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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