Post on 01-Jun-2018
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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$e&frinde=&ht#l
3o of electrons 1 1
3o of electrons 1 1
5EMB.%J.I; %A-%@ID%3;
http://www.quantum-physics.polytechnique.fr/index.htmlhttp://www.hqrd.hitachi.co.jp/em/doubleslit-f2.cfmhttp://www.hqrd.hitachi.co.jp/em/doubleslit-f2.cfmhttp://www.quantum-physics.polytechnique.fr/index.html8/9/2019 Quantum Mechanics-Dr.gagan Anand
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Jo#e ke" papers in the develop#ent of the do$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=
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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
++
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%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
+
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)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
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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
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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
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++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
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+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
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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 ; "
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d**=1
+UU
+olution
>< *
1
/+
/
=
*
a
having wave function O ; a& 2etween & ; " Q ! O ; "elsewhere.
d**a =1
,+
>< */
+a=
1perators
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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
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,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
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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
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%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
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+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
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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
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*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
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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 & ; "
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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=
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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
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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
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(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)( =
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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
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!. %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
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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 +, :
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$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