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Phy107 Fall 2006 1
From Last Time…
• Light waves are particles
and matter particles are waves!
• Electromagnetic radiation (e.g. light)
made up of photon particles
• Matter particles show wavelike properties like
interference
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Phy107 Fall 2006 2
• Light: Is quantized. Has energy and momentum:
• Electromagnetic radiation(light) has a dual nature.
It exhibits both wave and article characteristics
• The photoelectric effectshows the particle characteristics of light
• Interference and diffraction
Photon: particle and wave
E = hf = hc
λ =1240 eV − nm
λ
p = E
c= hf
c= h
λ
λ = h
p
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Phy107 Fall 2006 3
Wavelengths of massive objects
• de"roglie wavelength #
λ = h p
• #mv for a nonrelativistic
(v $$c) particle with mass.
λ = h
mv
λ = h p= hc
2 mc2 E kinetic
!ame
constant as
be"ore
#inetic energy
rest energy
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Phy107 Fall 2006 4
Wavelength of eV electrons
• %or an electron&
λ =1240 eV − nm
2 × 0.511 MeV
1
E kinetic
=1.23 eV
1/ 2 − nm
E kinetic
• ' e electron& λ#'.* nm• '+ e electron λ#+.*, nm• '++ e electron λ#+.' nm
constant
#inetic energy rest energy
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Phy107 Fall 2006 5
Wavelength of 100 eV objects
• %or an electron&
λ =1240 eV − nm
2 ×m0 MeV
1
100eV
=88 eV
1/ 2 − nm
m0 MeV
• '++ e electron& λ#+.' nm• '++ e proton λ#+.++, nm # ., pm• Electron .-'' Me& roton ,/+ Me
constant
#inetic energy rest energy
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Phy107 Fall 2006 6
Wave reflection from crstal
• If electron are waves the0 can interfere• Interference of waves reflecting from different
atomic la0ers in the cr0stal.
• 1ifference in path length 2 spacing etween atoms
side view
$e"lection "rom
to lane$e"lection
"rom next
lane
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Phy107 Fall 2006 8
Particle interference• 7sed this interference idea to to learn aout the structure of matter
• '++ e electrons4 λ # +.'nm8 9r0stals also the atom
• '+ :e electrons4
8 Inside the nucleus& *. fermi& '+;
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Phy107 Fall 2006 9
Let%s st&d electron waves
• =ere is a wave4
>where is the electron?8 @ave eAtends infinitel0 far in B x and %x direction
x
λ = h
p
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Phy107 Fall 2006 10
'nalog with so&nd
• Cound wave also has the same characteristics
• "ut we can often locate sound waves
8 E.g. echoes ounce from walls. 9an make a sound pulse
• EAample48 =and clap4 duration 2 +.+' seconds
8 Cpeed of sound # */+ mDs
8 Cpatial eAtent of sound pulse # *./ meters.
8 *./ meter long hand clap travels past 0ou at */+ mDs
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Phy107 Fall 2006 11
(eat fre)&enc: spatial locali*ation
• @hat does a sound particleF look like?
8 Gne eAample is a eat freuenc0F etween two notes
8 Two sound waves of almost same wavelength added.
QuickTime™ and a TIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and a TIFF (LZW) decompressor
are needed to see this picture.
&onstructiveinter"erence
Large
amplitude
&onstructiveinter"erence
Large
amplitude
'estructiveinter"erence
Cmall
amplitude
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Phy107 Fall 2006 12
+a,ing a particle o&t of waves
//+ =6 B
/*, =6
//+ =6 B
/*, =6 B
/*H =6
//+ =6 B
/*, =6 B
/*H =6 B
/*3 =6 B
/*< =6
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Phy107 Fall 2006 13
-patial e$tent
of locali*ed so&nd wave
∀ ∆ x spatial spread of wave packetF• Cpatial eAtent decreases as the spread in
included wavelengths increases.
-8
-4
0
4
8
-15 -10 -5 0 5 10 15J
∆ x
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Phy107 Fall 2006 14
-ame occ&rs for a matter wave
• 9onstruct a locali6ed particle 0 adding together
waves with slightl0 different wavelengths.
• Cince de "roglie sa0s λ # h D & each of these
components has slightl0 different momentum.8 @e sa0 that there is some uncertaint0F in the momentum
or the energ0
• nd still donFt know eAact location of the particle!
8 @ave still is spread over ∆ x (uncertaint0F in position)8 9an reduce ∆ x ut at the cost of increasing the spread in
wavelength (giving a spread in momentum).
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Phy107 Fall 2006 15
.nterpreting
• %or sound& we would Just sa0 that the sound pulse is
centered at some position& ut has a spread.
• 9anFt do that for a uantum;mechanical particle.
• Man0 measurements indicate that the electron is
indeed a point particle.• Interpretation is that the magnitude of electron wave;
pulseF at some point in space determines the
probabilit of finding the electron at that point.
-8
-4
0
4
8
-15 -10 -5 0 5 10 15J
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Phy107 Fall 2006 16
/eisenberg ncertaint Principle
• 7sing∆ x # position uncertaint0∆ momentum uncertaint0
• =eisenerg showed that the product
( ∆ x ) • ( ∆ ) is alwa0s greater than ( h D /π )
Gften write this as
where is pronounced h;arF
lanckFs
constant
∆ x( ) ∆ p( )~ h /2
h=
h
2π
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Phy107 Fall 2006 17
Thin,ing abo&t &ncertaint
%or a classical particle& #mv & so an
uncertaint0 in momentum corresponds to an
uncertaint0 in velocit0.
∆ x( ) ∆ p( )~ h /2
∆ x( ) ∆v( )~ h /2m
This sa0s that the uncertaint0 is small for massive oJects&
ut ecomes important for ver0 light oJects& such as
electrons.
Large& massive oJects donFt show effects of uantum
mechanics.
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Phy107 Fall 2006 18
ncertaint principle )&estion
Cuppose an electron is inside a oA ' nm in width.There is some uncertaint0 in the momentum of
the electron. @e then suee6e the oA to make
it +.- nm. @hat happens to the momentum?
. Momentum ecomes more uncertain
". Momentum ecomes less uncertain
9. Momentum uncertaint0 unchanged
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Phy107 Fall 2006 19
sing )&ant&m mechanics
• Kuantum mechanics makes astonishingl0
accurate predictions of the ph0sical world
• 9an appl0 to atoms& molecules& solids.
• n earl0 success was in understanding
8 Ctructure of atoms
8 Interaction of electromagnetic radiation with atoms
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Phy107 Fall 2006 20
Planetar model of atom
• ositive charge is concentrated inthe center of the atom ( nucleus )
• tom has 6ero net charge4
8 ositive charge in nucleus cancelsnegative electron charges.
• Electrons orit the nucleus like
planets orit the sun
• (ttractive) 9oulom force pla0srole of gravit0
nucleus
electrons
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Phy107 Fall 2006 21
!ifference between atoms
• 5o net charge to atom8 numer of oriting negative electrons same as
numer of positive protons in nucleus
8 1ifferent elements have different numer of
oriting electrons
• =0drogen4 ' electron
• =elium4 electrons
• 9opper4 , electrons• 7ranium4 , electrons!
• Grgani6ed into periodic tale of elements
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Phy107 Fall 2006 22
Elements in same
column have similar
chemical properties
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Phy107 Fall 2006 23
• 9ircular motion of oriting electronscauses them to emit electromagnetic radiationwith freuenc0 eual to orital freuenc0.
• Came mechanism 0 which radio waves are emitted
0 electrons in a radio transmitting antenna.• In an atom& the emitted electromagnetic wave
carries awa0 energ0 from the electron.
8 Electron predicted to continuall0 lose energ0.8 The electron would eventuall0 spiral into the nucleus
8 However most atoms are stable!
Planetar model and radiation
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Phy107 Fall 2006 24
'toms and photons
• EAperimentall0& atoms do emit electromagneticradiation& ut not Just an0 radiation!
• In fact& each atom has its own fingerprintF of
different light freuencies that it emits.
=0drogen
Mercur0
@avelength (nm)
/++ nm -++ nm
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Phy107 Fall 2006 25
/drogen emission spectr&m
• =0drogen is simplest atom
8 Gne electron oriting aroundone proton.
• The (almer -eries ofemission lines empiricall0
given 0
n # *& λ #
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Phy107 Fall 2006 26
/drogen emission
• This sa0s h0drogen emits onl0photons of a particular wavelength& freuenc0
• hoton energ0 # h" &
so this means a particular energ0.
• 9onservation of energ04
8 Energ0 carried awa0 0 photon is lost 0 theoriting electron.
h h h d
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Phy107 Fall 2006 27
The (ohr hdrogen atom
• etained planetar0F picture4 oneelectron orits around one proton
• Gnl0 certain orits are stale
• adiation emitted onl0 whenelectron Jumps from one staleorit to another.
• =ere& the emitted photon has anenerg0 of E initial%E final
Ctale orit
Ctale orit '
E initial
E "inal
hoton
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Phy107 Fall 2006 28
nerg levels
• Instead of drawing orits& we can Just indicate the energ0
an electron would have if it were in that orit.Nero energ0
n#'
n#
n#*n#/
E 1= −
13.6
12
eV
E 2= −
13.6
22
eV
E 3= −
13.6
32
eV
E n e r g 0
a A i s
Energ0 uanti6ed!
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Phy107 Fall 2006 29
mitting and absorbing light
hoton is emitted when electrondrops from one uantumstate to another
Nero energ0
n#'
n#
n#*n#/
E 1 = −13.6
12 eV
E 2= −
13.6
22
eV
E 3 = −
13.6
32
eV
n#'
n#
n#*n#/
E 1 = −13.612 eV
E 2= −
13.6
22
eV
E 3= −
13.6
32
eV
soring a photon of correct
energ0 makes electron Jump to
higher uantum state.
hoton
asored
h" #E *%E +
hoton
emitted
h" #E *%E +
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Phy107 Fall 2006 30
Photon emission )&estion
n electron can Jump etween the allowed uantum states
(energ0 levels) in a h0drogen atom. The lowest threeenerg0 levels of an electron in a h0drogen atom are;'*.< e& ;*./ e& ;'.- e.
These are part of the seuence E n %'*.
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Phy107 Fall 2006 31
• Each orit has a specific energ0
E n=-13.6/n2
• hoton emitted when electron
Jumps from high energ0 to low
energ0 orit.E i – E f = h f
• hoton asorption induces
electron Jump fromlow to high energ0 orit.
E f – E i = h f
• grees with eAperiment!
nerg conservation for (ohr atom
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Phy107 Fall 2006 32
$ample: the (almer series
• ll transitions terminate atthe n# level
• Each energ0 level has
energ0 E n#;'*.< D n) e
• E.g. n#* to n# transition
8 Emitted photon has energ0
8 Emitted wavelength
E photon = −13.6
32
− −
13.6
22
=1.89 eV
E photon = hf = hc
λ , λ =
hc
E photon
=1240 eV − nm
1.89 eV = 656 nm
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Phy107 Fall 2006 33
9ompare the wavelength of a photon produced from
a transition from n#* to n#' with that of a photon
produced from a transition n# to n#'.
-pectral 2&estion
n=2
n=3
n=1
. λ31 < λ21
". λ31 = λ21
9. λ31 > λ21
E31 > E21 so λ31 < λ21
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Ph 107 Fall 2006
(&t wh3
• @h0 should onl0 certain orits e stale?• "ohr had a complicated argument ased on
Ocorrespondence principleP
8 That uantum mechanics must agree with classicalresults when appropriate (high energies& large si6es)
• "ut incorporating wave nature of electron gives
a natural understanding of these
uanti6ed oritsF
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