische Daten] [SALVATORE DE VINCENZO] The topics of confinement, average forces, and the Ehrenfest theorem are examined for a particle in one spatial dimension. Two specific cases are considered: (i) A free particle moving on the entire real line, which is then permanently confined to a line segment or ‘a box' (this situation is achieved by taking the limit V 0 → ∞ in a finite well potential). This case is called ‘a particle-in-an-infinite-square-well-potential'. (ii) A free particle that has always been moving inside a box (in this case, an external potential is not necessary to confine the particle, only boundary conditions). This case is called ‘a particle-in-a-box'. After developing some basic results for the problem of a particle in a finite square well potential, the limiting procedure that allows us to obtain the average force of the infinite square well potential from the finite well potential problem is re-examined in detail. A general expression is derived for the mean value of the external classical force operator for a particle-in-an-infinite-square-well-potential, $\hat{F}$ . After calculating similar general expressions for the mean value of the position ( $\hat{X}$ ) and momentum ( $\hat{P}$ ) operators, the Ehrenfest theorem for a particle-in-an-infinite-square-well-potential (i.e., $\mathrm{d}\langle\hat{X}\rangle/\mathrm{d}t=\langle\hat{P}\rangle/M$ and $\mathrm{d}\langle\hat{P}\rangle/\mathrm{d}t=\langle\hat{F}\rangle$ ) is proven. The formal time derivatives of the mean value of the position ( $\hat{x}$ ) and momentum ( $\hat{p}$ ) operators for a particle-in-a-box are re-introduced. It is verified that these derivatives present terms that are evaluated at the ends of the box. Specifically, for the wave functions satisfying the Dirichlet boundary condition, the results, $\mathrm{d}\langle\hat{x}\rangle/\mathrm{d}t=\langle\hat{p}\rangle/M$ and $\mathrm{d}\langle\hat{p}\rangle/\mathrm{d}t=\mathrm{b.t.}+\langle\hat{f}\rangle$ , are obtained where b.t. denotes a boundary term and $\hat{f}$ is the external classical force operator for the particle-in-a-box. Thus, it appears that the expected Ehrenfest theorem is not entirely verified. However, by considering a normalized complex general state that is a combination of energy eigenstates to the Hamiltonian describing a particle-in-a-box with v(x) = 0 ( $\Rightarrow\hat{f}=0$ ), the result that the b.t. is equal to the mean value of the external classical force operator for the particle-in-an-infinite-square-well-potential is obtained, i.e., $\mathrm{d}\langle\hat{p}\rangle/\mathrm{d}t$ is equal to $\langle\hat{F}\rangle$ . Moreover, the b.t. is written as the mean value of a quantity that is called boundary quantum force, f B. Thus, the Ehrenfest theorem for a particle-in-a-box can be completed with the formula $\mathrm{d}\langle\hat{p}\rangle/\mathrm{d}t=\langle{{f_\mathrm{B}}}\rangle$ . Indian Academy of Sciences, 2013 Quantum mechanics nationallicence Schrödinger equation nationallicence confinement in one dimension nationallicence average forces nationallicence Ehrenfest theorem nationallicence Pramana Springer-Verlag 80/5(2013-05-01), 797-810 0304-4289 80:5<797 2013 80 12043 https://doi.org/10.1007/s12043-013-0517-4 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s12043-013-0517-4 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- DE VINCENZO SALVATORE Escuela de Física, Facultad de Ciencias, Universidad Central de Venezuela, A.P. 47145, 1041-A, Caracas, Venezuela aut NATIONALLICENCE 773 0- Pramana Springer-Verlag 80/5(2013-05-01), 797-810 0304-4289 80:5<797 2013 80 12043 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer