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

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      https://www.ias.ac.in/article/fulltext/pram/080/05/0797-0810

    • Keywords

       

      Quantum mechanics; Schrödinger equation; confinement in one dimension; average forces; Ehrenfest theorem.

    • 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 $V_{0} \rightarrow \infty$ 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., $d\langle \hat{X} \rangle /dt = \langle \hat{P} \rangle /M$ and $d\langle \hat{P} \rangle /dt = \langle \hat{F} \rangle$) 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, $d\langle \hat{x} \rangle /dt = \langle \hat{p} \rangle /M$ and $d\langle \hat{p} \rangle /dt = b.t. + \langle \hat{f} \rangle$, 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 eigen-states to the Hamiltonian describing a particle-in-a-box with $\nu(x) = 0(\Rightarrow \hat{f} = 0)$, 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., $d\langle \hat{p} \rangle /dt$ 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 $d\langle \hat{p} \rangle /dt = \langle f_{B} \rangle$.

    • Author Affiliations

       

      Salvatore De Vincenzo1

      1. Escuela de Física, Facultad de Ciencias, Universidad Central de Venezuela, A.P. 47145, Caracas 1041-A, Venezuela
    • Dates

       
  • Pramana – Journal of Physics | News

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