Download Study Guide Chap. 11

Study Guide for chapter 11
1.
Blackbodyradiation.
a. Whatisblackbodyradiation?Whatquantitydoesthe
totalenergyorenergydensity
oftheradiationdepend
on?Dotheydependentonthetypeofobjects?
b.
c.
Wediscussedseveraltheories:Stefan-Boltzmannlaw,
Wienequation,Planckequation,andRayleigh-Jeans
equation.Whichonesgivetheexactdescriptionofthe
energyofblackbodyradiation?Youneedtoknowthe
approximateformoftherelationships.Drawthecurvesof
radiationenergydensityvs.wavelengthorfrequency
overtemperatureatdifferenttemperaturesusing
Planck’sequation.HowdoesitchangeifWienequationor
Rayleigh-Jeansequationisused?Whatisultra-violet
catastrophe?Howdoyouconvertwavelengthto
frequency?
OriginofPlanck’sequation.Whatarethetwotheoretical
considerationsforderivingthePlanckequation?Under
whatassumptionsdoesPlanck’sequationreduceto
Rayleigh-Jeansequation?
2. Photoelectriceffect.Whataretheobservations?Howdoesthe
theorydevelopedbyEinsteinexplainthem?
3. Theparticlenatureoflightcanbeexplainedusingtheconcept
ofaphoton.Remembertherelationshipbetweenwavelength
(waveproperty)andlinearmomentum(particleproperty).
4. Electrondiffraction.WhatisthedeBrogliewavelengthof
particles?Whichparticleismorewave-like,a)electron;b)
neutron;orc)atom?
5. Wave-particleduality.Whatistherelationshipthatreconciles
thewaveandparticlenatureofmatter?
6. Heisenberg’suncertaintyprinciple.Intheelectrondiffraction
experiment,narrowingtheslitresultsinanincreasedaccuracy
inourknowledgeofthepositionofelectrons,butadecreased
accuracyintheknowledgeofthemomentuminthesame
direction.
7. Time-dependentSchrödingerequation.Writedownthe
equationinthegeneralform(inthreedimensions)andthe
equationforafreeelectron.Whatisthevariable?Whatarethe
operators?Whatistheunknown?Whatinformationdoesthe
solutionoffer?
8. Time-independentSchrödingerequation.
a. Derivethetime-independentSchrödingerequationfrom
thetime-dependentform
assumingthattheexternal
field(orpotential)istimeinvariant.Hint:expressthe
time-dependentwavefunctionasaproductofthetimeindependentwavefunctionandanexponentialfunction
(Eq.11.83).
b. Writedowntheequationinthreedimensionandpoint
outthevariable,operator.Whataretheunknowns?
c. Writedowntheprobabilitydensityandtheprobabilityof
findingaparticleinaspaceelementdxdydzandinthe
entirespace.Whatis|f|2forarealfunctionfanda
complexfunctionf=af1+ibf2?
9. Eigenvalueequation.Givenanexampleofeigenvalue
equationandpointouttheoperator,eigenfunctionand
eigenvalue.
10.
PostulatesIII.Quantummechanicaloperatorsperform
twotypesofoperations.Whatarethey?Remember:position
operator,linearmomentumoperator,kineticenergyoperator
andpotentialenergyoperator.
11.
Generalunderstandingof“particleinaonedimensionalbox”.WritedownthetwoSchrödingerequations
fordescribingtheparticleinaone-dimensionalbox(forboth
insideandoutsideofthebox)?Whatisthetotalenergyand
wavefunctionobtainedfromsolvingthesetwoequations?
12.
Moreon“particleinaone-dimensionalbox”.
a. Writedowntheprobabilitydensityoffindingthe
particleinsidethebox.
b. Drawaplotshowingthewavefunctionandprobability
densityforthestaten=2?
c. Whatisthetotalenergyoftheparticleinstaten=4?On
howmanypositionscanthe
particlevanishinsidethe
boxinthisstate?
d. Istheenergygapbetweentwoadjacentenergylevels,n
andn+1thesame?Givethe
expressionforthisenergy
gap.
e. Whatisprobabilityoffindingtheparticlebetween
positionxandx+dx?
f. Whatisthetotalprobabilityoffindingtheparticleinthe
box?
13.
Contrast“particleinaone-dimensionalbox”with
“tennisballinaone-dimensionalbox”.Supposewegivethe
tennisballaninitialvelocityinthexdirection.Nofrictionor
otherforcesarepresent.Pointouttwomajordifferences.Hint:
energylevelsandprobabilityoffindingtheparticleorball.
14.
Particleinabox–threedimensionalform.
a. Writedownthewavefunctionandenergyoftheparticle.
Thereisonequantum
numberforeachdimension.
b. Whatisthedegreeofdegeneracyfortheenergylevel
E2,3,4?Whatisthenumberof
wavefunctionsforthis
energylevel?
15.
HarmonicOscillator–classicalenergyandquantum
description.
a. Writedowntheclassicalexpressionforthetotalenergy
ofasystemconsistingoftwo
ballsconnectedbya
masslessspring.Pointoutthekineticenergyand
potential
energy.
b. WhatisthecorrespondingHamiltonian?
c. WhatistheSchrodingerequationthatdescribesthe
system?
d. Whatisthequantummechanicalenergyofaharmonic
oscillator?Explainallterms.
Whatisthelowestenergy?
Aretheenergylevelsdegenerate?
e. Pointouttwodifferencesbetweentheclassicaland
quantumdescriptionof
harmonicoscillator.Hint:energy
andprobabilitydensity.
16.
Vibrationalspectroscopy.
a. Whichenergylevelismostpopulatedatroom
temperature?Whatisthemost
probabletransition
betweenvibrationalenergylevels?Writedownthe
transition
energyandexplainallterms.Whatisthe
frequencyofemittedorabsorbedlight?
b. Nameoneusefulquantityonecanobtainfrommeasuring
thevibrationalspectrumof
amolecule.
c. Inwhichwavelengthregioncanavibrationaltransition
beobserved?a)Infrared;b)
Ultra-violet;c)X-ray;ord)
visible?
17.
Commutator.
a. Usethecommutationrelation(commutator)betweenthe
twooperators𝑧and𝑝!to
provethatzandpzcannotbe
simultaneouslydetermined,e.g.,∆𝑧∆𝑝!≠0.
b. Usethecommutationrelationtoprovethatenergyand
timecannotbe
simultaneouslydetermined,e.g.,∆𝑡∆𝐸≠
0.
c. Canwesimultaneouslydeterminezandpzoftwo
particles?
d. Bonus.Canxandkineticenergybesimultaneously
determined?
18.
Requirementsforanacceptablewavefunction.
a. Givethedefinitionforwavefunction.
b. Namethreerequirementsforanacceptablewave
function.
c. Whichofthefollowingfunctionscannotbeawave
function?a)kx;b)sin(ax);c)eax;
d)sin2(ax)
19. Quantumdescriptionofhydrogen-likeatoms
a. WritedowntheHamiltonianforasystemconsistingof
oneelectronandonenucleus.
Pointoutthekinetic
energyoperatorandpotentialenergyoperator.
b. WhatistheelectronicenergyofHatom?Whichquantum
numberdoesitdependon?
Whatthedegreeof
degeneracy?
c. Whatisthespacingbetweentwoadjacentlevels?Does
thespacingincrease,
decreaseorremainthesamewith
thequantumnumber?
20. Orbitals
a. Theelectronicwavefunctionisalsoknownasorbital.An
orbitalisspecifiedbywhat
quantumnumbers?
b. Writedownthequantumnumbersforthefollowing
orbitals:3s,4py,4dx2-y2,5dz2.
c. Howmanynodesdotheaboveorbitalshave?
d. DrawtheseorbitalsinCartesiancoordinate.
21. Radialfunction.
a. Whatisaradialfunction?Whatisaradialdistribution
function?
b. Fortheorbital5dz2,writedowntheexpressionforthe
radialdistributionfunction.
22.
Angularmomentum–classicalformandquantum
mechanicaloperator.
a. Writedowntheclassicalequationfortheangular
momentum(Lz)ofamassrotating
aboutzaxis.Explain
theterms.
b. Whatisthecorrespondingquantummechanical
operator?
23.
Rigidrotor–classicalenergyandquantumdescription.
a. Writedowntheclassicalexpressionforthekineticenergy
ofarigidrotor.
b. WhatisthecorrespondingHamiltonian?
c. WhatistheSchrodingerequationthatdescribesarigid
rotor?Whatarethequantum
numbers?
d. Writedownthequantummechanicalenergyofarigid
rotor.Aretheenergylevels
degenerate?Ifyes,whatis
thedegreeofdegeneracy?
24.
Rotationalspectroscopy.
a. Whataretheallowedtransitionsbetweenrotational
energylevels?Writedownthe
transitionenergyand
explainallterms.Whatisthefrequencyofemittedor
absorbed
light?
b. Nameoneusefulquantityonecanobtainfrommeasuring
therotationalspectrumof
amolecule.
c. Whatisthewavelengthrangeofrotationalspectra?
25.
Orbitalangularmomentumoperators
a. Writedowntheclassicalformandquantummechanical
operatorofangular
momentuminCartesiancoordinate.
b. Whatistheadvantageoftransformingtoaspherical
coordinate?
c. Whataretheeigenvaluesandeigenfunctionsof𝐿zand𝐿2
operators?Pointoutthe
quantumnumbersandthe
valuestheycantakeon.
d. WhatvaluescanLhave?
26.
Spinangularmomentumoperators
a. Noclassicalcounterpart.
b. Whataretheeigenvaluesandeigenfunctionsforof𝑆zand
𝑆2operators?
c. Whatisthespinforphotons,neutronsandelectrons?