Quantization rules for low dimensional quantum dots

Abstract

This paper applies the analytical transfer matrix method (ATMM) to calculate energy eigenvalues of a particle in low dimensional sharp confining potential for the first time, and deduces the quantization rules of this system. It presents three cases in which the applied method works very well. In the first quantum dot, the energy eigenvalues and eigenfunction are obtained, and compared with those acquired from the exact numerical analysis and the WKB (Wentzel, Kramers and Brillouin) method; in the second or the third case, we get the energy eigenvalues by the ATMM, and compare them with the EBK (Einstein, Brillouin and Keller) results or the wavefunction outcomes. From the comparisons, we find that the semiclassical method (WKB, EBK or wavefunction) is inexact in such systems.