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TheoryTheory of of thethe electronelectron

The electron (cathode rays) were “scientifically discovered” by J. J.

Thomson in 1896 at Cavendish Labs in Cambridge, UK. There were

some very brave assertions by Thomson and his group one of which later

proved to be incorrect.

These were:

1. Cathode rays are charged particles (which he called 'corpuscles').

2. These corpuscles are constituents of the atom.

3. These corpuscles are the only constituents of the atom.

Later on, in 1911, a brilliant experimentalist, Robert Millikan determined the

charge of the electron with his famous “oil drop” experiment. After many

attempts, he observed that the force due to the external field applied to each

droplet was always divisible by 1.602 x 10-19. This was the charge of one

electron in Coulombs.

Remember: F = E x q, in Millikan’s experiment, he knew E and F. F was

–m x g (gravitational force acting on the droplet).

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ThenThen camecame thethe wavewave interpretationinterpretation..

Niels Bohr constructed his model of the hydrogen atom assuming that electrons

were waves swirling-twirling around the positively charged nucleus. This way of

thinking combined with the “standing wave” concept gave rise to the prediction of

discrete energy levels for hydrogen. AndAnd it it workedworked !

Louis De Broglie then came up with the “wave-particle duality” interpretation for

electrons (same for photons).

ClassicalClassical limitlimit

Lorentz factor for effective mass correction (fyi)

Frequency relation to energy

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WaveWave--particleparticle dualityduality

hE ν=

πνω 2=

hω=E

π2

h=h

Physicists often use angular frequency, Physicists often use angular frequency,

ThusThus,,

where

is called the is called the reduced Planck constantreduced Planck constant

WhereWhere doesdoes thethe Planck Planck ConstantConstant comecome fromfrom ??????

Slope of electron energy-frequency curve

Despite a continuous variation of incoming

radiation, electrons are ejected at certain energies

in the “photoelectric effect” experiments.

DespiteDespite a a continuouscontinuous variationvariation of of incomingincoming

radiationradiation, , electronselectrons areare ejectedejected at at certaincertain energiesenergies

in in thethe ““photoelectricphotoelectric effecteffect”” experimentsexperiments..

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Time Time toto talk talk aboutabout thethe birthbirth of of quantumquantum physicsphysics conceptsconcepts

BlackBlack body body radiationradiation

HydrogenHydrogen emissionemission spectraspectra

EveryEvery energyenergy--exchangeexchange

happenshappens in in discretediscrete

amountsamounts !!

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UnderstandingUnderstanding thethe blackblack body body radiationradiation ((SoonSoon it it willwill be be appliedapplied toto electronselectrons!)!)

Black body radiation was the earliest puzzle to be solved. Max Planck made a

ad-hoc (full of assumptions-that he did not know whether they were correct or

not) attempt to explain it.

Nu

mb

er

Nu

mb

er

of

of m

od

es

mo

des

ForFor higherhigher frequenciesfrequencies, , moremore curvescurves can be fit can be fit withwith thethe constraintconstraint thatthat thethe

wavewave functionfunction willwill becomebecome zerozero at at thethe wallwall ((BoundaryBoundary conditioncondition).).

““If the mode is of shorter wavelength, there are more ways you caIf the mode is of shorter wavelength, there are more ways you can fit it into the cavity to meet that n fit it into the cavity to meet that

condition. Careful analysis by Rayleigh and Jeans showed that thcondition. Careful analysis by Rayleigh and Jeans showed that the number of modes was proportional to e number of modes was proportional to

the frequency squared.the frequency squared.”” ((http://hyperphysics.phyhttp://hyperphysics.phy--astr.gsu.edu/hbaseastr.gsu.edu/hbase))

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Energ

yE

nerg

yof

of ra

dia

tion

radia

tion

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WhatWhat is a is a ““standingstanding wavewave””??

“…a wave that neither goes left nor right (in 1D)”

A wave whose ends are fixed

The wave equation in 3D

General solution

When the general solution

is substituted into the wave

equation.

Simplify and you end up

with this

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MaxMax Planck Planck assumedassumed thatthat: : ThereThere areare oscillatorsoscillators on on thethe wallswalls of of thethe enclosureenclosure (inside (inside

thethe blackblack body). body). WhenWhen heatedheated upup, , theythey oscillateoscillate. . ThisThis oscillationoscillation producesproduces

electromagneticelectromagnetic radiationradiation ((justjust likelike thethe electronelectron oscillatingoscillating upup andand downdown on a on a

antennaantenna))

TheThe electricelectric fieldfield andand magneticmagnetic fieldfield has has toto be be zerozero at at thethe wallwall ((otherwiseotherwise wewe getget

chargingcharging on on thethe wallswalls of of thethe furnacesfurnaces –– whowho getsgets an an electricelectric shockshock whenwhen touchingtouching

a hot a hot surfacesurface?)?)

RememberRemember: : TheThe shortershorter thethe wavelengthwavelength, , thethe moremore numbernumber of of curvescurves couldcould be be

producedproduced toto fit inside fit inside thethe cubecube. . TheseThese curvescurves areare eacheach calledcalled ““modesmodes””. .

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MaxMax Planck Planck modificationmodification

((oror correctioncorrection))

AccordingAccording toto RayleighRayleigh andand JeansJeans, , thethe numbernumber of of

modesmodes possiblepossible toto fit inside fit inside thethe cubecube goesgoes toto infinityinfinity

withwith higherhigher oscillatoroscillator frequencyfrequency..

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DensityDensity of of modesmodes forfor a a givengiven wavelengthwavelength

3

3

4rV Sphere π=

n’s are positive, so:

3

3

4

8

1rV π=

2/12

3

2

2

2

1)( nnnr ++=

ThisThis waswas thethe solutionsolution of of thethe

wavewave equationequation

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UntilUntil nownow, , wewe calculatedcalculated ““howhow manymany standingstanding wavewave modesmodes wewe can fit can fit intointo a a

cavitycavity””

WeWe wantwant toto knowknow howhow manymany modesmodes wewe can fit can fit intointo a a smallsmall infinitesmalinfinitesmal changechange

in in thethe wavelengthwavelength of of radiationradiation ((radiationradiation thatthat is is emittedemitted byby thethe oscillatorsoscillators on on

thethe cavitycavity wallswalls))

EachEach modemode is is supposedsupposed toto havehave an an energyenergy kTkT

((SomeSome thermodynamicsthermodynamics herehere). ).

4

8

λ

πNumber of modes x kT

4

8

λ

πkT

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ElectronicElectronic--MagneticMagnetic--OpticalOptical PropertiesProperties areare allall aboutabout thethe behaviorbehavior of of electronselectrons in in

solidssolids..

A A fewfew examplesexamples::

--VeryVery weaklyweakly boundbound electronselectrons withwith ““availableavailable densitydensity of of statesstates ((emptyempty parkingparking lot)lot)””: :

ConductorsConductors

--UnpairedUnpaired electronselectrons in in termsterms of of spinsspins in in thethe outerouter shellsshells::

MagnetismMagnetism

--ElectronsElectrons in in ““bandsbands”” thatthat cannotcannot movemove anywhereanywhere ((fullfull parkingparking lot) but lot) but somesome can can

jumpjump toto thethe nextnext availableavailable//allowedallowed bandband andand thenthen theythey havehave plentyplenty of of statesstates toto hop hop

betweenbetween::

SemiconductorSemiconductor

--WhatWhat ifif somesome electronselectrons fallfall backback intointo thethe previousprevious bandband, , thethe energyenergy--statestate thatthat theythey

belongedbelonged toto?:?:

LightLight emittingemitting diodediode

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In the solution to the Schrodinger equation for the hydrogen atom, three

quantum numbers arise from the space geometry of the solution and a fourth

arises from electron spin. No two electrons can have an identical set of quantum numbers according to the Pauli exclusion principle , so the

quantum numbers set limits on the number of electrons which can occupy a

given state and therefore give insight into the building up of the periodic table

of the elements.

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SomeSome insightinsight aboutabout thethe subshellssubshells ((subsub--energyenergy levelslevels))

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ElectronElectron spinspin

Two types of experimental evidence which arose in the 1920s suggested an

additional property of the electron. One was the closely spaced splitting of the

hydrogen spectral lines, called fine structure. The other was the Stern-Gerlach

experiment which showed in 1922 that a beam of silver atoms directed through

an inhomogeneous magnetic field would be forced into two beams. Both of

these experimental situations were consistent with the possession of an

intrinsic angular momentum and a magnetic moment by individual electrons.

Classically this could occur if the electron were a spinning ball of charge, and

this property was called electron spin.

With this evidence, we say that the electron has spin 1/2. An angular

momentum and a magnetic moment could indeed arise from a spinning sphere

of charge, but this classical picture cannot fit the size or quantized nature of

the electron spin. The property called electron spin must be considered to be a

quantum concept without detailed classical analogy.

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SternStern--GerlachGerlach Experiment Experiment ((toto determinedetermine thethe electronelectron spinspin))

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SimpleSimple harmonicharmonic oscillatoroscillator

General solution:

InIn a a simplesimple harmonicharmonic oscillatoroscillator, not , not everyevery oscillationoscillation frequencyfrequency is is allowedallowed, ,

justjust likelike an an electronelectron in a in a potentialpotential wellwell. .