Section1:HowQMWorks,Part1

Intheseslideswewillcover:

• TheSchrödingerEquation

• Theprobabilityinterpretationofthewavefunction

• Thediscretenatureofobservables

• Thecorrespondencebetweenobservablesandoperators

• Eigenfunctions,eigenvaluesandtheirproperties

• MeasurementinQuantumMechanics

• Expectationvalues

Thewavefunction

Particlesandwaves

• Inclassicalphysics,weuseNewton’sLaws todeterminetheequationofmotion𝑥(𝑡) ofaparticleofmass𝑚movinginapotential𝑉(𝑥):

• Equivalently,wecanconservetheenergy oftheparticle:

𝐹 = 𝑚𝑑+𝑥𝑑𝑡+ = −

𝑑𝑉𝑑𝑥

12𝑚𝑣

+ + 𝑉 𝑥 = Energy

Thewavefunction

• ThispicturecannotapplyintheQuantumworld,becauseparticlesbehavelikewaves(see:thedouble-slitexperiment)

• Sinceawaveisanobjectextendedinspace,weneedtochangehowwedescribeaparticle

Particlesandwaves

Thewavefunction

TheSchrödingerequation

• InQuantumMechanics,theequationofmotionofaparticleinapotential𝑉(𝑥) isreplacedbytheSchrödingerequation:

• Thesymbolℏ = ℎ/2𝜋,whereℎ isPlanck’sconstant

• It’sanequationforthewavefunction oftheparticleΨ(𝑥, 𝑡).Thislookscomplicated,butwe’llsoonseeit’sthesameas:

• The𝑖 = −1� appearingintheSchrödingerequationlooksstrange– thewavefunction isacomplexnumberingeneral!

−ℏ+

2𝑚𝜕+Ψ 𝑥, 𝑡𝜕𝑥+ + 𝑉 𝑥 Ψ 𝑥, 𝑡 = 𝑖ℏ

𝜕Ψ(𝑥, 𝑡)𝜕𝑡

Kineticenergy + Potentialenergy = Totalenergy

Thewavefunction

Thewavefunction

• ThewavefunctionΨ – that’stheGreekletter“psi”– ishowwedescribethestateofaparticleinQuantumMechanics

• Atagiventime𝑡,aparticleisnotatafixedposition𝑥(𝑡),butisinastatedescribedasafunctionofposition,Ψ(𝑥, 𝑡)

• Thewavefunction dependsontheco-ordinatesofasystemandcontainsalltheinformationaboutthesystem

Classical: Quantum:

becomes

Thewavefunction

• Whatdoesthewavefunction mean?It’sconnectedtotheprobability oftheparticlebeinginaparticularposition:

• Theparticlemustbesomewhere!Hence,theseprobabilitiesmustsumto1.0,whichisknownasthenormalisation ofΨ:

• Theprobabilityinterpretationofthewavefunction impliesthatQuantumMechanicshasastatistical orindeterminate nature

Probabilityinterpretationofthewavefunction

Probabilityoffindingaparticleinarange𝑥 → 𝑥 + 𝑑𝑥 = Ψ +𝑑𝑥

Note:althoughΨ canbeacomplexnumber, Ψ + =Ψ L Ψ∗ isreal,asitshouldbeforaprobability!

N Ψ(𝑥, 𝑡) +𝑑𝑥O

PO= 1

Operators,eigenfunctions &eigenvalues

Discretenatureofobservables

• InQuantumMechanics,ameasurementofaquantitycanonlyproducediscrete(specific)outcomes,notanyvalue

• Youhavepreviouslystudiedaparticleinaninfinitepotentialwell,whichhascertainallowedenergylevels(seerecaponnextslide)

• AnotherexampleispoorSchrödinger’scat,whichonlyhas2possiblestates…

Operators,eigenfunctions &eigenvalues

Discretenatureofobservables

• InPhysics2AQM,youstudiedthataparticleenclosedinaninfinitepotentialwellhasdiscreteenergiesandwavefunctions

• We’llseethisexampleagaininSection3!

Imagecredit:https://opentextbc.ca/universityphysicsv3openstax/chapter/the-quantum-particle-in-a-box/

Operators,eigenfunctions &eigenvalues

Discretenatureofobservables

• Physicsisdescribedinthelanguageofmathematics;soweneedamathematicalstructureinwhichdiscretevaluesappear

• Welcometotheworldofoperators,eigenfunctions andeigenvalues!Pleasedonotturnback!

• WecandescribethemathematicalframeworkofQuantumMechanicsbythefollowingstatement:

• Whatdothesewordsmean??

Eachquantitywecanobserveisrepresentedbyacorrespondingoperator.Ifwemeasurethatobservable,wewillalwaysobtain

aresultwhichisoneoftheeigenvaluesoftheoperator

Operators,eigenfunctions &eigenvalues

Whatisanoperator?

• Anoperatorisamathematicalinstructionwhichactsonafunctiontoproduceanotherfunction:

• Example: QQR

isanoperatorwhichactsonafunction𝑓(𝑥) to

producethederivativefunction𝑔 𝑥 = QUQR

• Example:𝑥 L (“multiplyby𝑥”)isanoperatorwhichactsonafunction𝑓(𝑥) toproduceanotherfunction𝑔 𝑥 = 𝑥𝑓(𝑥)

Operator Function𝑓(𝑥) Function𝑔(𝑥)actson toproduce

Operators,eigenfunctions &eigenvalues

Eigenfunctions andeigenvalues

• Whenanoperatoractsonsomespecialfunctions – calledtheeigenfunctions oftheoperator– itreturnsthesamefunction,scaledbyanumber– calledaneigenvalue

𝐴W𝜙Y 𝑥 = 𝑎Y𝜙Y(𝑥)

𝐴W isanoperator –theseareusually

writtenwithlittlehats! 𝜙Y(𝑥) isaneigenfunction – thesubscript“𝑛”labelsthedifferenteigenfunctions(𝜙\, 𝜙+, 𝜙], … )

𝑎Y istheeigenvalue(number)correspondingtotheeigenfunction 𝜙Y(𝑥)

Operators,eigenfunctions &eigenvalues

Eigenfunctions andeigenvalues

• Whenanoperatoractsonsomespecialfunctions – calledtheeigenfunctions oftheoperator– itreturnsthesamefunction,scaledbyanumber– calledaneigenvalue

• Asanexample,let’sconsidertheoperator𝐴W = QQR

again

• 𝜙 𝑥 = 𝑒`R isaneigenfunction of𝐴W witheigenvalue𝑎

• Why? Because𝐴W𝜙 𝑥 = QaQR= 𝑎𝑒`R = 𝑎𝜙 𝑥 – theoperator

hasreturnedthesamefunction,scaledbyanumber

𝐴W𝜙Y 𝑥 = 𝑎Y𝜙Y(𝑥)

Operators,eigenfunctions &eigenvalues

Propertiesoftheoperatorsrepresentingobservables

Eachquantitywecanobserveisrepresentedbyacorrespondingoperator.Ifwemeasurethatobservable,wewillalwaysobtain

aresultwhichisoneoftheeigenvaluesoftheoperator

MomentumRepresentedbymathematical

operator

PositionRepresentedbymathematical

operator

EnergyRepresentedbymathematical

operator

AngularmomentumRepresentedbymathematical

operator

Operators,eigenfunctions &eigenvalues

Propertiesoftheoperatorsrepresentingobservables

• Theoperatorsrepresentingobservableshave3keyproperties:

1. Theireigenvaluesarereal(notcomplex)numbers,sotheycancorrespondtotheresultsofphysicalmeasurements

2. Differenteigenfunctions areorthogonal,whichisdefinedby:

3. Anyotherfunction𝑓(𝑥) canbeexpressedasalinearcombinationoftheeigenfunctions,whichwecanwriteas:

N 𝜙b∗ 𝑥 𝜙Y 𝑥 𝑑𝑥O

PO= c1, 𝑚 = 𝑛

0, 𝑚 ≠ 𝑛

𝑓 𝑥 =e𝑐Y𝜙Y(𝑥)�

Y

Note:𝜙∗ meansthecomplexconjugateof𝜙

Operators,eigenfunctions &eigenvalues

Propertiesoftheoperatorsrepresentingobservables

• Wecanusetheenergyeigenfunctions fortheinfinitepotentialwelltoillustrateorthogonality

• Thesesinefunctionsaveragetozeroif𝑚 ≠ 𝑛,∫ 𝜙b∗ 𝑥 𝜙Y 𝑥 𝑑𝑥OPO = 0

• If𝑚 = 𝑛,then∫ 𝜙Y(𝑥) +𝑑𝑥OPO = 1,whichisthesameasnormalisingtheeigenfunctions

Operators,eigenfunctions &eigenvalues

Linearcombinationsofeigenfunctions

• Wejustmentionedthatanyfunction𝑓(𝑥) canbeexpressedasalinearcombinationoftheeigenfunctions ofanoperator:

• Wecandeterminethecoefficients𝑐Y usingtheorthogonalityproperty.Wecanderivethembyconsidering:

𝑓 𝑥 =e𝑐Y𝜙Y(𝑥)�

Y

N 𝜙b∗ 𝑥 𝑓 𝑥 𝑑𝑥 =O

PON 𝜙b∗ 𝑥 e 𝑐Y𝜙Y(𝑥)

�

Y𝑑𝑥

O

PO

=e 𝑐Y�

YN 𝜙b∗ 𝑥 𝜙Y 𝑥 𝑑𝑥O

PO

= 𝑐bThisisequalto1 if𝑚 = 𝑛and0 otherwise

Changingtheorderoftheintegralandsum…

MeasurementinQuantumMechanics

Measurementif𝚿 isaneigenfunction

• AttheheartofQuantumMechanicsisthehowthewavefunction isrelatedtomeasurementofobservables

• Supposewemeasureaparticularobservableofasystem(e.g.momentum,position,energy,angularmomentum,etc.)

• Recapping… thisobservableisrepresentedbyacorrespondingoperator(wewillseesomeexamplesshortly)

• Ifthewavefunction ofthesystemisaneigenfunction ofthecorrespondingoperator(Ψ = 𝜙Y),thentheresultofthemeasurementisthecorrespondingeigenvalue,𝑎Y

MeasurementinQuantumMechanics

Measurementif𝚿 isnotaneigenfunction

• IfthewavefunctionΨ isnot aneigenfunction ofthecorrespondingoperator,itcanalwaysbeexpressedasalinearcombination oftheeigenfunctions:

• Inthiscase,theresultofthemeasurementmaybeanyoneoftheeigenvalues𝒂𝒏,withcorrespondingprobabilities 𝒄𝒏 𝟐

• Followingthemeasurement,thewavefunction “collapses”andbecomestheeigenfunction,Ψ(𝑥) = 𝜙Y(𝑥)

• Iftheobservableismeasuredagain,we’llfindvalue𝑎Y again

Ψ 𝑥 =e𝑐Y𝜙Y(𝑥)�

Y

MeasurementinQuantumMechanics

Measurementif𝚿 isnotaneigenfunction

Measurementchangesthewavefunction,causingitto“collapse”intotheeigenfunction correspondingtotheresultofthemeasurement.Thisensuresthatfuturemeasurementsof

thequantityproducethesameresult.

• ItisequivalenttoopeningtheboxcontainingSchrödinger’scat,andseeingtheresult!

[Yes,catphotosareanoccupationalhazardhere.]

MeasurementinQuantumMechanics

Measurementif𝚿 isnotaneigenfunction

• ThisgivesustherecipeformeasurementinQuantumMechanics,representedbythefollowingflowchart!

ThestateoftheparticleisdescribedbyitswavefunctionΨ(𝑥)

Wewanttomeasureanobservable𝐴

Whatistheoperator𝐴W correspondingtothisobservable?

WhataretheeigenfunctionsΨY(𝑥) andeigenvalues𝑎Y of

thisoperator𝐴W?

Expressthewavefunction asalinearcombinationoftheeigenfunctions,

Ψ 𝑥 = ∑ 𝑐Y𝜙Y(𝑥)�Y

Thepossibleresultsofthemeasurementaretheeigenvalues

𝑎Y,withprobabilities 𝑐Y +

Performthemeasurement

Obtainoneoftheeigenvalues,𝑎\

Thewavefunction collapsestothecorrespondingeigenfunction,𝜙\(𝑥)

MeasurementinQuantumMechanics

Measurementif𝚿 isnotaneigenfunction

• Thefactor \no� istoensureΨ(𝑥) isnormalised:∫ Ψ(𝑥) +𝑑𝑥O

PO = 1

• Wenoticethatthetermsinthesquarebracketarethe1st and2nd energyeigenfunctions.Substitutingthesein,wefindΨ 𝑥 = \

n�𝜙\ 𝑥 + +

n�𝜙+ 𝑥

• Hence,thepossiblemeasurementsoftheenergystateare𝐸\ (with

probability 𝑐\ + = \n�+= \

n)and𝐸+ (withprobability 𝑐+ + = +

n�+= q

n)

• Theprobabilities\nandq

nsumto1,astheyshould!

Example:aparticleinaninfinitepotentialwellintherange 𝑥 < 𝐿 ispreparedinthewavefunction𝛹 𝑥 = \

no� cos vR+o

+ 2 sin vRo

.Whatenergystatescanbemeasured,andwithwhatprobabilities?

MeasurementinQuantumMechanics

Expectationvalues

• Althoughwecan’tpredictexactlywhichvaluewillresultfromameasurement(onlytheirprobabilities),wecanpredictthemeanmeasurement,alsoknownastheexpectationvalue –whichhasthesymbolofangledbrackets, 𝑎

• Example:whatistheexpectationvaluewhenadiceisthrown?

𝑎 =eProb 𝑛 𝑎Y

�

Y

𝑎 =16 L 1 +

16 L 2 +

16 L 3 +

16 L 4 +

16 L 5 +

16 L 6 = 3.5

[Note:thevalue3.5 cannotbeobtainedforanyindividualthrowofthedice,butistheaverageovermanythrows!]

MeasurementinQuantumMechanics

Expectationvalues

• InQuantumMechanics,wehaveseenthatProb 𝑛 = 𝑐Y +

• Example:fortheparticleintheinfinitepotentialwell2slidesback, 𝐸 = \

n𝐸\ +

qn𝐸+

• Wecanalsofindageneralrelationbysubstitutingin𝑐Y =∫ Ψ 𝑥 𝜙Y∗(𝑥)OPO andusingtheorthogonalityrelation:

𝑎 =eProb 𝑛 𝑎Y

�

Y

=e 𝑐Y +𝑎Y

�

Y

𝑎 = N Ψ∗ 𝑥 𝐴WΨ 𝑥 𝑑𝑥O

PO

Summary

TherulesofQuantumMechanics

• ThestateofaparticleisdescribedbythewavefunctionΨ(𝑥, 𝑡),whichsatisfiestheSchrödingerequation,− ℏ|

+b}|~}R|

+ 𝑉 𝑥 Ψ = 𝑖ℏ }~}�

• Theprobabilityofaparticlebeinglocatedatapositionintherange[𝑥, 𝑥 +𝑑𝑥] attime𝑡 is Ψ(𝑥, 𝑡) +

• Physicalobservablesarerepresentedbyoperators,andthepossibleresultsofmeasuringanobservablearetheeigenvalues𝑎Y ofthoseoperators

• Anywavefunction canbeexpressedasalinearcombinationoftheeigenfunctions oftheseoperators:Ψ = ∑ 𝑐Y𝜙Y�

Y .Theprobabilityofmeasuringtheeigenvalue𝑎Y isthen 𝑐Y +

• Ifameasurementofanobservableyieldsaresult𝑎Y,thewavefunctioncollapsesintothecorrespondingeigenfunction,Ψ = 𝜙Y