Last Time…3-D particle in box: summary3-dimensional Hydrogen atomModified Bohr modelAngular momentum is quantized: orbital quantum number ℓOrbital magnetic dipole moment3D Hydrogen atom so far…Orbital mag. quantum number mℓQuestionSummary of quantum numbersHydrogen wavefunctionsFull hydrogen wave functions: Surface of constant probabilityn=2: next highest energyn=3: two s-states, six p-states and……ten d-statesElectron spinInclude spinPowerPoint PresentationSlide 19Putting electrons on atomOther elements: Li has 3 electronsMulti-electron atomsSlide 23The periodic tableExcited states of SodiumOptical spectrumThurs. Dec. 4 2008 Physics 208, Lecture 27 1Last Time…3-dimensional quantum states and wave functionsDecreasing particle sizeQuantum dots (particle in box)Course evaluations Tuesday, Dec. 9 in classOptional extra class: review of material since Exam 3 Fri. Dec. 12, 12:05 2103 ChThurs. Dec. 4 2008 Physics 208, Lecture 27 23-D particle in box: summaryThree quantum numbers (nx,ny,nz) label each statenx,y,z=1, 2, 3 … (integers starting at 1)Each state has different motion in x, y, zQuantum numbers determineMomentum in each direction: e.g.Energy:Some quantum states have same energy € px=hλnx= nxh2L€ E =px22m+py22m+pz22m= Eonx2+ ny2+ nz2( )Thurs. Dec. 4 2008 Physics 208, Lecture 27 33-dimensional Hydrogen atomBohr model:Considered only circular orbits Found 1 quantum number nEnergy , orbit radius€ En= −13.6n2 eVFrom 3-D particle in box, expect thatH atom should have more quantum numbersExpect different types of motion w/ same energy € rn= n2aoThurs. Dec. 4 2008 Physics 208, Lecture 27 4Modified Bohr modelDifferent orbit shapesBig angular momentumSmall angular momentumThese orbits have same energy, but different angular momenta:Rank the angular momenta from largest to smallest:ABCa) A, B, Cb) C, B, Ac) B, C, Ad) B, A, Ce) C, A, B € r L =r r ×r p ( )Thurs. Dec. 4 2008 Physics 208, Lecture 27 5Angular momentum is quantized: orbital quantum number ℓAngular momentum quantized , ℓ is the orbital quantum numberFor a particular n, has values 0, 1, 2, … ℓ n-1ℓ=0, most ellipticalℓ=n-1, most circular € r L = h l l +1( )For hydrogen atom, all have same energyThurs. Dec. 4 2008 Physics 208, Lecture 27 6Orbital magnetic dipole momentAngular momentum LPerpendicular to orbital plane € r r € r p € r L =r r ×r p Orbiting electron produces a current loopCurrent loop produces magnetic dipole moment along L € r μ =μBl l +1( ) € μB≡eh2m= 0.927 ×10−23A ⋅m2 € r μ =−μBr L /h( )IThurs. Dec. 4 2008 Physics 208, Lecture 27 73D Hydrogen atom so far…Each orbit hasSame energy:Different orbit shape (angular momentum): Different magnetic moment: € n,l( )€ En= −13.6 /n2 eV € L = h l l +1( ) € r μ =μBr L /h( )Thurs. Dec. 4 2008 Physics 208, Lecture 27 8Orbital mag. quantum number mℓDirections of ‘orbital bar magnet’ quantized.Orbital magnetic quantum number m ℓ ranges from - ℓ, to ℓ in integer steps (2ℓ+1) different valuesDetermines z-component of L: Also determines angle of LFor example: =1 gives 3 ℓstates: € Lz= mlh € Lz= L cosθ = mlhThurs. Dec. 4 2008 Physics 208, Lecture 27 9QuestionFor a quantum state with ℓ=2, how many different orientations of the orbital magnetic dipole moment are there?A. 1B. 2C. 3D. 4E. 5Thurs. Dec. 4 2008 Physics 208, Lecture 27 10Summary of quantum numbersn : describes energy of orbitℓ describes the magnitude of orbital angular momentumm ℓ describes the angle of the orbital angular momentumFor hydrogen atom:Thurs. Dec. 4 2008 Physics 208, Lecture 27 11Hydrogen wavefunctionsRadial probabilityAngular not shownFor given n, probability peaks at ~ same placeIdea of “atomic shell”Notation:s: ℓ=0p: ℓ=1d: ℓ=2f: ℓ=3g: ℓ=4Thurs. Dec. 4 2008 Physics 208, Lecture 27 12Full hydrogen wave functions: Surface of constant probability Spherically symmetric.Probability decreases exponentially with radius.Shown here is a surface of constant probability € n =1, l = 0, ml= 01s-stateThurs. Dec. 4 2008 Physics 208, Lecture 27 13n=2: next highest energy € n = 2, l =1, ml= 0 € n = 2, l =1, ml= ±1 € n = 2, l = 0, ml= 02s-state2p-state2p-stateSame energy, but different probabilitiesThurs. Dec. 4 2008 Physics 208, Lecture 27 14 € n = 3, l = 1, ml= 0 € n = 3, l = 1, ml= ±13s-state3p-state3p-state € n = 3, l = 0, ml= 0n=3: two s -states, six p-states and…Thurs. Dec. 4 2008 Physics 208, Lecture 27 15…ten d-states € n = 3, l = 2, ml= 0 € n = 3, l = 2, ml= ±1 € n = 3, l = 2, ml= ±23d-state3d-state3d-stateThurs. Dec. 4 2008 Physics 208, Lecture 27 16Electron spinNew electron property:Electron acts like a bar magnet with N and S pole.Magnetic moment fixed… …but 2 possible orientations of magnet: up and downSpin downNS€ ms= −1/2Described by spin quantum number msNS€ ms= +1/2Spin upz-component of spin angular momentum € Sz= mshThurs. Dec. 4 2008 Physics 208, Lecture 27 17Include spinQuantum state specified by four quantum numbers: Three spatial quantum numbers (3-dimensional)One spin quantum number € n, l , ml, ms( )Thurs. Dec. 4 2008 Physics 208, Lecture 27 18Quantum Number QuestionHow many different quantum states exist with n=2?A. 1B. 2C. 4D. 8ℓ = 0 :ml = 0 : ms = 1/2 , -1/2 2 statesℓ = 1 :ml = +1: ms = 1/2 , -1/2 2 statesml = 0: ms = 1/2 , -1/2 2 statesml = -1: ms = 1/2 , -1/2 2 states2s22p6There are a total of 8 states with n=2Thurs. Dec. 4 2008 Physics 208, Lecture 27 19QuestionHow many different quantum states are in a 5g (n=5, ℓ =4) sub-shell of an atom?A. 22B. 20C. 18D. 16E. 14 ℓ =4, so 2(2 ℓ +1)=18. In detail, ml = -4, -3, -2, -1, 0, 1, 2, 3, 4and ms=+1/2 or -1/2 for each.18 available quantum states for electronsThurs. Dec. 4 2008 Physics 208, Lecture 27 20Putting electrons on atomElectrons obey Pauli exclusion principleOnly one electron per quantum state (n, ℓ, mℓ, ms)Hydrogen: 1 electron one quantum state occupiedoccupiedunoccupiedn=1 statesHelium: 2 electronstwo quantum states occupiedn=1 states € n =1,l = 0,ml= 0,ms= +1/2( ) € n =1,l = 0,ml= 0,ms= +1/2( ) € n =1,l = 0,ml= 0,ms= −1/2( )Thurs. Dec. 4 2008 Physics 208, Lecture 27 21Other elements: Li has 3 electronsn=1
View Full Document
Unlocking...