atomic spectra and structure: bohr model of hydrogensmithb/website/course... · ·...
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Atomicspectraandstructure:Bohrmodelofhydrogen
• Atomicstructure• Greeks: Indivisibleconstituentsofallmatter• J.J.Thomsonfindselectron• Rutherford’s planetarymodel
• Atomicspectra• Bohr’squantizedmodel• Franck-Hertzexperiment:Furtherevidenceofquantizationofatomicstructure
Atomicstructure– Briefhistory• Greeks:Atomsdeterminepropertiesofmatter– Indivisibleconstituents
• J.J.Thomsondiscoverselectron(1897)– Proposesplum-puddingmodelofatom
Atomicstructure– NuclearAtom
• Geiger-Marsden(Rutherfordgold-foil)experiment(1909)– Demonstratesmassive,positivelychargednucleus
• Rutherfordplanetarymodelofatom(1911)– Positivelychargednucleussurroundedbynegativeelectrons
Atomicspectra
• Emissionandabsorptionspectra
Atomicspectra– hydrogen• Emissionspectrum– Observenarrowemissionwavelengthsaccording toempirical formula (Rydberg formula)
• Absorptionspectrum– Notalllinesofemissionarefound inabsorption
Rydberg formula
R = 13.6 eV
Rydberg constant
� =hc
R
✓1
m2� 1
n2
◆�1
Atomic'spectra'–'hydrogen'
• Emission'spectrum'– Observe'narrow'emission'wavelengths'according'to'empirical'formula'(Rydberg'formula)'
• Absorp<on'spectrum'– Not'all'lines'of'emission'are'found'in'absorp<on'
Rydberg'formula'
R = 13.6 eV
Rydberg'constant'
� =hc
R
✓1
m2� 1
n2
◆�1
ProblemswithplanetarymodelClassicaltreatmentpredicts• Electronorbitingthenucleusundergoesconstantaccelerationandthusshouldcontinuallyemitradiation
• Electronshouldspiralintonucleusinapproximatelymicroseconds
• Radiationemittedshouldhaveacontinuousspectrumoffrequencies
Bohrmodelofhydrogen• Electronorbitsnucleusincircularorbit• ElectronandnucleusboundbyCoulombforce
• Thisgivesthekineticenergy
• PotentialenergyisjustCoulombpotential
• Totalenergyis
Classicaluptothispoint...
F =1
4⇥�0
e2
r2=
mev2
r
U = � 1
4⇥�0
e2
r
K =1
2mev
2 =1
8⇥�0
e2
r
E = � 1
8⇥�0
e2
r
Centripetalacceleration
Bohrmodelofhydrogen
Bohrpostulates:• Electronscanonlyoccupycertainstationarystates,inwhichradiationisnotemittedandangularmomentumisquantized
• Radiationisonlyemittedorabsorbedwhentheelectronmakesatransitionfromonestationarystatetoanother.
~ = h/2�L = mevr = n~
Bohrmodelofhydrogen• Solvingforvelocity
• Usethisresultinforceequation
• Thiscanbesolvedforallowedradii
v =
✓n~mer
◆L = mevr = n~ =)
F =1
4⇥�0
e2
r2=
mev2
r=
me
r
✓n~mer
◆2
rn =4⇥�0~2mee2
n2 = a0n2
Bohrradius
a0 ⇡ 0.53 A
• Allowedenergylevels(subrn intoclassicalexpressionfortotalenergy)
E1 = 13.6 eV
En = � mee4
32⇥2�20~21
n2= � e2
8⇥�0a0
1
n2= �E1
n2
SameasRydberg constant!
Bohrmodelofhydrogen
n iscalledthequantumnumber
Emissionspectra
• Photonenergygivenbydifferencebetweeninitialandfinalatomicenergylevels
Eph = Em � En = E1
✓1
n2� 1
m2
◆
�ph =E1
h
✓1
n2� 1
m2
◆
Absorptionspectra
• Photonenergygivenbydifferencebetweeninitialandfinalatomicenergylevels
Startingroundstateatroomtemperature!
Eph = Em � En = E1
✓1
n2� 1
m2
◆
�ph =E1
h
✓1
n2� 1
m2
◆
Hydrogen-likeions
• NuclearchargenowZe• Changealle2 toZe2 inprevioustreatment• Thisgives
rn =a0n2
ZEn = �E1
Z2
n2
Ze
�e
�ph =E1Z2
h
✓1
n2� 1
m2
◆
Franck-Hertzexperiment– 1914• FurtherevidenceforBohrmodel
Measuredcurrentdropsevery4.9V
Electronsexperienceaninelastic collisionandlose4.9eV ofenergy
LimitationsofBohrmodel• Cannotbeappliedtomulti-electronatoms.• Doesnotpredictfinestructureofatomicspectrallines.
• Doesnotprovideamethodtocalculaterelativeintensitiesofspectrallines.
• Predictsthewrongvalueofangularmomentumfortheelectronintheatom.
• ViolatestheHeisenberguncertaintyprinciple(althoughBohr'smodelprecededthisbymorethanadecade).
SummaryofBohrmodel
• Startwithclassical,circularorbits• Angularmomentumisquantizedinallowedstationarystates
• Givesallowedradiiandenergylevels
• Emissionandabsorptionoflightbytransitionsbetweenstationarystatesonly
L = mevr = n~
rn =4⇥�0~2mee2
n2 = a0n2 En = �E1
n2
Wavelikebehaviorofmatter• Doubleslitexperimentwithphotons• deBrogliehypothesis• Davisson-Germer experiment• Diffractionandinterferencewithlargersystems:atoms,molecules•Wave-particleduality:Wavefunction
Doubleslitexperimentrevisited
• Atlowlightlevelsinterference patternrequiresfinitetimetocollectlight
• Photonsdetectedoneatatimeatalocalizedpoint!
WAVE
PARTICLE
deBrogliehypothesis
• Iflighthasbothwaveandparticleproperties,cannotmatteralsohavewaveproperties?
• Specialrelativityimplies:• Foraphoton:• UsingthePlanckrelation:
�dB =h
p
E2 = p2c2 + (mc2)2
E = pc
E = h⇥ =hc
�
deBrogliewavelength
deBrogliewaves?
• deBrogliewentbeyondlightandsuggestedthisequationholdsforallmatter
• deBrogliewavelength• Whynotobservedineverydaylife?
�dB =h
p
m=92kgv =44.72 km/h~12.42m/s
m =0.16kgv =161.26 km/h~44.8m/s�cricket =
h
mv
=6.626⇥ 10�34 Js
(0.16 kg)(44.8 m/s)
= 9.2⇥ 10�35 m
�Bolt
=h
mv
=6.626⇥ 10�34 Js
(92 kg)(12.4 m/s)
= 5.8⇥ 10�37 m
deBroglieonBohratom• deBroglielookedatBohr’satomicmodellikeamusicalinstrument.Electronsguidedby“pilotwaves”ineachorbit.Orbitcircumference=integernumberofwavelengths
�dB =h
p
2⇥rn = n�dB
pn =~
a0n
Analogytooptics• Whendoeswavelengthof“matterwave”becomerelevant?
• Similartodiffractioninoptics:Whenscatteringobjects(ofsized)becomecomparabletowavelength
Wavelength Optics Matterwaves
RayOptics ParticleTrajectories
WaveOptics WaveMechanics
� ⌧ d
� & d
Momentummatters• Whendoeswavelengthof“matterwave”becomerelevant?
• Similartodiffractioninoptics:Whenscatteringobjects(ofsized)becomecomparabletowavelength. �dB =
h
p
ElectronswithkineticenergyK=50eV
canusenon-relativistic formofKsinceK<<mec2 =0.511MeV
K ⇡ p2
2mp ⇡
p2mK
pc ⇡p2mc2K
=)
pc ⇡p2(0.511⇥ 106 eV)(50) eV
= 7.15⇥ 103 eV
� =hc
pc=
1240 eV nm
7.15⇥ 103 eV
= 0.173 nm
Latticespacingincrystals
Momentummatters• Whendoeswavelengthof“matterwave”becomerelevant?
• Similartodiffractioninoptics:Whenscatteringobjects(ofsized)becomecomparabletowavelength. �dB =
h
p
NeutronswithkineticenergyK=0.00024eV
canusenon-relativistic formofKsinceK<<mnc2 =940MeV
K ⇡ p2
2mp ⇡
p2mK
pc ⇡p2mc2K
=)Nanowires
pc ⇡p2(940⇥ 106 eV)(2.4⇥ 10�4 eV)
= 672 eV
� =hc
pc=
1240 eV nm
672 eV
= 1.85 nm
Electrondiffraction:Davisson-Germerexperiment(1925) Firstdirectdemonstrationofwave
propertiesofmatter
d sin � = n⇥
ElectronsacceleratedthroughpotentialdifferencegivingK=54eV
� =h
mv= 0.167 nm
Constructiveinterference
Nickel latticespacingd =0.215nm
d sin � = (0.215 nm) sin(50�)
= 0.165 nm
Dataandtheoryingoodagreement!
Doubleslitexperimentwithelectrons
A.Tonomura,etal,Am.J.Phys.57,117–120(1989).
• Two-pathsaroundachargedwire
(Mach-Zehnder interferometer)• Electronmicroscopesusedinmanyapplications– resolution~0.1nm
Matter-wavediffractionwithfree-standinggratingsovertheyears
1961 1999 2012
Jönssen (Cugrating) Arndt(Si3N4 grating) Arndt (Si3N4 grating)
Classicaluncertaintyinbeamenergy• Spreadinelectronbeamenergyleadstospreadinwavelengthandthusdiffractionpattern
• Considerdiffractiongrating
�dB =h
p
p ⇡p2mK
=)
d sin �n = n⇥Constructiveinterference:
�✓
✓⇡ ��
�� ⇡ n⇥
d
Forsmallangles:
� =h
p��
�=
�p
p
Fractionaluncertaintyinscatteringangle
Fractionaluncertaintyinwavelengthequaltothatofmomentum
�p =
����dp
dK
�����K =1
2(2mK)�1/22m�K =
1
2p�K=)
Non-relativisticmomentum:
Classicaluncertaintyinbeamenergy• Spreadinelectronbeamenergyleadstospreadinwavelengthandthusdiffractionpattern
• Considerdiffractiongrating
�dB =h
p
=)
d sin �n = n⇥Constructiveinterference:
� ⇡ n⇥
d
Forsmallangles:
� =h
p
Fractionaluncertaintyinscatteringangle
��
�⇡ �⇥
⇥=
�p
p=
1
2
�K
KFornon-relativisticmatter
��
�=
�Eph
EphForaphoton(notefactorof2difference)
Summary:Diffractionandinterferenceofmatterwaves• Interferenceofmatterwaves– Electrons– Neutrons– Atoms– Molecules
• C60• Phthalocyanine
• Spreadinkineticenergyofparticlesleadstospreadindiffractionangle!
• Braggscattering• Doubleslit• Diffractiongrating
�dB =h
p
deBrogliewavelength
��
�=
1
2
�K
K
��
�=
�Eph
Eph
Wave-particleduality
• Doubleslit(again)• Superpostion•Wavefunction• Probability• Complementarity• Uncertaintyprinciple
Doubleslitinterference• Detectionofaparticleatpointx onthescreengovernedbyinterferenceofpathwaystheparticlecouldtake(Feynman)
• Twopossiblepaths(r1 orr2),withequalamplitudesA,andphasesφ1 andφ2
⇤1 = kr1 =2⇥
�
L2 +
✓x+
d
2
◆2!1/2
⇤2 = kr2 =2⇥
�
L2 +
✓x� d
2
◆2!1/2
Doubleslitinterference(Superposition)• Particleflux(numberperunittime)atapointx onthescreenisgivenbythetotalamplitudesquared
N = |Atot
|2
N = A2|ei�1 + ei�2 |2 = 2A2(1 + cos(��))
�⇤ = ⇤1 � ⇤2 ⇡ 2⇥xd
�L
N(x) = 4A2cos
2
✓⇥xd
�L
◆
Atot
= A1
+A2
= A(ei�1 + ei�2)
Doubleslitwith“which-path”information• Looktoseethroughwhichslittheelectronpasses(putawirelooparoundeachslit)
• Anyknowledgeof“whichslit”localizestheparticleanddestroysthesuperpositionofpossiblepaths– hencenointerference.
Complementarity• Waveorparticle?Youdecide!Dependsonhowyoulookatthesystem…
• Wavenatureofsystemcanbeobservedwhenperforminga“wave-like”experiment
• Particlenatureofsystemisobservedwhenperforminga“particle-like”(which-path)experiment
WAVE
PARTICLE
Wave-particleduality• Waveè wavefunction,e.g.aplanewave
–Well-definedwavelength(frequencyorenergy)andmomentum
– Completelydelocalized(nospatialortemporalinformation)
• Particleè trajectory–Well-definedposition–Wavelength(andtherefore“momentum”)undefined
�(x, t) = Aei(k·x��t)
x(t) =
Z t
�1v(t0)dt0
Innaturebothwaveandparticlepropertiesarepresentuntilyoulook!
Wave-functioninterpretation• Borninterpretation:Wavefunctiondescribesprobabilitytofind“particle”inasmallregionaboutx attimet
P (x)dx = |�(x, t)|2dx
Superpositionandwavepackets• Addingtwoormoreplanewavesgivesabeatfrequencyandlocalizedwavepacket
sin(k1x) + sin(k2x)
10X
m=1
sin(kmx)
Superpositionandwavepackets• Addingtwoormoreplanewavesgivesabeatfrequencyandlocalizedwavepacket
• Takentoacontinuumsum->integral• Gaussianwavepacket
Z 1
�1e�ax
2+bx
dx =
⇣�a
⌘1/2exp
✓b2
4a
◆
⇥(x) =
Z 1
�1�(k)eikxdk =
Z 1
�1e�k
2/2�k
2
eikxdk
|�(k)|2
�kp2
Superpositionandwavepackets• Addingtwoormoreplanewavesgivesabeatfrequencyandlocalizedwavepacket
• Takentoacontinuumsum->integral• Gaussianwavepacket
⇥(x) =
Z 1
�1�(k)eikxdk =
Z 1
�1e�k
2/2�k
2
eikxdk
⇥(x) =�kp�e�x
2�k
2/2
|�(k)|2
�kp2
�x =1p2�k
Superpositionandwavepackets• Addingtwoormoreplanewavesgivesabeatfrequencyandlocalizedwavepacket
• Takentoacontinuumsum->integral• Gaussianwavepacket
|�(k)|2
�kp2
�x =1p2�k
�x�p
x
=~2
px
= ~k�x�k = 1/2
Heisenberguncertaintyprinciple• Positionandmomentumarenotsimultaneouslywell-defined
• Singleslitorconfinedparticle:Example–electronconfinedtoanatom
p = ~k
�x ⇡ 0.5A
p ⇡ �p ⇡ ~�x
E =(pc)2
2mc
2=
(~c)22�x
2mc
2= 15 eV
Givesnearlythesameenergyasthegroundstateenergyofhydrogen!
�x�k = 1/2�x�p
x
=~2
Wavicles summary• WavefunctiondescribesdeBrogliewaves
• Wavefunctiondescribesprobabilitieswhereonewillfinda“particle”
• Superpositionkey– Superposition ofplanewavesleadstolocalizedwavepackets
– Measuringpositioncollapsessuperposition– Complementarity – Canonlyobservewaveorparticlenatureinagivenexperiment
• HeisenbergUncertaintyRelation
�(x, t) = A(x, t)ei�(x,t)
P (x)dx = |�(x, t)|2dx
�x�p = h