1Phy107 Fall 20061From Last Time…Today• Superposition of wave functions• Indistinguishability• Electron spin: a new quantum effect• The Hydrogen atom and the periodic table• Hydrogen atom quantum numbers• Quantum jumps, tunneling and measurementsPhy107 Fall 20062Hydrogen Quantum Numbers• Quantum numbers, n, l, ml• n: how charge is distributed radially around thenucleus. Average radial distance.– This determines the energy• l: how spherical the charge distribution– l = 0, spherical, l = 1 less spherical…•ml: rotation of the charge around the z axis– Rotation clockwise orcounterclockwise andhow fast•Small energydifferences forl and ml states n = 2, l =1, ml=±1 n =1, l = 0, ml= 0Phy107 Fall 20063Measuring which slitMeasure induced current frommoving charged particle• Suppose we measure which slit the particle goes through?• Interference pattern is destroyed!• Wavefunction changes instantaneously over entire screen whenmeasurement is made.• Before superposition of wavefunctions through both slits. After onlythrough one slit.Phy107 Fall 20064A superposition state• Margarita or Beer?• This QM state has equal superposition of two.• Each outcome(drinking margarita, drinking beer)is equally likely.• Actual outcome not determined untilmeasurement is made (drink is tasted).Phy107 Fall 20065What is object before themeasurement?• What is this new drink?• Is it really a physical object?• Exactly how does the transformation from thisobject to a beer or a margarita take place?• This is the collapse of the wavefunction.• Details, probabilities in the collapse, dependon the wavefunction, and sometimes themeasurementPhy107 Fall 20066Not universally accepted• Historically, not everyone agreed with thisinterpretation.• Einstein was a notable opponent– ‘God does not play dice’• These ideas hotly debated in the early part ofthe 20th century.• However, one more set of crazy ideas needed tounderstand the hydrogen atom and the periodictable.2Phy107 Fall 20067Spin: An intrinsic property• Free electron, by itself in space, notonly has a charge, but also acts like abar magnet with a N and S pole.• Since electron has charge, could explainthis if the electron is spinning.• Then resulting current loops wouldproduce magnetic field just like a barmagnet.• But as far as we can tell the electron isnot spinningNSPhy107 Fall 20068Electron magnetic moment• Why does it have a magnetic moment?• It is a property of the electron in the same waythat charge is a property.• But there are some differences.– Magnetic moment is a vector: has a size and a direction–It’s size is intrinsic to the electron– but the direction is variable.– The ‘bar magnet’ can point in different directions.Phy107 Fall 20069Quantization of the direction• But like everything in quantum mechanics,this magnitude and direction are quantized.• And also like other things in quantum mechanics,if magnetic moment is very large,the quantization is not noticeable.• But for an electron, the moment is very small.– The quantization effect is very large.– In fact, there is only one magnitude and two possibledirections that the bar magnet can point.– We call these spin up and spin down.– Another quantum number: spin up: +1/2, down -1/2Phy107 Fall 200610Electron spin orientationsSpin downNSNSSpin upThese are two different quantum statesin which an electron can exist.Phy107 Fall 200611Other particles• Other particles also have spin• The proton is also a spin 1/2 particle.• The neutron is a spin 1/2 particle.• The photon is a spin 1 particle.• The graviton is a spin 2 particle.Phy107 Fall 200612Particle in a box• We labeled the quantum states with an integer• The lowest energy state was labeled n=1• This labeled the spatial properties of the wavefunction(wavelength, etc)• Now we have an additional quantum property, spin.– Spin quantum number could be +1/2 or -1/2= 2LOne half-wavelengthp =h=h2LmomentumLThere are two quantum states with n=1Can write them asn = 1, spin =+1/2n = 1, spin = 1/23Phy107 Fall 200613Spin 1/2 particle in a boxWe talked about two quantum statesIn isolated space, which has lower energy?n = 1, spin =+1/2n = 1, spin = 1/2A.B.C. Both samen = 1, spin =+1/2n = 1, spin = 1/2An example of degeneracy: twoquantum states that haveexactly the same energy.Phy107 Fall 200614Indistinguishability• Another property of quantum particles– All electrons are ABSOLUTELY identical.• Never true at the macroscopic scale.• On the macroscopic scale, there is always someaspect that distinguishes two objects.• Perhaps color, or rough or smooth surface• Maybe a small scratch somewhere.• Experimentally, no one has ever found anydifferences between electrons.Phy107 Fall 200615• Quantum Mechanics says that electrons areabsolutely indistinguishable.Treats this as an experimental fact.– For instance, it is impossible to follow an electronthroughout its orbit in order to identify it later.• We can still label the particles, for instance– Electron #1, electron #2, electron #3• But the results will be meaningfulonly if we preserve indistinguishability.• Find that this leads to some unusual consiquensesIndinstinguishability and QM123Phy107 Fall 200616Example: 2 electrons on an atom• Probability of finding an electron at a locationis given by the square of the wavefunction.• We have two electrons,so the question we would is ask is– How likely is it to find one electron at location r1and the other electron at r2?Probability large hereProbability small herePhy107 Fall 200617• Suppose we want to describe the state withElectron#13s stateone electron in a 3s state… and one electron in a 3d stateElectron#23d statePhy107 Fall 200618• Must describe this with a wavefunction that says– We have two electrons– One of the electrons is in s-state, one in d-state• Also must preserve indistinguishabilityOn the atom, theylook like this. (Bothon the same atom).4Phy107 Fall 200619QuestionWhich one of these states doesn’t ‘change’ whenwe switch particle labels.A.B.1Electron1 ins-stateElectron2 ind-state2C.12+Electron1 ins-stateElectron2 ind-state21Electron2 ins-stateElectron1 ind-state12+Electron1 ins-stateElectron2 ins-state21Electron1 ind-stateElectron2 ind-statePhy107 Fall 200620Preserves indistinguishabilityWavefunction unchanged12+Electron 1 ins-stateElectron 2
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