Post on 07-Apr-2018
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Wave Properties of Particles
Serway/Jewett chapters 38.5; 40.4 40.7
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Photons and Waves Revisited
Some experiments are best explained bythe photon model
Some are best explained by the wavemodel
We must accept both models and admit thatthe true nature of light is not describable interms of any single classical model
The particle and wave models complementone another
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Dual Nature of EM radiation
To explain all experiments with EM radiation(light), one mustassume that light can bedescribed both as wave (Interference, Diffraction)andparticles (Photoelectric Effect, Frank-Hertz
Experiment, x-ray production, x-ray scatteringfrom electron) To observe wave properties must make
observations using devices with dimensions
comparable to the wavelength. For instance, wave properties of X-rays were observed
in diffraction from arrays of atoms in solids spaced by afew Angstroms
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Louis de Broglie
1892 1987
French physicist
Originally studied
history
Was awarded theNobel Prize in 1929
for his prediction ofthe wave nature ofelectrons
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De Broglies Hypothesis
Louis de Broglie postulated that the dualnature of the light must be expanded toALL matter
In other words, all material particles possess
wave-like properties, characterized by thewavelength,B, related to the momentump ofthe particle in the same way as for light
p
hB de Broglie
wavelength of the
particle
Plancks Constant
Momentum of the
particle
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Wave Properties of Particles
Louis de Broglie postulated that because photons haveboth wave and particle characteristics, so too all forms ofmatter have both properties
For photons:
De Broglie hypothesized that particles of well definedmomentum also have a wavelength, as given above, thede Broglie wavelength
p
hOr
hchcEp
hE
,
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Frequency of a Particle
In an analogy with photons, de Brogliepostulated that a particle would also have afrequency associated with it
These equations present the dual nature of
matter Particle nature, pand E
Wave nature, and (and k)
fhhfEhEf 2
2
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Photons and Waves Revisited
Some experiments are best explained bythe photon model
Some are best explained by the wave
model
We must accept both models and admitthat the true nature of light is not
describable in terms of any single classicalmodel
Also, the particle model and the wave
model of light complement each other
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Complementarity
The principle of complementarity statesthat the wave and particle models of eithermatter or radiation complement each other
Neither model can be used exclusively todescribe matter or radiation adequately
No measurements can simultaneouslyreveal the particle and the wave propertiesof matter
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The Principle of Complementarity
and the Bohr Atom
How can we understand electron orbits inhydrogen atom from wave nature of the electron?
Remember: An electron can take only certainorbits: those for which the angular momentum, L,takes on discrete values
How does this relate to the electrons de
Broglies wavelength?
nmvrL
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The Principle of Complementarity
Only those orbits are allowed,which can fit an integer(discrete) number of the electrons
de Broglies wavelength Thus, one can replace 3rdBohrs
postulate with the postulatedemanding that the allowed orbitsfit an integer number of theelectrons de Broglies wavelength
This is analogous to the standingwave condition for modes inmusical instruments
B
BB
B
B
nr
nh
nr
hmv
mv
h
p
h
nmvrL
2
2
Bnr 2
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De Broglies Hypothesis predictsthat one should see diffractionand interference of matter waves
For example we should observeElectron diffractionAtom or molecule diffraction
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Estimates for De Broglie wavelength
Bullet:
m = 0.1 kg; v = 1000 m/s B ~ 6.6310-36 m
Electron at 4.9 V potential: m = 9.1110-31 kg; E~4.9 eV B ~ 5.510-10 m = 5.5
Nitrogen Molecule at Room Temperature: m ~ 4.210-26 kg; E = (3/2)kBT0.0375 eV B~2.810-11 m = 0.28
Rubidium (87) atom at 50 nK:
B ~ 1.210-6
m = 1.2 mm = 1200
mE
h
mv
h
p
hB
2
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Diffraction of X-Rays by Crystals
X-rays are electromagnetic waves of relatively shortwavelength ( = 10-8 to 10-12 m = 100 0.01 )
Max von Laue suggested that the regular array ofatoms in a crystal (spacing in order of several
Angstroms) could act as a three-dimensionaldiffraction grating for x-rays
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X-ray Diffraction Pattern
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X-Ray Diffraction
This is a two-dimensionaldescription of thereflection (diffraction) ofthe x-ray beams
The condition forconstructive interference is
where n = 1, 2, 3
nd sin2 This condition is
known as Braggs law
This can also be usedto calculate the spacing
between atomic planes
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Davisson-Germer Experiment
If particles have a wave nature, then underappropriate conditions, they should exhibitdiffraction
Davisson and Germer measured thewavelength of electrons
This provided experimental confirmation ofthe matter waves proposed by de Broglie
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Davisson and Germer Experiment Electrons were directedonto nickel crystals Accelerating voltage is
used to control electronenergy: E = |e|V The scattering angle
and intensity (electron
current) are detected is the scattering angle
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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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Davisson and Germer Experiment
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Davisson and Germer Experiment
Observations:
Intensity was strongerfor certain angles forspecific accelerating
voltages (i.e. for specificelectron energies) Electrons were reflected
in almost the same way
that X-rays ofcomparable wavelength
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Davisson and Germer Experiment
Observations: Current vs accelerating
voltage has a maximum,i.e. the highest numberof electrons is scatteredin a specific direction
This cant be explainedby particle-like nature of
electrons
electronsscattered on crystalsbehave as waves
For ~ 50
the maximum is at ~54V
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Davisson and Germer Experiment
For X-ray Diffraction on Nickel
5065
A65.1;A91.0
sin2
o
ray-X
o
111
d
d
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Davisson and Germer Experiment
(Problem 40.38) Assuming the wave nature
of electrons we can use de Brogliesapproach to calculate wavelengths of amatter wave corresponding to electrons inthis experiment
V= 54 V E = 54 eV = 8.6410-18J
A67.1J106.8kg101.92
sec-J1063.6
2,2,
2
1831
34
2
B
BmE
hmEp
m
pE
This is in excellent agreement with wavelengths of
X-rays diffracted from Nickel!
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In previous experiments many electrons
were diffracted Will one get the same result for a single
electron?
Such experiment was performed in 1949 Intensity of the electron beam was so low that
only one electron at a time collided withmetal
Still diffraction pattern, and not diffusescattering, was observed, confirming that
Thus individual electrons behave as waves
Single Electron Diffraction
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Two-slit Interference
Thomas YoungThe intensity is obtained bysquaring the wave,I1 ~ , I2 ~ ,I12 = = ,where < > is average over time of
the oscillating wave.h1h2 ~ cos(2p/) and reflects theinterference between wavesreaching the point from the twoslits.When the waves arriving fromslits 1 and 2 are in phase, p = n,and cos(2p/) = 1.For = , I12 = 4I1.When the waves from slits 1 and 2are out of phase, = n + /2, and
cos(2/) = -1 and I12 = 0.
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Electron Diffraction, Set-Up
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Electron Diffraction, Experiment
Parallel beams of mono-energeticelectrons that are incident on a double slit
The slit widths are small compared to the
electron wavelength An electron detector is positioned far from
the slits at a distance much greater than
the slit separation
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Electron Diffraction, cont.
If the detector collectselectrons for a longenough time, a typicalwave interference
pattern is produced This is distinct evidence
that electrons areinterfering, a wave-likebehavior
The interference patternbecomes clearer as thenumber of electronsreaching the screen
increases
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Active Figure 40.22
Use the active figureto observe thedevelopment of the
interference pattern Observe the
destruction of thepattern when you
keep track of whichslit an electron goesthrough PLAY
ACTIVE FIGURE
http://../Active_Figures/active_figures/AF_4022.htmlhttp://../Active_Figures/active_figures/AF_4022.htmlhttp://../Active_Figures/active_figures/AF_4022.html8/3/2019 5 Wave Properties of Particles
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Electron Diffraction, Equations
A maximum occurs when This is the same equation that was used for
light
This shows the dual nature of the electron The electrons are detected as particles at a
localized spot at some instant of time
The probability of arrival at that spot isdetermined by calculating the amplitudesquared of the sum of all waves arriving at apoint
sind m
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Electron Diffraction Explained
An electron interacts with both slitssimultaneously
If an attempt is made to determineexperimentally through which slit the electrongoes, the act of measuring destroys theinterference pattern It is impossible to determine which slit the
electron goes through
In effect, the electron goes through both slits The wave components of the electron are
present at both slits at the same time
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Neutrons He atoms
C60
molecules
Other experiments showed wave nature forneutrons, and even big molecules, which
are much heavier than electrons!!
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Example of Electron Diffraction Electrons from a hot filament are incident upon a crystal at an angle = 30
from the normal (the line drawn perpendicular) to the crystal surface. Anelectron detector is place at an angle = 30 from the normal. Atomiclayers parallel to the sample surface are spaced by d = 1.3 A.
Through what voltage V must the electron be accelerated for a maximum inthe electron signal on the detector?
Will the electrons scatter at other angle?
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Addition of Two Waves
)2
sin()2
cos(2)sin()sin(),(
tkxAtkxAtkxAtxy
tkxAtkxAtkxAtxy cossin2)sin()sin(),(
Two sine waves traveling in the same direction:
Constructive and Destructive Interference
Two sine waves traveling in opposite directions create a standing wave
Two sine waves with different frequencies: Beats
]2
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txkk
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txkAtxkAtxy
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Beat Notes and Group Velocity, vg
This represents a beat note with the amplitude of the beat moving at speed
dk
d
v
vkv
g
g
:wavesofondistributicontinuousofionsuperpositFor
/)2//()2/(
]2
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sin[])2/()2/cos[(2),(
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tx
kk
txkAtxy
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Beats and Pulses
Two tuning forks are struck simultaneously. The vibrate at 512 and 768 Hz.(a) What is the frequency of the separation between peaks in the beat envelope?(b) What is the velocity of the beat envelope?
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Beats and Pulses
Two tuning forks are struck simultaneously. The vibrate at 512 and 768 Hz.(a) What is the separation between peaks in the beat envelope?(b) What is the velocity of the beat envelope?
(a)
The rapidly oscillating wave is multiplied by a more slowly varying envelopewith wave vector
]2
(
2
(sin[])2/()2/cos[(2),(
)21)21tx
kktxkAtxy
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Construction Particles From Waves Particles are localized in space
Waves are extended in space. It is possible to build localized entities from a
superposition of number of waves with differentvalues of k-vector. For a continuum of waves, thesuperposition is an integral over a continuum ofwaves with different k-vectors. The wave then has a non-zero amplitude only within a
limited region of space
Such wave is called wave packet
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Wave Picture of Particle
Consider a wave packet made up of waves with adistribution of wave vectors k,A(k), at time t. Asnapshot, of the wave in space along the x-direction is obtained by summing over waves
with the full distribution of k-vectors. For acontinuum this is an integral. The spatial distribution at a time t given by:
0
)cos()(),( dktkxkAtx
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Wave Picture of Particle
1. A(k)is spiked at a given k0, and
zero elsewhere only one wave with k = k0 ( = 0)
contributes; thus one knowsmomentum exactly, and thewavefunction is a traveling wave
particle is delocalized2. A(k)is the same for all k No distinctions for momentums, so
particles position is well defined -the wavefunction is a spike,representing a very localized
particle3. A(k)is shaped as a bell-curve
Gives a wave packet partiallylocalized particle
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Wave Picture of Particle
The greater the range of wave numbers (andtherefore s) in the mix, the narrower thewidth of the wave packet and the morelocalized the particle
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Group Velocity for Particles and Waves
The group velocity in term of particle parameters is
Consider a free non-relativistic particle. The total,energy for this particle is, E= Ek = p2/2m
dpdE
pd
Ed
dk
dvg
particle
particle
g
k
g
um
mu
m
pv
m
p
m
p
dp
d
dp
dE
dp
dE
v
2
2
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Group Velocity
The group speed of wave packet is identical to the
speed of the corresponding particle,
Is this true for photon, for which u = c? For photon total energy E =pc
cpcdpd
dpdEvg
dp
dEvu gparticle
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Group Velocity in Optical Fiber
A pulse of light is launched in an optical fiber. The amplitude A(k) of the pulsesis peaked in the telecommunications band at the wavelength in air, = 1,500nm.The optical fiber is dispersive, with n = 1.50 + 102/, near = 1,500nm, where isexpressed in nm. What is the group velocity?
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Group Velocity in Optical Fiber
A pulse of light is launched in an optical fiber. The amplitude A(k) of the pulsesis peaked in the telecommunications band at the wavelength in air, = 1,500nm.The optical fiber is dispersive, with n = 1.50 + 102/, near = 1,500nm, where isexpressed in nm. What is the group velocity?
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