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Ntial well E,V(x) x - only discrete energies E allowed - wave functions are standing waves b) Free Electrons - plane wave solutions - any energy allowed: jkx (x) Ce j(kx- t) (x,t) Ce 2 k2 E 2m - Parabolic E versus k E k Lecture 10 Kronig Penny Model 10/12/00 4 c) Periodic Potential E,V(x) Energy bands x - ability of electrons to tunnel between barrier walls spreads out the discrete energy levels seen for isolated wells into bands - We fine a number of bands of energi.

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The Kronig–Penney model (named after Ralph Kronig and William Penney) is a simple, idealized quantum-mechanical system that consists of an infinite periodic array of rectangular potential barriers.

The Kronig–Penney (KP) model is one of the fundamental models of solid state physics and has since its inception[1] received significant attention. It combines predictive power with accessibility and has, in fact, become a standard model that is taught in almost all solid state classes for undergraduate students.

Hence we conclude that the total number of possible wave functions in any energy band is equal to the number of unit cells N. We have concluded in the Kronig-Penny model that the energy discontinuities in a monatomic one-dimensional lattice occur when k = (nπ/a), where n is any positive or negative integer.

Purpose: to demonstrate that in solids, where many atoms stay closely, the interference between atoms will create allowed and forbidden bands of energy for electrons. To simplify the analysis, we only consider a one-dimensional system where atoms are aligned and equally spaced.

The Kronig-Penney model [1] is a simplified model for an electron in a one-dimensional periodic potential. The possible states that the electron can occupy are determined by the Schrödinger equation, In the case of the Kroning-Penney model, the potential V(x) is a periodic square wave.

where k1=√2mE/ℏ and k2=√2m(E−V0)/ℏ, the appropriate free particle wavevectors, thus for E<V0,k2 is imaginary.

The basic assumptions of the Kronig-Penney model are that the system is in a quantum mechanical state and the Schrödinger equation is solved. The boundary conditions are given by the equation of motion and the energy equation.

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