# Basic Quantum Mechanics 20 and 22 January 2016

Basic Quantum Mechanics 20 and 22 January 2016 What Is An Energy Band And How Does It Explain The Operation Of Classical Semiconductor Devices and Quantum Well Devices? To Address These Questions, We Will Study: Introduction to quantum mechanics (Chap.2) Quantum theory for semiconductors (Chap. 3) Allowed and forbidden energy bands (Chap. 3.1) Also refer to Appendices: Table B 2 (Conversion Factors), Table B.3

(Physical Constants), and Tables B.4 and B.5 Si, Ge, and GaAs key attributes and properties. We will expand the derivation Schrdingers Wave Equation as summarized in Appendix E. Will use a transmission line analogy for discussing Schrdingers wave equation solutions 1 Classical Mechanics and Quantum Mechanics

Mechanics: the study of the behavior of physical bodies when subjected to forces or displacements Classical Mechanics: describing the motion of macroscopic objects. Macroscopic: measurable or observable by naked eyes Quantum Mechanics: describing

behavior of systems at atomic length scales and smaller . 2 Incident light with frequency Emitted electron kinetic energy = T

Tmax Photoelectric Effect 0 Metal Plate The photoelectric effect ( year1887 by Hertz)

o Experiment results Inconsistency with classical light theory According to the classical wave theory, maximum kinetic energy of the photoelectron is only dependent on the incident intensity of the light, and independent on the light frequency; however, experimental results show that the kinetic energy of the

photoelectron is dependent on the light frequency. Concept of energy quanta 3 Energy Quanta Photoelectric experiment results suggest that the energy in light wave is contained in discrete energy packets, which are called energy quanta or photon

The wave behaviors like particles. The particle is photon Plancks constant: h = 6.62510-34 J-s Photon energy = h Work function of the metal material = ho Maximum kinetic energy of a photoelectron: Tmax= h(-o) 4

5 Electrons Wave Behavior Nickel sample =0 Electron beam

Scattered beam =45 =90 Detector

Davisson-Germer experiment (1927) Electron as a particle has wave-like behavior 6 Wave-Particle Duality Particle-like wave behavior (example, photoelectric effect) Wave-like particle behavior (example, Davisson-Germer experiment)

Wave-particle duality Mathematical descriptions: The momentum of a photon is: The wavelength of a particle is: h p

h p is called the de Broglie wavelength 7 The Uncertainty Principle The Heisenberg Uncertainty Principle (year 1927): It is impossible to simultaneously describe with absolute accuracy the

position and momentum of a particle p x It is impossible to simultaneously describe with absolute accuracy the energy of a particle and the instant of time the particle has this energy E t The Heisenberg uncertainty principle applies to electrons and states that we can not determine the exact position of an electron. Instead, we could determine the probability of finding an electron at a particular

position. 8 Quantum Theory for Semiconductors How to determine the behavior of electrons )and holes) in the semiconductor? Mathematical description of motion of electrons in quantum mechanics Schrdingers Wave Equation Solution of Schrdingers Wave Equation energy band structure and probability of finding a electron at a particular

position 9 Schrdingers Wave Equation One dimensional Schrdingers Wave Equation: 2 2 ( x, t ) ( x, t ) V ( x ) ( x, t ) j

2 2m x t ( x, t ) : Wave function 2

( x, t ) dx ( x, t ) V (x ) : m: 2 , the probability to find a particle in (x,

, the probability density at location x+dx) at time t x and time t Potential function Mass of the particle 10

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