Abstract
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 \(\mathrm{d}\langle\hat{p}\rangle/\mathrm{d}t=\langle{{f_\mathrm{B}}}\rangle\).
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DE VINCENZO, S. Confinement, average forces, and the Ehrenfest theorem for a one-dimensional particle. Pramana - J Phys 80, 797–810 (2013). https://doi.org/10.1007/s12043-013-0517-4
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DOI: https://doi.org/10.1007/s12043-013-0517-4