ELEC$1908$QM$$

Fundamentals$of$Quantum$Theory$and$wells$$

Atomic%Theory%•  Maler%consists%of%par;cles%(electron,%

protons,%neutrons).%These%par;cles%have%quali;es%(mass,%charge).%%

•  Par;cles%are%dis;nguishable%(we%can%label%them).%%

•  They%have%very%specific%posi;ons,%trajectories%and%predictable%ac;ons%(determinis;c).%%

•  They%are%subject%to%forces%and%governed%by%laws.%%

•  Materials%are%atoms%held%together%by%forces%(a%bit%unspecified)%and%described%by%``bulk''%proper;es%(conduc;vity,%hardness,%etc).%%

Others%QQ%primarily%derived%from%the%previous%and%atomic%theory%Wave Equation Sound, EM, light, etc

@2u@t2 = c2r2u

Di↵usion Equation Heat, gasses, electrons, holes etc.

@u@t = Dr2u+ S(~r)

Fluid Flow Naiver-Stokes Equation

⇢�@v@t + v ·rv

�= �rp+r · T+ f

Stress-Strain Continuum mechanics

⇢̇+ ⇢r · v = 0 (one of a number of linked equations)

The%following%are%described%as%forming%Classical%mechanics:%

• Newtonian mechanics, the original theory of motion (kinematics) and

forces (dynamics)

• Hamiltonian mechanics, a theoretical formalism, based on the principle of

conservation of energy

• Lagrangian mechanics, another theoretical formalism, based on the prin-

ciple of the least action

• Celestial mechanics, the motion of heavenly bodies: planets, comets, stars,

galaxies, etc.

• Astrodynamics, spacecraft navigation, etc.

• Solid mechanics, elasticity, the properties of (semi-)rigid bodies

• Acoustics, sound ( = density variation propagation) in solids, fluids and

gases.

• Statics, semi-rigid bodies in mechanical equilibrium

And%• Fluid mechanics, the motion of fluids

• Soil mechanics, mechanical behavior of soils

• Continuum mechanics, mechanics of continua (both solid and fluid)

• Hydraulics, mechanical properties of liquids

• Fluid statics, liquids in equilibrium

• Applied mechanics, or Engineering mechanics

• Biomechanics, solids, fluids, etc. in biology

• Biophysics, physical processes in living organisms

• Statistical mechanics, assemblies of particles too large to be described in

a deterministic way

• Relativistic or Einsteinian mechanics, universal gravitation

Most%of%engineering%is%(or%has%been)%based%on%classical%physics.%

Classical%versus%Modern%or%Quantum%Physics%

•  The%following%are%categorized%as%being%part%of%Quantum%mechanics:%–  Par;cle%physics,%the%mo;on,%structure,%and%reac;ons%of%par;cles%–  Nuclear%physics,%the%mo;on,%structure,%and%reac;ons%of%nuclei%–  Condensed%maler%physics,%quantum%gases,%solids,%liquids,%etc.%–  Quantum%sta;s;cal%mechanics,%large%assemblies%of%par;cles%

•  Philosophically,%some%very%big%changes%with%QM.%%

Classical%versus%Modern%or%Quantum%Physics%

•  Classical%mechanics%is%determinis;c.%%•  If%you%know%the%star;ng%point%and%all%the%

physics%you%know%where%everything%is%going.%No%Free%Will?%%

•  QM%is%stochas;c%and%random.%Par;cles%are%never%precisely%pinned%down%to%a%place/;me%or%state.%Back%comes%Free%Will?%Perhaps?%%

•  Some%people%think%that%QM%and%consciousness%are%very%closely%linked.%A%lot%of%weird%results%and%a%lot%of%arguments%(Einstein%and%Bohr).%%

•  S;ll%not%closed%or%understood.%

Classical%versus%Modern%or%Quantum%Physics%

•  There%is%a%natural%tendency%to%draw%a%line%between%the%two.%However,%it%is%difficult%and%leads%to%more%problems.%Get%into%SciFi%stuff%such%as%many%worlds,%teleporta;on%etc.%

•  We%have%a%very%good%intui;on%for%CM.%It%is,%basically,%the%physics%of%our%;me%and%length%scale.%%

•  We%understand%Newton's%laws%because%we%evolved%to%understand%them!%If%we%did%not%we%would%die!%%

•  QM%is%the%physics%of%the%small%and%the%low%energy.%This%is%not%our%natural%domain.%We%have%a%very%poor%feel%for%the%physics.%%

•  Everything%is%vague;%oGen%counterQintui;ve%and%some;mes%down%right%bizarre.%%

Classical%versus%Modern%or%Quantum%Physics%

•  However,%QM%is%correct%QQ%never%been%shown%to%be%wrong,%CM%is%an%approxima;on%of%QM.%%

•  Science%proceeds%by%a%series%of%approxima;ons%to%the%truth.%We%have%an%agreed%set%of%facts,%which%grows%over%;me%as%our%experimental%techniques%improve.%%

•  As%these%facts%invalidate%a%theory%another%theory%replaces%it.%

•  %Paradigm%shiGs%are%used%to%describe%this%phenomena.%Usually,%the%previous%theory%is%found%to%be%a%limited%approxima;on%of%the%latest%one.%%

Classical%versus%Modern%or%Quantum%Physics%

•  A%theory%makes%predic;ons%based%on%premises%and%facts%and%the%applica;on%of%logic%and%math.%If%the%predic;ons%are%found%to%be%in%error%then%the%theory%is%lacking%or%incorrect.%%

•  No%predic;ons,%then%no%theory,%just%specula;on.%Useful%but%not%defini;ve.%%

Problems%with%Classical%Physics%•  These%simple%laws%with%extensions%and%modifica;ons%were%found%to%

correctly%predict%a%vast%array%of%effects%and%proper;es.%%•  However,%a%number%of%effects%became%experimentally%confirmed%that%

could%not%be%explained%by%these%classical%theories.%•  %Three%of%the%most%prominent%were:%%

–  The%photoelectric%effect%–  Black%body%radia;on%–  Heat%capacity%of%solids%%

PhotoQELECTRIC%EFFECT%•  Suppose%a%light%with%frequency%f%is%incident%on%a%

solid,%then%experimentally%electrons%are%emiled%from%the%surface%

•  According%to%the%classical%analysis,%light%is%like%a%classical%EM%wave,%therefore%the%higher%intensity%we%have,%the%more%energy%is%delivered%to%the%solid%surface.%

•  Model%of%electron%is%ball%on%spring%(spring%is%alrac;ve%force)%and%the%EM%wave%will%make%the%electron%oscillate%un;l%it%has%enough%energy%to%“break”%the%spring.%%

•  More%energy%the%more%electron%emission%is%predicted!%

•  Should%occur%at%any%frequency,%just%take%longer%at%low%energies.%

PhotoQElectric%Effect%•  Experimentally%this%was%not%found.%%•  It%was%found%that%only%when%

%%%%was%there%any%emission%of%electrons%%

~! > � (1)

• ! = 2� f0 and f0 is the threshold light frequency.

• � is a function of the material called the work function.

• ~ = h / 2⇡ and h is a fundamental constant in physics called Planksconstant, h = 6.626068⇥ 10�34 j.s

Photons%•  In%1900%Max%Planck%announced%a%new%

formula%that%filed%the%experimental%evidence%well:%

•  Einstein%proposed%the%photon%theory%of%light%in%photoelectric%effect.%%

•  Light%as%both%a%par;cle%and%wave!%

E = ~! = hf p = ~k

Photons%•  Using%Planck’s%idea%of%discrete%energy,%Einstein%showed%that%the%quan;zed%

energy%E%released%from%the%solid%is%propor;onal%to%the%frequency%f%of%the%incident%light.%

•  Only%when%a%photon%with%energy%greater%than%the%work%func;on%of%the%metal%(the%amount%of%energy%needed%to%remove%an%electron%from%a%solid)%collided%with%an%electron%there%would%be%enough%energy%to%emit%an%electron.%

•  If%the%energy%of%a%photon%was%less%than%the%work%func;on%of%the%metal%then%no%electrons%would%be%emiled.%

•  This%explained%the%phenomena%and%was%the%start%of%QM%

Heat%and%QM%•  A%black%body%is%a%theore;cal%object%

that%absorbs%100%%of%the%radia;on%that%hits%it.%

•  Black%body%radia;on%is%the%Radia;on%emiled%by%an%opaque%or%hot%body.%%

•  Classical%physics%failed%in%explaining%the%form%of%the%light%intensity%curves%as%a%func;on%of%wavelength%

•  Heat%capacity%of%a%solid%is%the%rate%of%the%increase%of%internal%energy%per%unit%temperature.%

•  Dulong%and%Pe;t%law%stated%that%the%heat%capacity%at%constant%volume%of%all%elementary%solids%is%approximately%3R%

•  Which%was%found%to%be%wrong%at%low%temperatures%

The$Start$of$QM$•  In$1924$De$Broglie$proposed$that$

waveRparAcle$duality$did$not$belong$to$light$only$and$that$$ma?er$(parAcles)$could$also$exhibit$waveRparAcle$duality.$$

•  The$famous$de$Broglie's$wavelength$relaAon$is:$–  $λ$=$h/p$=$h/mv$

•  Inspired$by$Einstein’s$theory$in$photoelectric$effect$i.e.$E$=$hf.$He$described$an$electron$as$a$standing$wave$around$the$circumference$of$an$atomic$orbit$$

Bohr’s$theory$

•  Also$at$that$Ame,$Bohr$was$working$on$a$planetary$model$for$an$atom$in$which$electronRorbits$could$only$have$certain$discrete$quanAzed$energies.$

•  $Bohr's$second$postulate$states:$

$!Instead!of!the!infinity!of!orbits!which!would!be!possible!in!classical!mechanics,!it!is!only!possible!for!an!electron!to!move!in!an!orbit!for!which!its!orbital!angular!momentum!is!an!integral!mul:ple!of!Plank’s!constant$

•  From$which$we$obtain:$

Bohr$+$de$Broglie$

•  However,$using$an$assumpAon$of$electronic$standing$waves$around$a$nucleus$and$wavelength$relaAonship$we$have$by$combining$the$two$ideas$

•  And$we$see$the$two$ideas$are$compaAble$as$h/2π$=$“h$bar”.$

•  We$have$wave$parAcle$duality$–$as$for$the$photon/EM$wave.$

Electron$wave$length$•  Calculate$the$de$Broglie$wavelength$of$an$electron$which$has$a$

kineAc$energy$of$4$eV.$

Note:$The$wavelength$has$the$dimensions$of$interRatomic$spacing.$•  What’s$the$de$Broglie$wavelength$of$a$tennis$ball$having$a$mass$of$

50$g$and$traveling$with$a$velocity$of$200$km/h?$

•  Does$this$explain$why$tennis$balls$don't$diffract?$

Electrons$as$waves$•  If$parAcles$have$wavelengths$we$

should$expect$that$they$should$have$waveRlike$behavior.$$

•  DiffracAon$is$an$easily$observed$and$enArely$waveRlike$effect.$$

•  The$waveRlike$behavior$of$electrons$was$experimentally$confirmed$by$Davisson$and$Germer.$$

•  Bragg$diffracAon$pa?ern$was$produced$by$electrons$with$the$de$Broglie$wavelength$determining$the$wave$behaviour$

What’s$waving?$•  Quantum$mechanics$is$based$on$

Planck's$quantum$and$De$Broglie’s$waveRparAcle$duality.$$

•  It$is$a$revision$of$the$``laws$of$mechanics''$designed$to$extend$the$subject$into$the$realm$of$atomic$and$nuclear$phenomena.$

•  A$big$quesAon$was$“what$is$waving?”.$

•  Well$the$funcAon$that$waves$is$called$the$waveRfuncAon$and$is$denoted$ψ$R$what$it$“is’’$is$difficult.$

•  The$wave$amplitude$is$related$to$the$probability$of$finding$a$parAcle$in$a$given$region$of$space$or$state.$

XKCD$$

Schrodinger’s$EquaAon$•  An$equaAon$was$need$to$solve$for$ψ$

and$Schrodinger's$equaAon$is$the$fundamental$equaAon$of$quantum$mechanics.$

•  Schrodinger$used$classical$ideas$of$energy$conservaAon$to$obtain$a$likely$SCE.$$$

•  From$classical$mechanics$a$funcAon$known$as$the$Hamiltonian$had$been$derived$which$represented$the$total$energy$of$a$system.$

•  P$is$the$momentum$operator$•  We$get$a$waveRlike$equaAon$•  Then$assume$harmonic$in$t$•  Obtain$SS$equaAon$

SCE$in$free$space$•  In$free$space$we$have$no$forces$and$the$

potenAal$energy$can$be$defined$as:$$ $ $$V(r)$=$0.$$

•  For$simplicity$we$assume$the$1D$case.$•  $SoluAon$is$the$equaAon$of$a$traveling$

wave.$

•  Note$velocity$is$not$what$we$might$expect$from$½$mv2$

•  Geqng$the$probability$of$electron$being$present$we$have:$

•  Electron$is$everywhere$??$

InterpretaAon$of$the$WaveRfuncAon$

•  The$physical$interpretaAon$of$the$waveRfuncAon$has$been$problemaAc$since$the$development$of$QM.$$

•  The$difficulAes$stem$from:$$–  The$interpretaAon$of$probabilisAc$nature$of$the$results.$The$waveRfuncAon$

only$tells$you$the$probability$of$what$a$system$might$do$not$what$it$does.$–  The$uncertainty$relaAonship$RR$the$indeterminacy$of$reality.$–  The$apparent$contradicAon$of$macroscopic$(our)$reality$and$the$QM$

descripAon$

–  Entanglement$RR$acAon$at$a$distance.$

Quantum$Randomness$•  The$SCE$provides$a$soluAon$for$ψ$which$

is$used$to$predict$the$probability$of$outcomes.$$

•  For$example$an$electron$is$sca?ered$off$an$atom;$it$has$a$probability$X$of$going$to$the$right$and$Y$to$leu.$

•  $QM$can$calculate$X$and$Y$but$does$(and$can$not)$say$which$it$does.$$

•  Very$disturbing$to$classical$physicists.$

Measurements$–$“the$line$between$QM$and$classical”$

•  We$can,$however,$watch$an$electron$sca?er$using$experimental$measurements.$$

•  What$we$see$is$an$electron$go$either$to$the$right$or$leu.$$

•  This$seems$to$mean$that$the$waveRfuncAon$“collapses”$to$a$probability$of$one$and$is$in$a$parAcular$state$auer$a$system$is$measured,$however,$QM$has$fundamentally$no$explanaAon$for$this$phenomena.$$

•  This$is$known$as$the$measurement$problem.$

Uncertainty$RelaAon$•  QM$also$shows$that$certain$

variables,$for$example$posiAon$and$momentum,$are$coupled$in$way$that$if$you$try$to$measure$one$you$disturb$the$other.$$

•  This$leads$to$a$limit$to$our$knowledge$of$the$system$i.e.$an$uncertainty$in$posiAon$and$momentum.$

•  MathemaAcally$very$clear$(as$we$will$see)$but$very$unRclassical$for$parAcles!$$$

Entanglement$•  In$a$QM$system$of$two$objects$if$the$objects$interact$their$$waveR

funcAons$become$coupled$and$inseparable.$$•  Quantum$entanglement$(nonRlocal$connecAon)$is$when$the$

quantum$states$of$the$consAtuAng$objects$are$linked$together$so$that$one$object$can$no$longer$be$described$without$knowing$the$state$of$the$other$object.$

•  Even$if$the$individual$objects$become$spaAally$separated$they$are$sAll$coupled!$$

•  Photons$and$electron$etc$all$couple$

Spin$coupling$•  For$example$two$electrons$in$close$proximity$can$be$coupled$so$one$must$be$spin$up$and$the$other$spin$down$(spin$is$a$quality$of$an$electron).$$

•  If$the$two$electrons$are$then$separated$and$driu$apart$the$net$spin$must$sAll$be$zero.$$

•  However,$QM$does$not$tell$you$which$electron$is$up$and$which$is$down$it$can$only$calculate$the$probability$of$each$electron$being$spin$up$or$down.$$

Mixed$States$•  Fundamentally,$each$electron$is$both$spin$

up$and$down$at$the$same$Ame$(in$a$“mixed”$state).$$

•  When$we$measure$one$of$the$electron's$spin$and$find$up$we$now$know$that$the$other$must$be$down$as$the$net$spin$is$zero.$$

•  However,$the$electrons$could$be$separated$by$light$years$of$distance!$$

•  By$measuring$one$electron$we$forced$the$other$to$be$the$complementary$spin$RR$acAon$at$a$distance$

•  $Not$Good!$Very$strange.$Einstein$very$unhappy.$

Schrodinger's$Cat$•  A$thought$experiment$was$developed$in$a?empt$to$show$the$ludicrous$nature$of$QM$(by$

this$Ame$Einstein$and$Schrodinger$were$beginning$to$regret$the$whole$idea).$•  Cat$is$in$mixed$state$(dead/alive)$unAl$we$look$inside.$•  At$that$point$the$wavefuncAon$“collapses”$and$the$cat$is$either$dead$or$alive$•  What/who$is$an$observer$(the$cat,$the$Geiger$counter,$…)$•  If$the$cat$is$decomposing$when$we$look$then$we$know$that$he$died$in$the$past,$but$he$

was$not$either$dead$or$alive$unAl$we$looked.$That$means$the$observer$determines$the$past!$$

*ParAally$from:$h?p://en.wikipedia.org/wiki/Schrodingers_cat$

Schrodinger's$Cat$•  Schrödinger$wrote:$

One$can$even$set$up$quite$ridiculous$cases.$A!cat!is!penned!up!in!a!steel!chamber,!along!!with!the!following!device!(which!must!be!secured!against!direct!interference!by!the!cat):!in!a!Geiger!counter,!there!is!a!:ny!bit!of!radioac:ve!substance,!so!small!that!perhaps!in!the!course!of!the!hour,!one!of!the!atoms!decays,!but!also,!with!equal!probability,!perhaps!none;!if!it!happens,!the!counter!tube!discharges,!and!through!a!!relay!releases!a!hammer!that!shaEers!a!small!flask!of!hydrocyanic!acid.!If!one!has!leH!!this!en:re!system!to!itself!for!an!hour,!one!would!say!that!the!cat!s:ll!lives!if!meanwhile!no!atom!has!decayed.!The!psiJfunc:on!of!the!en:re!system!would!express!this!by!having!in!it!the!living!and!dead!cat!(pardon!the!expression)!mixed!or!smeared!out!in!!equal!parts.!It!is!typical!of!these!cases!that!an!indeterminacy!originally!restricted!to!the!atomic!!domain!becomes!transformed!into!macroscopic!indeterminacy,!which!can!then!be!resolved!by!direct!observa:on.!That!prevents!us!from!so!naively!accep:ng!as!valid!a!"blurred!model"!for!represen:ng!reality.!In!itself,!it!would!not!embody!anything!unclear!or!!contradictory.!There!is!a!difference!between!a!shaky!or!outJofJfocus!photograph!and!a!!snapshot!of!clouds!and!fog!banks.!

InterpretaAons:$Copenhagen$RR$the$classical$and$original$one$

•  In$the$Copenhagen$interpretaAon$of$quantum$mechanics,$a$system$stops$being$a$superposiAon$of$states$and$becomes$either$one$or$the$other$when$an$observaAon$takes$place.$$

•  This$experiment$makes$apparent$the$fact$that$the$nature$of$measurement,$or$observaAon,$is$not$wellRdefined$in$this$interpretaAon.$$

•  Some$interpret$the$experiment$to$mean$that$while$the$box$is$closed,$the$system$simultaneously$exists$in$a$superposiAon$of$the$states$"decayed$nucleus/dead$cat"$and$"unRdecayed$nucleus/living$cat",$and$that$only$when$the$box$is$opened$and$an$observaAon$performed$does$the$waveRfuncAon$collapse$into$one$of$the$two$states.$$

•  More$intuiAvely,$some$feel$that$the$"observaAon"$is$taken$when$a$parAcle$from$the$nucleus$hits$the$detector.$

•  $Problem$defining$observer$and$observing$system$RR$where$to$draw$the$line$between$macro$and$microscopic.$

InterpretaAons:$Copenhagen$RR$Many$Worlds/Many$minds$RR$Collapse$who$needs$it!$

•  In$1957,$Hugh$Evere?$formulated$the$manyRworlds$interpretaAon$of$quantum$mechanics,$$which$does$not$single$out$observaAon$as$a$special$process.$$

•  In$the$manyRworlds$$interpretaAon,$both$alive$and$dead$states$of$the$cat$persist,$but$are$independent$of$each$other.$

•  When$the$box$is$opened,$that$part$of$the$universe$containing$the$observer$and$cat$is$split$into$two$separate$universes:$one$containing$an$observer$$looking$at$a$box$with$a$dead$cat,$and$one$containing$an$observer$looking$at$a$box$with$a$$live$cat.$

•  $Since$the$dead$and$alive$states$are$independent,$there$is$no$effecAve$communicaAon$or$interacAon$between$them.$$

•  When$an$observer$opens$the$box,$he$becomes$entangled$with$the$cat,$so$"observer$states"$corresponding$to$the$cat's$being$alive$and$dead$are$formed,$$and$each$can$have$no$interacAon$with$the$other.$

•  $$Another$version$has$a$spliqng$of$the$``mind$of$the$observer''$not$the$universe.$

InterpretaAons:$Many$forms$of$these$and$others$

–  Ensemble$interpretaAon,$or$staAsAcal$interpretaAon$–  The$Copenhagen$interpretaAon$–  ParAcipatory$Anthropic$Principle$(PAP)$–  Consistent$histories$–  ObjecAve$collapse$theories$–  Many$worlds$–  Many$minds$–  StochasAc$mechanics$–  The$decoherence$approach$

–  Quantum$logic$–  The$Bohm$interpretaAon$–  TransacAonal$interpretaAon$–  RelaAonal$quantum$mechanics$–  Modal$interpretaAons$of$quantum$theory$–  Incomplete$measurements$

For$engineers$does$it$ma?er?$Not$really$just$use$it.$Calculate$and$move$on.$Nothing$to$see$here.$$But$it$is$really,$really$interesAng.$

Using$QM$

•  Electron$in$well$–  Infinite$well$– Finite$well$– Hydrogen$atom$

Quantum$Wells$

•  Quantum$wells$are$very$important$– Materials$–$electrons$in$solids$

– Devices$•  Lasers$•  Solar$cells$•  Quantum$dots$(biomedical)$

Electron$in$a$box$

•  Infinite$potenAal$well$•  Solve$SCERTI$•  BC$–$no$electron$outside$the$

box$ψ$=$0$

•  V(x$=$0:a)$$=$0$•  Define$k$•  Simple$wave$equaAon$$•  We$can$expect$modes$for$

soluAons$

2$

QuanAzaAon$•  Use$complex$exponenAal$

wave$soluAon.$

•  Apply$BC$–  sin$soluAon$due$to$odd$

soluAon$being$needed$–  QuanAzaAon$due$to$BC$

•  NormalizaAon$ensures$1$electron$in$the$box$$$

2$

E$missing$

Modes$•  Set$of$valid$soluAons.$$•  SuperposiAon$can$be$used$to$

form$wave$packet$•  Non$classical$behavior$

Finite$Box$

•  If$the$box$is$finite$in$height$then$the$soluAon$is$a$li?le$more$complex.$

•  Two$cases:$–  Bound$(E$<$V0)$

–  Unbound$(E$>$V0)$

•  Bound$is$very$similar$to$infinite$box.$

•  Unbound$is$similar$to$free$electron$

Bound$finite$well$soluAons$

•  Very$similar$to$infinite$well.$•  Modes$and$quanAzaAon.$

•  PenetraAon$into$the$sidewall$areas.$Very$nonRclassical$as$$–  E$<$V0$

•  Maximum$En$for$which$the$quanAzaAon$is$maintained.$

•  Again$superposiAon$allows$for$waveRpacket$from$summaAon$of$modes.$Electron$with$mulAple$energies!$$$$

Unbound$states$E$>$V0$

•  Unbound$states$are$purely$oscillatory.$

•  Two$spaAal$frequencies$k1$and$k2$

•  $Not$quanAzed$$

Finite$Well$–$Energy$states$$•  Two$disAnct$regimes:$

–  Bound$states:$•  E$<$V0$

•  Modes$for$waveRfuncAon$

•  QuanAzed$energy$and$momentums$

–  Unbound$states:$•  E$>$V0$

•  Travelling$wave$soluAon$•  ConAnuous$energies$and$momentums$

The$Hydrogen$AtomR$Spherical$Symmetry$

•  A$special$and$3D$case$of$the$bound$electron$in$a$well$is$an$atom.$$$

•  For$an$isolated$Hydrogen$atom,$has$one$electron$bounded$to$its$nucleus$by$coulomb$force$

The$Hydrogen$AtomR$Spherical$Symmetry$

•  Using$Schrodinger’s$equaAon$we$can$analyAcally$find$the$eigenRstates$for$the$Hydrogen$atom.$

The$Hydrogen$AtomR$Spherical$Symmetry$

•  Examples$of$the$waveRfuncAons$that$describe$the$chemical$states$(orbitals)$

ImplicaAons$of$QuanAzaAon$

•  Because$of$the$fact$that$the$electron$of$an$isolated$atom$can$assume$only$certain$energy$levels,$it$follows$that$the$energies$which$are$excited$or$absorbed$also$posses$only$discrete$values.$

•  The$discrete$E's$produce$spectral$lines$

ApplicaAons$of$QuanAzaAon$

•  Semiconductor$materials$and$devices$•  Photonics,$Lasers$•  Memory$storage$by$storing$informaAon$as$excited$states$in$molecules$

•  Input/Output$signals$using$excitaAon$and$sAmulated$emission$

•  Quantum$Computers.$Use$photons$to$switch$states.$