1Phy107 Fall 20061From Last Time…Today• Molecules, metals and semiconductors• Important new Quantum Mechanical Concepts– Indistinguishability:– Symmetries of the wavefunction:Symmetric and Antisymmetric– Pauli exclusion principle: only one fermion per state–Spin• Final concepts needed to understand the hydrogenatom and the periodic tablePhy107 Fall 20062Atoms and Molecules• Have talked about atoms– Atoms are a central nucleus with some number ofelectrons orbiting around it.– Number of orbiting electrons determines what element• Molecules:– One or more atoms bonded together.Phy107 Fall 20063Simple molecules•Water: H2O• Carbon dioxide: CO2• Even these simple moleculescan be quite complex.• Many nuclei, many electrons.• However some properties can be determinedwithout worrying too much about the details.Phy107 Fall 20064Symmetry• The symmetries of fermions and bosonswere a little subtle.• Symmetries show up in many situations,many times in more direct ways.• Both water and carbon dioxidehave spatial symmetries:Phy107 Fall 20065Molecular Symmetries• These symmetries can determine many physicalproperties.• Can be related to microscopic quantummechanical properties such as the wavefunctionand the probability.• These are easiest to see if we start with a verysimple molecule– Two protons and one electron.Phy107 Fall 20066Molecule questionA simple molecule consists of two protons andone electron orbiting around them.This molecule isA. Helium moleculeB. Hydrogen moleculeC. Lithium moleculeH2electronionized hydrogen moleculeElectron must be described as a wave.Use a wavefunction to do this.The square of the wavefunction is the probability offinding the electron.2Phy107 Fall 20067A two atom moleculeWhat do we expect for the charge density?If atoms are identical, do we expect morecharge on right, left?No reason to expect electron to reside on oneatom over the other.One electron orbiting two atomsWhat wavefunction is consistent with electronnot preferring one atom over the other?Phy107 Fall 20068• Two ways to superimpose statesTwo possible wavefunctionsElec onleft atomElec onright atom+Elec onleft atomElec onright atom— Symmetric Antisymmetric• In quantum mechanics, we can have BOTH.• Wavefunction is an equal superposition ofelectron on left atom and electron on right atom.Phy107 Fall 20069• These are obtained by adding or subtractionquantum states on either atom.• Both give symmetric charge density……but details slightly different.Phy107 Fall 200610QuestionWhich state has the lower energy? Symmetric AntisymmetricA. SymmetricB. AntisymmetricC. Both samePhy107 Fall 200611Two-atommoleculeSymmetry of the wavefunctionCompareparticle ina box= LTwo half-wavelengthsp =h=hLmomentum= 2LOne half-wavelengthp =h=h2LmomentumSymmetricAnti-symmetricPhy107 Fall 200612Quantum state energies• Symmetric state is the ‘ground state’.• Antisymmetric state is the excited state.– Wavelength half as large– momentum twice as large– Larger momentum -> larger kinetic energy• Since momentum dependson ‘size of box’ (atomicseparation)…• … energy differenceincreases as atomseparation decreases.Atom separationAnti-symmetricstateSymmetric state3Phy107 Fall 200613A three-atom moleculeEnergy levelsWavefunctionsPhy107 Fall 200614A six-atom moleculeWavefunctionsEnergy levelsCan see differentwavelengths forthe differentmolecular statesPhy107 Fall 200615Solids• Solids consist of many atoms bonded together• Many possible ways to combine atomicwavefunctions to get charge density with correctsymmetry.• All these quantum states have slightly differentenergies.• Solid is similar to atom or molecule,except quantum states are extremely closetogether in energy.Phy107 Fall 200616Energy levels in a solid• Solids consists of ~1024 atoms• Energy levels spaced extremely close together3-atom molecule 6-atom molecule 1024-atom‘molecule’Phy107 Fall 200617Energy bands in a solid• This energy region of densely packedquantum states in called an energy band.• Each quantum state on an individual atom(for instance, 1s, 2s, etc) leads to one ofthese energy bands.• The detailed arrangement of these energybands is called the band structure.Phy107 Fall 200618Band structureTwo atommoleculeFive atom‘molecule’Many atomsolidAtomic separation Atomic separation Atomic separation• Atomic orbitals interact to form new energy levels4Phy107 Fall 200619Occupation of quantum states• These quantum states are filled with electronsjust as atomic states get filled one by one, lowestenergy first, just like an atom.• Dramatically more electrons to fill the states!• But since each band arises from an atomicquantum state,• But due to details in which atomic states broadeninto bands, sometimes bands overlap and and arenot completely full or empty.Phy107 Fall 200620Solid sodium (metal)Sodium atom Sodium metal1s2s3s2p3pNa = [Ne]3s1empty1 electron6 electrons2 electrons2 electronsFullFullFull3p3s2p2s1sPartially FullThis band notcompletelyoccupiedPhy107 Fall 200621Electrical conductivity• This little detail turns out todramatically effect theelectrical properties ofmaterials.• In particular whether they willcarry an electrical currentOnly a partially full band will carry electrical current!Phy107 Fall 200622Metals, insulators, semiconductors• Only partially full bands carry current• Completely full, or completely empty bands, carry no currentMetal(at least onepartially full band)Insulator(all bands completelyfull or empty)(empty)(full)Semiconductor(insulator with smallenergy gap)(full)(empty)Phy107 Fall 200623Properties of some elementsMetalsCopperZincAluminumGoldPlatinumSodiumCalciumInsulatorsDiamond (Carbon)SulfurSemiconductorsSiliconGermaniumWider variety of insulators, semiconductors can belisted if compounds are includedPhy107 Fall 200624Doped semiconductors• If semiconductors are insulators,why is my computer made out of them?• An impurity atom (such as Arsenic) and besubstituted for one of the Si atomsThis dramaticallychanges theelectrical properties5Phy107 Fall 200625Doping a semiconductor• Elements in samecolumn have samenumber of ‘outer’electrons.• Substitution ofelement fromanother columnchanges band fillingOne fewerelectron(acceptors)One extraelectron(donors)Phy107 Fall 200626Example: Phosphorus-doped siliconSi = [Ne]3p2P = [Ne]3p3• P has one extra
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