1Chapter-2Theory of solidsQuantum MechanicsIn physics, quantum mechanics is the study of the relationship between energy quanta (radiation) and matter. Quantum theory generalizes all classical theories, including mechanics, electromagnetism (except general relativity), and provides accurate descriptions for many previously unexplained phenomena such as black body radiation and stable electron orbits.The effects of quantum mechanics are typically not observable on macroscopic scales, but become evident at the atomic and subatomic level.Why Quantum Mechanics?To progress the modeling of carriers in crystal, one should be able to describe the electronic situation inside an isolated semiconductor atom.Classical Mechanics = Newtonian Mech. + Wave Mech. Æincorrect in the atomic dimensional systemQuantum Mechanics!Maxwell’s eq.QM is more precise description of nature that reduces to classical mechanics in the limit where the Masses and Energies of the particles involved are large.Wave propagation.aviElectromagnetic wavesClassical Mechanics?2James Clerk Maxwell1831-1879The Maxwell EquationsMaxwell’s Equation0BDJDHBE=⋅∇=⋅∇+∂∂=×∇∂∂−=×∇ρttχEPM,HµHBP,EεED000εµε=+==+==Classical MechanicsMaxwell’s Electromagnetic wave equation2200222222tEzEyExEr∂∂=∂∂+∂∂+∂∂µεε2v1Classical Mechanics History of Quantum Mechanics1900 Blackbody radiation explained by Plank1905 Photoelectric effect explained by Einstein1906 Thomson model of the atom 1911 Rutherford model of the atom1913 Bohr model of the atomHydrogen atom spectrum explanation1924 de-Broglie matter waveWave-particle duality of electron1925 Heigenberg matrix mechanics 1926 Uhlenbeck and Goudsmitelectron spin ¼Relativistic Q.M.1924 Schödinger wave mechanicsMathematical unify of classical wave theory+de Broglie material wave1928 Dirac Unification of relative and non-relative Q.M.non-relativistic Q.M.Atomic Models3Atomic ModelsThomson model1906Rutherford model1911Bohr model1913Rutherford scattering experiment showed it to be incorrect.---------electronpositively charged matter---------electronpositively nucleus+αAu foilTiny, massive, positively charged nucleus surrounded by electrons.+-Classically, spiral rapidly into the nucleus as it radiates energy due to its accelerationDiscrete electron energy+--Bohr Model of the Atom Bohr atomic model resolved ‘Rutherford atom dilemma’ and explained the discrete nature of the spectra emitted by heated gases.Electrons in an atom were restricted to certain well-defined orbit, or orbiting electrons could take on quantized values of angular momentum Ln.Assumption...,,nnvrmLnn3210=== hCentrifugal force = Coulombic attractionnrvm02024nrq⋅πεStable+n = 1n = 2n = 3n = 4-13.6 eV-3.40 eV-1.51 eV-0.85 eV-0.54 eVn = 5EnergyIntensityAn electron can circle a nucleusonly if its orbit contains an integral number of de Broglie wavelengthλ= h/mv.Bohr Model of the AtomHydrogen atomBohr Model of the Atom()20204Zemnrnhπε=Bohr radiusKinetic energy = nrqvm022042121πε=Potential energy = nrq024πε−(P.E set =0 at r=∞)Total energy = Kinetic energy + Potential energy Energy level()eVn.nqmEn2204061342−=−=hπεnrq02421πε−=[eV] = 1.6 x 10-19joules4Bohr Model of the AtomHydrogen atom()20204Zemnrnhπε=()eVn.nqmEn2204061342−=−=hπεEnergy difference between two state −⋅=−=∆2220211421iffinnqEEEhπεnf= 1: Lyman seriesnf= 2: Balmer seriesnf= 3: Paschen seriesnf= 4: Braket seriesnf= 5: Pfund seriesExercise• Determine the first three electron energies in the hydrogen atom.Theenergyoftheelectroninthehydrogenatom(Z =1).Electron energy,En.0E=KE-15-5-101-13.6 eVGround state2-3.403-1.514-0.855-0.54nn=∞Ionizationenergy,EIn=1Continuum of energy. Electron is freeExcited statesAtomic SpectraPhotonλλEmission spectrum+λPhotonλAbsorption spectrum+(a)(b)The physical origin of (a) emission and (b) absorption spectra.Atomic Spectra5n=2Photonn=1(b)Justaftercollision (c)Photonemission(a) Before collisionn=1AtomAtomAn Atom can become excited by a collision with another atom.When it returns to its ground energy state, it emits a photon.Atomic SpectraEnergy01s12s2p23s3p33d4s4p44d4f5s5p55d5fl=0l=1l=2l=3-13.6eVnlPhotonAn illustration of the allowed photon emission processes.Photon emission involves∆l= ±1.Atomic SpectraBrief Introduction to Quantum Mechanics Nature of Light Wave Property of Light Blackbody Radiation Photoelectric Effect Compton Effect  De Broglie Material Wave Wave-particle duality Uncertainty principle Schrödinger equationNature of LightReflectionWhat is Light?Light has a Wave-Particle Duality; in some cases it behaves as a wave, and in other cases it behaves as a particle.The name of the particle associated to light is called a photon. The light is reflected from the mirror-pond and reaches our eyes: we see an image of these objects. The pond gives a symetrical image of the objects, with respect to the mirror surface RefractionDiffractionThe white rim of the cloud is caused by diffraction. The light coming from the sun is bent by the water droplets present in the clouds. The waves interferebetween each other. When two waves canceleach other ( destructive interference) we see dark. When they combine (constructive interference), the light appears brighterScatteringThe blue haze is due to a physics phenomenon called scattering. Eucalyptus trees release oil droplets into the atmosphere. The sunlight hits these droplets and the different wavelengths get deflected in different directions. DispersionThe sun is refracted by the water dropletspresent in the atmosphere. White light which is a mixture of colors, is then separated into its different wavelengths: this phenomenon is called dispersion. Halos are natural phenomena: they are a ring of light in the sky. Halos form when light coming from the moon or the sun is refracted or bent by ice crystalspresent in the atmosphere. The wavelengths corresponding to the blueare deflected more than the wavelengths corresponding to the red. Our eyes see a blue haze. Interference6Young’s Double-slit Interference ExperimentPS1S2Constructive interferenceDestructive interferencePhotographic film showingYoung's fringesWave Property of LightWave OpticsWave Property of LightX-Ray DiffractionX-rays with singlewavelengthPowdered crystal orpolycrystalline materialScattered X-raysPhotographic filmX-rays with allwavelengthsSingle crystalPhotographic