Chem 442 Review for Exam 1. Hamiltonian operator in 1 dimension: ˆ d first term is kinetic energy, second term is potential energy

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1 Chem 44 Review for Eam 1 Eergies are quatized Wave/partice duaity de Brogie waveegth: h p Eergy: E h mometum operator: pˆ positio operator: ˆ d i d potetia eergy operator: V ˆ( ) pˆ d kietic eergy operator i 1 dimesio: m m d Hamitoia operator i 1 dimesio: ˆ d H Vˆ( ) md first term is kietic eergy, secod term is potetia eergy Hamitoia operator i 3 dimesios: ˆ H Vˆ(, y, z ) m i Cartesia coordiates y z spherica ad cyidrica coordiates: covert to ad from Cartesia, spherica: r,, ; cyidrica: r,, z kow coversios; do t eed to memorize Eigevaue equatio: ˆ operator ˆ, eigefuctio, eigevaue epressios Time-idepedet Schrödiger equatio: Ĥ E eigevaue equatio with eigefuctio ad eigevaue E; quatized eergies Time-depedet Schrödiger equatio: ˆ d Hi dt * Bor iterpretatio: is the probabiity desity for ormaized rea ad o-egative d is the probabiity of fidig partice betwee ad d 1

2 Normaizatio: tota probabiity of fidig the partice somewhere is 1 If a wavefuctio is ormaized, d 1 If a wavefuctio is ot ormaized, mutipy by costat N: N d 1 N d i oe dimesio 1 straightforward to eted to three dimesios Coditios o wavefuctio: sige vaued, square-itegrabe, cotiuous, ad smooth (athough eceptios to smooth if potetia has siguarity eacty caceig) iet Time-depedet wavefuctio: ( t, ) ( e ) iet phase: e probabiity desity is time-idepedet: ( ) Hermitia operator: ˆ d ˆ Eampes: ˆ, pˆ, H ˆ * * * b a a b Properties of Hermitio operator: ˆ a a a - its eigevaues are rea * - its eigefuctios are orthogoa: d a b 0 (eceptio for eigefuctios with same eigevaue, but these ca be made orthogoa) - its eigefuctios are compete: ay fuctio that coforms to the aowed forms of wavefuctios ca be epressed as a iear combiatio of the eigefuctios d Postuates of quatum mechaics: - The physica state of a partice ca be described by a wavefuctio. ik - ca be ormaized. (Note: two wavefuctios differig oy by factor of e are cosidered the same.) - is the probabiity desity. - A Hermitia operator eists for each observabe physica property. - Sovig the eigevaue equatio, ˆ, eads to a compete ad orthogoa set of eigefuctios ad eigevaues. - If the wavefuctio is equa to oe of the eigefuctios, the a measuremet gives a defiite vaue that is the correspodig eigevaue. - If the wavefuctio is NOT equa to oe of the eigefuctios, it ca be writte as a iear combiatio of eigefuctios, 1 c, ad a measuremet wi give oe of the eigevaues with probabiity c for eigevaue. * - The epectatio vaue or average vaue of a property is ˆ d.

3 Commutator: Aˆ, Bˆ AB ˆ ˆ BA ˆˆ Two operators commute whe their commutator is zero: AB ˆ, ˆ 0. Whe two operators share the same eigefuctios, the they commute. If two operators commute, the the two observabes associated with these two operators ca be determied eacty simutaeousy. Ucertaity pricipe: Whe two operators do ot commute, the observabes associated with them are compemetary observabes ad caot be determied eacty simutaeousy. The errors i the two observabes caot be simutaeousy zero. Specific eampe: Because positio ad mometum do ot commute, it is impossibe to specify simutaeousy, with arbitrary precisio, both the mometum ad the positio of a partice: p. Partice i a oe-dimesioa bo: V( ) 0 for 0 L, V( ) esewhere ( ) si L L h E 8mL 1,,3,... Eergies: positive zero poit eergy, quatized eergy eves, o-degeerate eves, spacigs icrease as icreases, spacigs icrease as m ad L decrease Wavefuctios: symmetric ( odd) or atisymmetric ( eve) about midde of bo, -1 odes, zero outside bo Caot have = 0 because zero everywhere; caot have < 0 because this eads to the same wavefuctios as > 0 Ucertaity pricipe: as L 0, 0, ad pk. As L,, ad pk 0. Need zero poit eergy or woud vioate ucertaity pricipe because woud have zero mometum ad aso competey defied positio as L 0. Cassica-quatum correspodece pricipe: as approaches, the probabiity desity becomes more uiform, as i cassica case. As L ad m approach, the eergy eves decrease (become eary cotiuous, as i cassica case). 1, k L. i L Each of these terms is a eigefuctio of p ˆ, so the measuremet of mometum gives either k or k with equa probabiity. Wavefuctio ca aso be writte as: ( ik ik ) e e 3

4 Partice i a two-dimesioa we: Hamitoia separabe: mutipy 1D wavefuctios, add 1D eergies y y, (, ) si si y y L L L y L y h E h y, y 8mL 8mLy, 1,,3,... y Square: degeerate eergy eves (same eergies, differet wavefuctios) Straightforward to eted to three-dimesioa bo (separatio of variabes) Oe-dimesioa harmoic osciator 1 potetia eergy: V( ) k k frequecy: m E 1 v v v NvH v( y)*ep( y / ) H v ( y) :Hermite poyomia, y, mk v 0,1,,... Eergies: positive zero poit eergy ( ), quatized eergy eves, evey spaced by, o-degeerate Wavefuctios: symmetric (v eve) or atisymmetric (v odd) about =0, v odes, fiite probabiity for a, probabiity beyod cassica turig poits Permeatio: probabiity desity has o-zero vaue where it is cassicay forbidde Quatum-cassica correspodece: as v gets higher, probabiity i cassicay forbidde regio gets smaer ad probabiity ear the turig poit icreases (as i cassica case, where greatest probabiity is at turig poits). 14 Partice i a rig: Hamitoia: Hˆ mr 0 Wavefuctio: m 1 ep( im ) 4

5 Eergy: E m mr0 m 0, 1,, Eergies: zero poit eergy is zero (m = 0), each o-zero eergy eve is douby degeerate (m ), spacigs icrease as m icreases m = 0 aowed because wavefuctio o-zero (partice eists) m < 0 aowed because wavefuctio is distict from correspodig positive vaue Wavefuctios: itegra umber of fu waves must fit i rig, cycic boudary coditios Equa probabiity desity everywhere o rig for a m The ack of zero poit eergy does ot vioate ucertaity pricipe because the positio o rig is competey ukow. Aguar mometum: ˆz i ˆ zm m m These states are eigefuctios of both aguar mometum ad the Hamitoia so both observabes ca be determied simutaeousy ad eacty. Therefore, these states have defiite aguar mometum, ad a measuremet of mometum for state m wi aways produce m. m positive couter-cockwise aroud rig, z poitig i positive z directio m egative cockwise aroud rig, z poitig i egative z directio ˆz ad ˆ do ot commute; ucertaity pricipe hods because z is eacty kow ( m ) but age φ is ukow (uiform probabiity aroud etire rig) Partice o the surface of a sphere: Hamitoia: ˆ H m Use i spherica coordiates, radius fied so oy fuctio of (, ) Wavefuctio: m, (, ) N,, ( ) ( ),, (, ) m m m N my m 1 m ( ) ep( im ) m, ( ): associated Legedre poyomias Y, (, ): spherica harmoics Eergy: m ( 1) E mr 0 (Note: eergy depeds oy o, ot o m ) = 0,1,, m so m =, +1,, 0,, 1, 5

6 Eergies: zero poit eergy is zero (=0), degeeracy of each eve is +1 Aguar mometum: The tota aguar mometum is quatized i both its magitude ad orietatio it caot poit i ay arbitrary directio (space quatizatio) ˆ z, m (, ) m, (, ) m so observabe z m ˆ (, ) ( 1) (, ) so observabe tota ( 1) tota m, m, m, (, ) are ot eigefuctios ˆ ad ˆy ˆ, ˆ z tota 0 but ˆz does ot commute with ˆ ad ˆy For a give eigefuctio, we ca determie eacty simutaeousy the magitude ad z compoet of aguar mometum ad the eergy, but NOT the ad y compoets of aguar mometum. Spi: A eectro has spi quatum umber s 1 3 The tota spi aguar mometum is ss ( 1) 4 The spi magetic quatum umber m s has vaues s,,s. For a eectro ms 1, 1. The z-compoet of spi aguar mometum is ms, or Idistiguishabe partices: ( r1, r) ( r, r 1) Fermios: haf-iteger spi quatum umbers; eectros, protos, eutros; atisymmetric (Paui ecusio pricipe): ( r1, r) ( r, r 1) Bosos: iteger spi quatum umbers; photos; symmetric: ( r1, r) ( r, r 1) Tueig: A partice of eergy E ca tue through a barrier of potetia V > E. A partice of eergy E ca permeate ito the potetia wa of V > E. The probabiity of tueig icreases as the barrier gets thier, as the barrier gets ower, ad as the partice gets ighter. 6

Lecture #5: Begin Quantum Mechanics: Free Particle and Particle in a 1D Box

Lecture #5: Begin Quantum Mechanics: Free Particle and Particle in a 1D Box 561 Fall 013 Lecture #5 page 1 Last time: Lecture #5: Begi Quatum Mechaics: Free Particle ad Particle i a 1D Box u 1 u 1-D Wave equatio = x v t * u(x,t): displacemets as fuctio of x,t * d -order: solutio