Generalized One-Dimensional Periodic Potential Wells Tending to the Dirac Delta Potential

Physics Pub Date : 2024-01-09 DOI:10.3390/physics6010006
F. Mendoza-Villa, Juan A. Ramos-Guivar, R. M. Espinoza-Bernardo
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Abstract

The solution of a quantum periodic potential in solid state physics is relevant because one can understand how electrons behave in a corresponding crystal. In this paper, the analytical solution of the energy formulation for a one-dimensional periodic potential that meets certain boundary conditions is developed in a didactic and detailed way. In turn, the group speed and effective mass are also calculated from the transcendental energy equation of a potential V=V(x). From this, a comparison is made with periodic potentials with known analytical solutions, such as the Dirac delta, as well as rectangular and triangular potentials. Finally, some limits are proposed in which these periodic potentials of the form V=V(x) approach the periodic Dirac delta potential of positive intensity. Therefore, the calculations described in this paper can be used as the basis for more-complex one-dimensional potentials and related simulations.
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趋向于狄拉克三角势的广义一维周期势阱
量子周期势的求解与固体物理学息息相关,因为人们可以从中了解电子在相应晶体中的行为方式。本文以说教的方式详细介绍了满足特定边界条件的一维周期势的能量公式的解析解。反过来,群速度和有效质量也是由势能 V=V(x) 的超越能量方程计算得出的。在此基础上,与已知分析解的周期势(如狄拉克三角势)以及矩形和三角形势进行了比较。最后,提出了一些极限,在这些极限中,V=V(x) 形式的周期势接近正强度的周期性狄拉克三角势。因此,本文所述计算可作为更复杂的一维势和相关模拟的基础。
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