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