Confinement, average forces, and the Ehrenfest theorem for a one-dimensional particle

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) -> aEuro parts per thousand a 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, . After calculating similar general expressions for the mean value of the position () and momentum () operators, the Ehrenfest theorem for a particle-in-an-infinite-square-well-potential (i.e., and ) is proven. The formal time derivatives of the mean value of the position () and momentum () 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, and , are obtained where b.t. denotes a boundary term and 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 (), 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., is equal to . 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 .