Investigation on Threshold Conditions Bound States Band Structure and Transmission by Periodic Delta Potentials

Abstract

Study of bound states, scattering states and resonances by delta function potentials remains a continuing area of interest in quantum mechanics both from pedagogical and application point of view. Delta function potential finds application in almost all fields of physics like optics, atomic, molecular and condensed matter physics. Its role in the study of heterostructure, resonant tunneling diodes and low dimensional semiconductor devices has increased to a great extent in recent years. In view of this in this thesis a systematic quantum mechanical investigation of different types of locally periodic delta potentials is undertaken. The number of bound states Nb generated by N attractive delta potentials satisfies the condition 1 and#8804; Nb and#8804; N in one dimension (1D) whereas in the case of three dimension( 3D) the corresponding condition is 0 and#8804; Nb and#8804; N. Algebraic equations that govern the generation of threshold energy bound states are obtained by solving the Schrodinger equation using appropriate boundary conditions for N attractive delta potentials. In particular for a set of uniformly placed delta function potentials of same strength which is called Dirac comb, very elegant polynomial expressions in terms of dimensionless parameter g = and#955;Va govern the threshold conditions for the occurence of bound states. The parameter and#955; is used to compensate the dimensionality of delta term, V and a are the potential strength and spacing parameter respectively. For a given N both in 1D and 3D cases the numerical values of gn, n = 1, 2...., N are obtained such that when g = gn, n bound states occur including one threshold energy bound state. Infact gn are the roots of the polynomials having integer coefficient. Based on numerical calculations, it can be easily seen when N becomes large, gN and#8594; 4 and average value lt gn gt = 2. An empirical formula is constructed to study the variation of gn as a function of n. One of the main accomplishments of the present work is the heuristic inductive approach formulated for the construction ....