5 Wave Properties of Particles

8/3/2019 5 Wave Properties of Particles

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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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9/49

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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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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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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For X-ray Diffraction on Nickel

5065

A65.1;A91.0

sin2

o

ray-X

o

111

d

d


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(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.html


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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?


36/49


37/49

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

(

2

(sin[])2/()2/cos[(2

]2

(

2

(sin[]

2

(

2

(cos[2

)sin()sin(),(

)21)21

)21)21)21)21

2211

txkk

txkA

txkk

txkk

A

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

(

2

(

sin[])2/()2/cos[(2),(

)21)21

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?


40/49

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

phasebeat

beat

beatbeat

beat

phasephase

phasephase

phase

phase

vvsothofwavelengtindependenissoundofspeedceresultExpected

kvb

mknotesbeatbetweenceDis

mkkk

mvfvk

mvfvk

mphsmsoundofspeedtheisv

ffkv

kkk

sin

344)35.903.14/()512768(2)2//()2/()(

70.233.2/2/2:tan

33.22/)35.90.14(2/)(

35.9344/5122/2/

03.14344/7682/2/

)770(/344,

,)/2/(2/

2/)(2/

1

12

1

111

1

222

12


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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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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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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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5 Wave Properties of Particles

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