Quantum Mechanics-Dr.gagan Anand

8/9/2019 Quantum Mechanics-Dr.gagan Anand

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X-ray Interaction withX-ray Interaction with

MatterMatter Electromagnetic Radiation interactsElectromagnetic Radiation interacts

with structures with similar size towith structures with similar size to

the wavelength of the radiation.the wavelength of the radiation. Interactions have wavelike andInteractions have wavelike and

particle like properties.particle like properties.

X-rays have a very small wavelengthX-rays have a very small wavelengthno larger than !"no larger than !"-#-# to !"to !"-$-$..


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X-ray Interaction withX-ray Interaction with

MatterMatter%he higher the energy of the &-ray the%he higher the energy of the &-ray the

shorter the wavelength.shorter the wavelength.

'ow energy &-rays interact with whole'ow energy &-rays interact with wholeatoms.atoms.

Moderate energy &-rays interact withModerate energy &-rays interact with

electrons.electrons. (igh energy &-rays interact with the(igh energy &-rays interact with the

nuclei.nuclei.


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)ive forms of &-ray)ive forms of &-ray

InteractionsInteractions *lassical or *oherent +cattering*lassical or *oherent +cattering *ompton E,ect*ompton E,ect

hotoelectric E,ecthotoelectric E,ect air productionair production

hotodisintegrationhotodisintegration


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%wo )orms of X-ray%wo )orms of X-ray

InteractionsInteractions

Important toImportant toiagnostic X-rayiagnostic X-rayPhotoelectric EfectPhotoelectric Efect

Compton EfectCompton Efect


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%he hoto Electric E,ect%he hoto Electric E,ect

iscovery implications andiscovery implications and

current technologycurrent technology


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Discovery:Discovery:Heinrich Hertz and PhillipHeinrich Hertz and Phillip

LenardLenard

(ertz clari/ed Ma&well(ertz clari/ed Ma&wells electromagnetic theory ofs electromagnetic theory oflight0light0

1 roved that electricity can 2e transmitted inroved that electricity can 2e transmitted inelectromagnetic waves.electromagnetic waves.

1 Esta2lished that light was a form of electromagneticEsta2lished that light was a form of electromagneticradiation.radiation.

1 )irst person to 2roadcast and receive these waves.)irst person to 2roadcast and receive these waves.

Back in 1887


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'enard 3oes )urther4'enard 3oes )urther4

(is assistant hillip 'enard e&plored the(is assistant hillip 'enard e&plored thee,ect further. (e 2uilt his own apparatuse,ect further. (e 2uilt his own apparatuscalled acalled a phototu2ephototu2eto determine theto determine the

nature of the e,ect0nature of the e,ect0


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'enard5s hotoelectric 6pparatus0


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%he E&periment0%he E&periment07y varying the voltage on a7y varying the voltage on a negatively charged gridnegatively charged grid

2etween the e8ecting surface and the collector plate2etween the e8ecting surface and the collector plate

'enard was a2le to0'enard was a2le to0

1 etermine that the particles had a negativeetermine that the particles had a negativecharge.charge.

1 etermine the kinetic energy of the e8ectedetermine the kinetic energy of the e8ectedparticles.particles.


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'enard5s )indings0

%hus he theorized that this voltage must 2e e9ual to thema&imum kinetic energy of the e8ected particles or0

KEKEmaxmax= eV= eVstoppingstopping

erple&ing :2servations0

%he intensityof light had no e,ect on energy

%here was a threshold requencyfore8ection

Classical physics ailed to explain this,Classical physics ailed to explain this,

Lenard won the Nobel Prie in Physics in !"#$%Lenard won the Nobel Prie in Physics in !"#$%


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hotoelectric E,ect

%he electron removed from thetarget atoms is called aphotoelectron%

%he photoelectronescapes with&inetic energy equal to thediference between the energy o

the incident x'ray and thebinding energy o the electron%


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hotoelectric E,ect

'ow anatomic num2er target atomssuch as soft tissue have low 2indingenergies.

%herefore the photoelectric electronis released with kinetic energy nearlye9ual to the incident &-ray.

(igher atomic num2er target atomswill have higher 2inding energies.


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hotoelectric E,ect

%herefore the kinetic energy of thephotoelectron will 2e proportionallylower%

*haracteristic &-rays are producedfollowing a photoelectric interaction tothose produced in the &-ray tu2e.

%hese characteristic &-rays are alsosecondary radiation and acts likescatter.


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hotoelectric E,ect

(he probability o a photoelectricinteraction is a unction o thephoton energy and the atomic

number o the target atom%

) photoelectric interaction cannot occur unless the incident x'

ray has energy equal to orgreater than the electron bindingenergy%


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hotoelectric E,ect

%he pro2a2ility of photoelectricinteraction is inversely proportional tothe third power of thephoton

energy%

%he pro2a2ility of photoelectricinteraction is directly proportional to

the third power of theatomicnumber o the absorbing material


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*ompton E,ect

Moderate energy &-ray photon through outthe diagnostic &-ray range can interact withouter shell electron.

%his interaction not only changes thedirection 2ut reduced its energy and ionizesthe atom as well.

%he outer shell electron is e8ected. %his is

called Compton Efect or Compton*cattering%


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Incident hoton

E; hD> ; m>c?@p>c>

p>c>; AE>@ >mc>AE

7ut AE ; h< - hc>; Ch1 >ChChc>; Ch1 >Ch; Ch1 >ChCh< - hmc>Ch< - hChc>

we have0

mc <


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mc ! ! <


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*ompton E,ect

hotons scattered 2ack towards theincident &-ray 2eam are called7ackscatter Radiation.

Bhile important in radiation therapy2ackscatter in diagnostic &-ray issometimes responsi2le for the hinges

on the 2ack of the the cassette to 2eseen on the &-ray /lm

i l i. i


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Experimental -eri.cation

MonochromaticX-ray Source

photon

Graphitetarget

Braggs X-ray

Spectrometer

1. One peak is foun at sameposition. !his is unmoifie raiation

". Other peak is foun at higher

#a$e%ength. !his is moifie signa% of%o# energy.

&. increases #ith increase in .


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)eatures of *ompton+cattering

Most likely to occur

6s &-ray energy increases

6s atomic num2er of the

a2sor2er increases

6s mass density of

a2sor2er increases

Bith outer-shell electrons

Bith loosely 2oundelectrons.

Increased penetrationthrough tissue w=ointeraction.

Increased *ompton relativeto photoelectric scatter.

Reduced total *omptonscattering.

Fo e,ect on *ompton+catter

roportional increase in*ompton +catter.


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air roduction

If the incident &-rayhas suHcientenergy it may

escape the electroncloud and comeclose enough to thenucleus to come

under the inuenceof the strongelectrostatic /eld ofthe nucleus.


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air roduction

%he interactionwith the nucleusstrong electrostatic

/eld causes thephoton todisappear and inits place appear

two electrons.


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air roduction

:ne is positivelycharged and calleda positron while

the other remainsnegativelycharged. %his iscalled Pair

Production%


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!he rest mass energy of an e%ectron or positron is '.(1Me) *accoring to + , mc".

!he minimum energy reuire for pair prouction is 1.'"

Me).

/ny aitiona% photon energy 0ecomes the kinetic energyof the e%ectron an positron.

!he corresponing maimum photon #a$e%ength is 1." pm.+%ectromagnetic #a$es #ith such #a$e%engths are ca%%egamma rays .)(


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air roduction

It take a photonwith !."> MeJ toundergo air

roduction. %herefore it is not

important todiagnostic &-ray.


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Pair Annihilation

2hen an e%ectron an positron interact #ith each other ue

to their opposite charge3 0oth the partic%e can annihi%atecon$erting their mass into e%ectromagnetic energy in theform of t#o - rays photon.

++ + ee

4harge3 energy an momentum are again con$erse. !#o- photons are prouce *each of energy '.(1

Me) p%us ha%f the 5.+. of the partic%es to conser$e the

momentum.


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6ouis e Brog%ie *in 17"&

If light can behave

both as a wave and a

particle I wonder if aparticle can also

behave as a wave!

!he eistence of e Brog%ie#a$es #as eperimenta%%yemonstrate 0y 17"83 an !heua%ity princip%e they representpro$ie the starting point for

Schroingers successfu%


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6ouis e Brog%ie

Ill tr" #essing aro$nd with

so#e of %insteins for#$lae and

see what I can co#e $p with&

De Broglie


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'e have seen that light co#es in discrete $nits (photons) withparticle properties (energ" and #o#ent$#) that are related to the

wavelike properties of fre$enc" and wavelength&

M/!!+9 2/)+S

h

p=

In 1*+, -rince .o$is de Broglie post$lated that ordinar" #atter can have

wavelike properties with the wavelengthrelated to #o#ent$#pin the sa#e wa" as for light

de Broglie wavelength

de Broglie relation

,/0&0, 1 2sh =

-lancks constant

Prediction:'e sho$ld see diffraction and interference of #atter waves

De Broglie

3B wavelength depends on #o#ent$# not on the ph"sical si4e of the particle


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Davisson G.P. Thomson

5avisson 6& 2&

re %lectrons

'aves! 9ranklin

Instit$te 2o$rnal

205 :*7 (1*+8)

;he 5avissonscattering a bea# of electrons fro#

a 3i cr"stal& 5avisson got the 1*,7

3obel pri4e&

t fi=ed accelerating voltage (fi=ed

electron energ") find a pattern of sharpreflected bea#s fro# the cr"stal

t fi=ed angle find sharp peaks in

intensit" as a f$nction of electron energ"


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;a$isson-Germer +periment


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Davisson and Germer Experiment


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Davisson and Germer Experiment

If electrons are just particles, we expect a smooth

monotonic dependence of scattered intensity onangle and voltage because only elastic collisions areinvolved

Diffraction pattern similar to X-rays would beobserved if electrons behave as waves


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42

(.&> +%ectron Scattering

;a$isson an Germer eperimenta%%y o0ser$e that e%ectrons #ere iffractemuch %ike rays in nicke% crysta%s.

George ?. !homson *1@7"A178(3 son of . .!homson3 reporte seeing the effects of e%ectron

iffraction in transmission eperiments. !he firsttarget #as ce%%u%oi3 an soon after that go%3a%uminum3 an p%atinum #ere use. !he ranom%yoriente po%ycrysta%%ine samp%e of SnO"prouces

rings as sho#n in the figure at right.


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=

4urrent $s acce%erating $o%tage has a maimum *a 0ump orkink notice in the graph3 i.e. the highest num0er of e%ectronsis scattere in a specific irection.

The bump becomes most prominent for 54 V at ~ 50


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/ccoring to e Brog%ie3 the #a$e%ength associate #ith ane%ectron acce%erate through ) $o%ts is

o

AV

+8&1+

=

Cence the #a$e%ength for (D ) e%ectron

o

A07&1:/

+8&1+

==

=rom X-ray ana%ysis #e kno# that the nicke% crysta% acts as ap%ane iffraction grating #ith grating space , '.71 E


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Cere the iffraction ang%e3 F ('

!he ang%e of incience re%ati$e to the fami%y of Braggs p%ane

=rom the Braggs euation


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=rom the Bragg s euation

#hich is eui$a%ent to the H ca%cu%ate 0y e-Brog%ieshypothesis.

sin+d=o

oo

AA 0:&10:sin)*1&(+ ==


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Electron !icroscope" #nstrumental Application of !atter $aves

@esolving power of an" optical

instr$#ent is proportional to the

wavelength of whatever (radiation or

particle) is $sed to ill$#inate the

sa#ple& n optical #icroscope $ses

visible light and gives :A

#agnification+ n# resol$tion&

9ast electron in electron #icroscope

however have #$ch shorter

wavelength than those of visible

light and hence a resol$tion of C&1

n##agnification 1A can

be achieved in an %lectron

Dicroscope&


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9o%e of an O0ser$er

!he o0ser$er is o0Iecti$e an passi$e

?hysica% e$ents happen inepenent%y of#hether there is an o0ser$er or not

!his is kno#n as o0Iecti$e rea%ity


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;ou0%e-S%it +periment>act of o0ser$ation affects 0eha$iour of e%ectron

9 % f O0 i


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9o%e of an O0ser$er inJuantum Mechanics

!he o0ser$er is noto0Iecti$e an passi$e

!he act of o0ser$ation changes thephysica% system irre$oca0%y

!his is kno#n as su0Iecti$e rea%ity


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sind

Eriginall" perfor#ed b" Fo$ng (181) to de#onstrate the wavenat$re of light&

Gas now been done with electrons ne$trons Ge ato#s a#ong others&

D

?d

5etecting

screen

Inco#ing coherent

bea# of particles

(or light)

y

lternative

#ethod of

detection> scan a

detector across

the plane and

record n$#ber of

arrivals at each

point

!C+ ;OKB6+-S6

For particles we expect two peaks, for waves an interference pattern

T%o slit #nterference Experiment


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T%o&slit #nterference Experiment

6aser

Source

S%it

S%it ;etector

9ate of photon arri$a% , " 1'Lsec

9ate of photon etection , 1'(sec !ime %ag , '.( 1'-Lsec

Spatia% separation 0et#een photons , '.( 1'-Lc , 1(' m

1 meter

! % i t *17'@ 0% %it i t ith i


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A !ay%ors eperiment *17'@> ou0%e s%it eperiment #ith $ery im%ight> interference pattern emerge after #aiting for fe# #eeks

A interference cannot 0e ue to interaction 0et#een photons3 i.e.

cannot 0e outcome of estructi$e or constructi$e com0ination ofphotons

interference pattern is ue to some inherent property of eachphoton - it Ninterferes #ith itse%f #hi%e passing from source toscreen

A photons ont Nsp%it A%ight etectors a%#ays sho# signa%s of same intensity

A s%its open a%ternating%y> get t#o o$er%apping sing%e-s%it iffractionpatterns A no t#o-s%it interference

A a etector to etermine through #hich s%it photon goes>no interference

A interference pattern on%y appears #hen eperiment pro$iesno means of etermining through #hich s%it photon passes

;ou0%e s%it eperiment JM interpretation


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;ou0%e s%it eperiment A JM interpretation

A patterns on screen are resu%t of istri0ution of photons

A no #ay of anticipating #here particu%ar photon #i%% strikeA impossi0%e to te%% #hich path photon took A cannot assign

specific traIectory to photon

A cannot suppose that ha%f #ent through one s%it an ha%f throughother

A can on%y preict ho# photons #i%% 0e istri0ute on screen *oro$er etector*s

A interference an iffraction are statistica% phenomena associate#ith pro0a0i%ity that3 in a gi$en eperimenta% setup3 a photon #i%%

strike a certain pointA high pro0a0i%ity 0right fringesA %o# pro0a0i%ity ark fringes

Double slit expt %ave vs (uantum


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Double slit expt' && %ave vs (uantum

pattern of fringes>

A


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3e$trons Heilinger

et al. 1*88Reviews of

Modern Physics 6010717,

Ge ato#s> E 6arnal and 2 Dl"nek

1**1Physical Review Letters 66

+08*+0*+

60#olec$les> D

rndt et al. 1***

Nature 0! 08

08+'ith

#$ltipleslit

grating

'itho$t grating

+X?+9


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Double-slit Experiment for Electrons

Jee also this 2ava si#$lation> http>www&$ant$#ph"sics&pol"techni$e&frinde=&ht#l

3o of electrons 1 1

3o of electrons 1 1

5EMB.%J.I; %A-%@ID%3;
http://www.quantum-physics.polytechnique.fr/index.htmlhttp://www.hqrd.hitachi.co.jp/em/doubleslit-f2.cfmhttp://www.hqrd.hitachi.co.jp/em/doubleslit-f2.cfmhttp://www.quantum-physics.polytechnique.fr/index.html


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Jo#e ke" papers in the develop#ent of the do$bleslit e=peri#ent d$ring the +th cent$r">

K-erfor#ed with a light so$rce so faint that onl" one photon e=ists in the apparat$s at an" one ti#e

< I ;a"lor 1**Proceedings of the Ca!ridge Philosophical "ociety!5 11/11:K-erfor#ed with electrons

6 2Lnsson 1*01#eitschrift f$r Physi% !6! /://7/

(translated 1*7/Aerican &ournal ofPhysics 2 /11)K-erfor#ed with single electrons

;ono#$ra et al.1*8*Aerican &ournal of Physics 5" 1171+K-erfor#ed with ne$trons

Heilinger et al. 1*88Reviews of Modern Physics 60 10717,K-erfor#ed with Ge ato#s

E 6arnal and 2 Dl"nek 1**1Physical Review Letters 66 +08*+0*+K-erfor#ed with 60 #olec$les

D rndt et al. 1***Nature 0! 0808+K-erfor#ed with 67 #olec$les showing red$ction in fringe visibilit" as te#perat$re rises

and the #olec$les give awa" their position b" e#itting photons

.& Gacker#ller et al +/Nature 2" 71171/K-erfor#ed with 3a Bose%instein 6ondensates

D @ ndrews et al.1**7 "cience2"50,70/1

n e=cellent s$##ar" is available inPhysics 'orld(Jepte#ber ++ iss$e page 1:)

and at http())physicswe!.org)(readers voted the do$bleslit e=peri#ent the #ost bea$tif$l in ph"sics)&

5EMB.% J.I; %A-%@ID%3;

BIB.IE


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Ceisen0erg rea%ise that ...


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easur ng t e pos t on anmomentum

of an e%ectron Shine %ight on e%ectron an etect ref%ecte%ight using a microscope

Minimum uncertainty in position

is gi$en 0y the #a$e%ength of the%ight

So to etermine the position

accurate%y3 it is necessary to use%ight #ith a short #a$e%ength


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Measuring the position an momentumof an e%ectron *cont

By ?%ancks %a# E, hc3 a photon #ith ashort #a$e%ength has a %arge energy

!hus3 it #ou% impart a %arge Pkick to the

e%ectron But to etermine its momentum

accurate%y3

e%ectron must on%y 0e gi$en a sma%% kick !his means using %ight of %ong #a$e%engthQ


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=unamenta% !rae Off R

Kse %ight #ith short #a$e%ength>

A accurate measurement of position 0ut not

momentum

Kse %ight #ith %ong #a$e%ength>

A accurate measurement of momentum 0ut not

position

Ceisen0ergs Kncertainty


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Ceisen0erg s Kncertainty?rincip%e

The more accurately you know the position (ie!the smaller xis)! the less accurately you know themomentum (ie! the larger pis)" an# $ice $ersa


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+amp%e of Base0a%% *cont

!he uncertainty in position is then

:o #oner one oes not o0ser$e the

effects of the uncertainty princip%e ine$eryay %ifeQ


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+amp%e of +%ectron

Same situation3 0ut 0ase0a%% rep%ace 0yan e%ectron #hich has mass 7.11 1'-&1kg

So momentum , &.L 1'-"7kg msan its uncertainty , &.L 1'-&1kg ms

!he uncertainty in position is then

not er onseuence o


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not er onseuence oCeisen0ergs Kncertainty

?rincip%e / uantum partic%e can ne$er 0e in a stateof rest3as this #ou% mean #e kno# 0oth its

position an momentum precise%y

!hus3 the carriage #i%%

0e Iigg%ing aroun the0ottom of the $a%%eyfore$er

C+


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?hotons transfer momentum to the partic%e #hen they scatter.

Magnitue of p is the same 0efore an after the co%%ision. 2hy

?+

p

p

*

y

hp

Bp h =

yp y h

G%IJ%3B%@< M36%@;I3;F -@I36I-.%&

y

Kncertainty in photony-momentum, Kncertainty in particley-momentum

( ) ( )sin B + sin B +yp p p

( )+ sin B +yp p p =

e Brog%ie re%ation gi$es

Sma%% ang%e approimation

an so

=rom 0efore hence

C+


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2e #i%% sho# forma%%y *section D

B +

B +

B +

*

y

+

* py p

+ p

h

h

h

2e cannot ha$e simu%taneous kno#%egeof PconIugate $aria0%es such as position an momenta.

G%IJ%3B%@< M36%@;I3;F -@I36I-.%&

y* p :ote3 ho#e$er3

/r0itary precision is possi0%e in princip%e for

position in one irection an momentum in another

etc

e sen erg s ncerta nty


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e sen erg s ncerta nty?rincip%e in$o%$ing energy an

time

!he more accurate%y #e kno# the energy of a 0oy3the %ess accurate%y #e kno# ho# %ong it possessethat energy

!he energy can 0e kno#n #ith perfect precision *E, '3on%y if the measurement is mae o$er an infinite perio oftime *t, U

C+


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!here is a%so an energy-time uncertainty re%ation

!ransitions 0et#een energy %e$e%s of atoms are not perfect%ysharp in freuency.

B +, t

h

n , &

n , "

n , 1

,+, h=

,+

6essons from


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Summary> 6essons fromCeisen0erg

!he iea of a perfect%y preicta0%euni$erse cannot 0e true

!here is no such thing as an iea%3o0Iecti$e o0ser$er

4O:46KS


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4O:46KS


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Jelocity

(ow fast is the wavetravelingK

Jelocity is a referencedistance

divided 2y a reference time.

;he phase velocit" is the wavelength period> v O

JincefO 1 >

In ter#s of k %O + andthe ang$lar fre$enc" O + this is>

v O f

v O %

%he 3roup


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pJelocity

;his is the velocit" at which the overall shape of the waves a#plit$des or the

wave Penvelope propagates& (Osignal velocity)

Gere phase velocit" O gro$p velocit" (the #edi$# is non-dispersive)

ispersion0 phase=group velocity depends on


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fre9uency

Black dot #oves at phase velocit"& @ed dot #oves at gro$p velocit"&

;his is noral dispersion(refractive inde= decreases with increasing )


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'aves> -hase and gro$p velocities of a wave packet


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;he speed at which a given phase propagates does not coincide with the

speed of the envelope&

3ote that the phase velocit" is

greater than the gro$p velocit"&

'aves> -hase and gro$p velocities of a wave packet


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;he gro$p velocit"is the velocit" with which the envelope of thewave

packet propagates thro$gh space&

;hephase velocit"is the velocit" at which the phase of an" one

fre$enc" co#ponent of the wave will propagate& Fo$ co$ld pick one

partic$lar phase of the wave (for e=a#ple the crest) and it wo$ld appearto travel at the phase velocit"&

Q$estion> Is the -wave speed a phase or a gro$p velocit"!


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E&pectation values

**P**i

ii )(=

=== d*t**d***P**+

)()(

;h$s if we know (* t) (a sol$tion of ;5J%) then knowledgeof Rd*allows the averageposition to be calc$lated>

In the li#it that * then the s$##ation beco#es>

== d*t**d**P**++++ )()(Ji#ilarl"

;he average is also know as the e*pectation valueand are ver"

i#portant in $ant$# #echanics as the" provide $s with theaverage val$es of ph"sical properties beca$se in #an" cases

precise val$es cannot even in principle be deter#ined S see later&

li i


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Formalisation

== 1)()(+d*t*d**P

;otal probabilit" of finding a particle an"where #$st be 1>

;his re$ire#ent is known as theNoralisation condition& (;his

condition arises beca$se the J% is linear in and therefore if isa sol$tion of ;5J% then so is cwhere cis a constant&)Gence if original $nnor#alised wavef$nction is (*t) then thenor#alisation integral is>

= d*t*N++ )(

nd the (rescaled) nor#alised wavef$nction norO (1N) &

#xample !> 'hat val$e ofNnor#alises the f$nctionN * (* L)of *L!

7 d di i f


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7oundary conditions for

In order for to be a sol$tion of the JchrLdinger e$ation torepresent a ph"sicall" observable s"ste# #$st satisf" certainconstraints>

1& D$st be a singleval$ed f$nction of*and tT

+& D$st be nor#alisableT ;his i#plies that the as*T

,&(*) #$st be a contin$o$s f$nction of*T/& ;heslopeof #$st be contin$o$s specificall" d(*)d*

#$st be contin$o$s (e=cept at points where potential is

infinite)&(*)

*

(*)

*

(*)

*

(*)

*

/ave unction


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RUU + =

%he 9uantity with which Luantum Mechanics is concernedis the wave function of a 2ody.

VWV" is proportiona% to the pro0a0i%ity of fining a partic%e at aparticu%ar point at a particu%ar time.


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*chrodinger0s time independent waveequation


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equation

*A +sin=

:ne dimensional wave e9uation for the waves associatedwith a moving particle is

)rom CiD

*A

*

+sin

/+

+

+

+

=

is the #a$e amp%itue for a gi$en . where

6 is the ma&imum amplitude.

H is the #a$e%ength

CiD

+

+

+

+ /=

*

CiiD

h=


88/112

vo=

+

++

+

1

h

vo= +

+

+

1

+

h

v oo

=

++

+1

h

/o

= where A is the A.E. for the non-relativistic

case

CiiiD

+uppose E is the total energy of the particle

and J is the potential energy of the particle

)(+1

++V,

h

o =

E9uation CiiD now 2ecomes

++


89/112

%his is the time independent Csteady stateD +chrodinger5s

wave e9uation for a particle of mass mo total energy Epotential energy J moving along the &-a&is.

If the particle is moving in -dimensional spacethen

)(+/

+

+

+

+

V,h*

o =

)(+

++

+

=+

V,

*

o

)(+++

+

+

+

+

+

=++

+

V,

+y*

o

+


90/112

)or a free particle J ; " so the +chrodinger e9uationfor a free particle

++

+ =+ ,o

)(+

+

+ =+ V,o

%his is the time independent Csteady stateD +chrodinger5swave e9uation for a particle in -dimensional space.

*chrodinger0s time dependent waveequation


91/112

equation

)( p*,ti

Ae

=

Bave e9uation for a free particle moving in @& direction is

CiiiD

+

+

+

+

p

*=

where E is the total energy an p is the momentum of theparticle i,erentiating CiD twice w.r.t. &

CiD

+

+++

*p

= CiiD

i,erentiating CiD w.r.t. t

i,t

=

t

i,

=

)or non-relativistic case


92/112

Nsing CiiD and CiiiD in CivD

CivD

V*t

i +

=

+

++

+

E ; A.E. @ otentialEnergy

t*Vp,

+

+ +=

V

p, +=

+

+

%his is the time dependent +chrodinger5s wave e9uationfor a particle in one dimension.

Linearity and *uperposition


93/112

++11 aa +=

If O1an "are t#o so%utions of any Schroinger euation of a

system3 then %inear com0ination of 1an " #i%% a%so 0e a so%ution

of the euation..

(ere are constants

62ove e9uation suggests0

+1V aa

is also asolution

CiD %he linear property of +chrodinger e9uation

CiiD 1an " follow the superposition principle

If 1is the pro0a0i%ity ensity corresponing to 1an ?"is the

pro0a0i%ity ensity corresponing to


94/112

+1 +%hen %otal pro2a2ility will 2e

+

+1

+ UUUU +==P

due to superposition

principle

)()( +1R

+1 ++=

))(( +1R+R1 ++=1

R

++

R

1+

R

+1

R

1 +++=

1

R

++

R

1+1 +++= PPP

+1 PPP + ro2a2ility density can5t 2e added linearly

pro0a0i%ity ensity corresponing to "

Expectation values


95/112

d**f

= +UU)(

E&pectation value of any 9uantity which is a function ofP&5 say fC&D is given 2y

for normalized O

%hus e&pectation value for positionP&5 is

>< )(*f

d**

= +UU>< *

E&pectation value is the value of P&5 we would o2tain if wemeasured the positions of a large num2er of particles

descri2ed 2y the same function at some instant Pt5 andthen averaged the results.

L. )ind the e&pectation value of position of a particlehaving wave function O ; a& 2etween & ; " Q ! O ; "


96/112

d**=1

+UU

+olution

>< *

1

/+

/

=

*

a

having wave function O ; a& 2etween & ; " Q ! O ; "elsewhere.

d**a =1

,+

>< */

+a=

1perators


97/112

pi*

=

C6nother way of /nding the e&pectation valueD

)or a free particle

6n operator is a rule 2y means of which from a given

function we can /nd another function.

)( p*,ti

Ae

= %hen

(ere

*ip

=

W

is called the momentum operator

CiD

,

i+imilarly


98/112

,t

=

(ere

ti,

=

W

is called the %otal Energy operator

CiiD

E9uation CiD and CiiD are general results and theirvalidity is the same as that of the +chrodinger e9uation.

If a particle is not free then

W

W+W


99/112

0*it

i +

= +

+

1

%his is the time dependent +chrodinger e9uation

WWW

&& 0,/, +=W+W

+0

p,

o

+=

00= W

0*t

i += +++

+

0

*t

i +

=

+

++

+

If :perator is (amiltonian


100/112

%hen time dependent +chrodinger e9uation can 2ewritten as

0

*

1 +

=

+

++W

+

,1 =W

%his is time dependent +chrodinger e9uation in(amiltonian form.

Eigen values and Eigen unction


101/112

+chrodinger e9uation can 2e solved for some speci/cvalues of energy i.e. Energy Luantization.

a=W

+uppose a wave function COD is operated 2y an operator P5

such that the result is the product of a constant say Pa5 andthe wave function itself i.e.

%he energy values for which +chrodinger e9uation can 2esolved are called PEigen values5 and the correspondingwave function are called PEigen function5.

then

O is the eigen function of

ais the eigen value of

W

W

L. +uppose is eigen function of operator*e+=

+

+

d

d


102/112

then /nd the eigen value.

%he eigen value is ?.

+olution.

+d*

+

+W

d*d2=

+

+W

d*

d2

= )( +

+

+*e

d*

d=

*e2 +W

/=

/W

=2

Particle in a 2ox


103/112

*onsider a particle of rest mass mo enclosed in a one-

dimensional 2o& Cin/nite potential wellD.

%hus for a particle inside the 2o& +chrodinger e9uation is

7oundary conditions forotential

JC&

D;

" for " S & S'

Tfor " U & U ' 7oundary conditions for O

V ;

" for & ; "

T" for & ; '

+

++

+

=+

,

*

o

& ; " & ; '

=V =V

particle

=V

=V inside CiD

h= += Ck is the propagation


104/112

E9uation CiD 2ecomes

p=

%=

p%=

,o+=

+

+ +

,% o=

Ck is the propagationconstantD

CiiD

++

+

=+

%*

CiiiD

3eneral solution of e9uation CiiiD is

%*.%*A* cossin)( += CivD

7oundary condition says O ; " when & ; "


105/112

E9uation CivD reduces to

CvD

&cos&sin)( %.%A +=

1& .+= = .

%*A* sin)( =

7oundary condition says O ; " when & ; '

L%AL &sin)( =

L%A &sin =A &sin = L%

nL% sin&sin =

n%L=


106/112

ut this in E9uation CvD

CviD

L

n%

=

L

*nA*

sin)( =

Bhen n W " i.e. n ; ! > 4. this gives O ; "everywhere. ut value of k from CviD in

CiiD

+

+ +

,% o=

+

++

,

L

n o=

%,

++

++hn


107/112

Bhere n ; ! > 4.

E9uation CviiD concludes

o,

+=

+8 Lo=

Cvi

iD

!. Energy of the particle inside the 2o& can5t 2e e9ualto zero.

%he minimum energy of the particle is o2tained forn ; !

+

+

18 L

h,

o

= 34ero PointEnergy5

If momentum i.e.1, p * 7ut since the particle is con/ned in the 2o& of

dimension '.

L* = #a=

%hus zero value of zero point energy violates the(eisen2erg5s uncertainty principle and hence zero


108/112

(eisen2erg s uncertainty principle and hence zerovalue is not accepta2le.

>. 6ll the energy values are not possi2le for a particlein

potential well.

Energy is 6uantied . En are the eigen values and Pn5 is the 9uantum

num2er. ?. Energy levels CEnD are not e9ually spaced.

n ;!

n ;

n ;>

,,

1,

+,

L

*nA*n

sin)( =


109/112

Nsing Formalization

condition

L

1sin

++ =

d*

L

*nA

L

1U)(U + =

d**n

1+

+ =

L

AL

A +

=

%he normalized eigen function of the particle are

L

n*

L*n

sin

+)( =

ro2a2ility density /gure suggest that0


110/112

!. %here are some positions CnodesD in the 2o& that willnever 2e occupied 2y the particle.

>. )or di,erent energy levels the points of ma&imumpro2a2ility are found at di,erent positions in the 2o&.

O!>is ma&imum at '=> Cmiddle of the 2o&D

V"V"is Yero 6".

Particle in a (hree 7imensional 2ox


111/112

Eigen

function +y*

=

L

+n

L

yn

L

*nAAA +

y*+y*

sinsinsin=

L

+n

L

yn

L

*n

L

+y* sinsinsin+

,

=

+

++++

8)(L

hnnn, +y* ++=

+y* ,,,, ++= Eigen energy

References0 ti < ff 2 +: Ph ,l i ,ff @ t i d 1 +, : d Phili L d i h @ t i d 1 +, :


112/112

$stin www&ee$als#cs$ared&a$ckland&ac&n4sitese#c+tlpeeovervi

ew&cf#

%instein lbert& (1*:)& En a Ge$ristic Xiewpoint 6oncerning the

-rod$ction and ;ransfor#ation of .ight&Annalen der Physi% Xol 17

1,+&

%lert h"perte=tbook&co#ph"sics#odernphotoelectric

Ga#akawa Foshihiro& (+/)& ;hin9il# Jolar 6ells> 3e=t generation

photovoltaics and its application& 3ew Fork> Jpringer&

.enardic 5enis&A 'al% 3hrough 3ie& @etrieved 111+:&

http>www&pvreso$rces&co#enhistor"&php

M&J& 5E% -hotovoltaics -rogra#& (+:)&Photovoltaics 3ieline&

@etrieved 1+7:& http:$$inventors%a&out%com$li&rary$inventors$&lsolar2%html

n&a& n&d&Philipp Lenard 4 iography.@etrieved 1+,:&

http>nobelpri4e&orgph"sicsla$reates1*:lenardbio&ht#l

n&a& n&d& 3he Photo ,lectric ,ffect& @etrieved 10:&

http>www&lancs&ac&$k$gNackso#+

n&a& n&d& 3he ,lectric 5ield 6n Action& @etrieved 111+:&

http>www&sandia&govpvdocs-X9%ff%lectricY9ield&ht#

n&a& n&d& 3ieline of "olar Cells& @etrieved 1+7:&

http>www&nation#aster&co#enc"clopedia;i#elineofsolarcel

ls

@obertson % 9& E6onner 2 2&A history of 7uantu Mechanics.

@etrieved 1+::&

http>wwwgro$ps&dcs&st

and&ac&$kChistor"Gist;opics;heYQ$ant$#YageYbegins&ht#l

J#ith 'illo$ghb"& (187,)& %ffect of .ight on Jeleni$# d$ring

the passage of an %lectric 6$rrent&Nature Xol ! ,,&

vailable M@.>http:$$histv2%free%fr$selenium$smith%htm
http://www.pvresources.com/en/history.phphttp://www.sandia.gov/pv/docs/PVFEffElectric_Field.htmhttp://www.nationmaster.com/encyclopedia/Timeline-of-solar-cellshttp://www.nationmaster.com/encyclopedia/Timeline-of-solar-cellshttp://www.nationmaster.com/encyclopedia/Timeline-of-solar-cellshttp://www.nationmaster.com/encyclopedia/Timeline-of-solar-cellshttp://www.sandia.gov/pv/docs/PVFEffElectric_Field.htmhttp://www.pvresources.com/en/history.php

Quantum Mechanics-Dr.gagan Anand

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