physics 222

Physics 222Physics 222

David D. AllredDavid D. Allred

Adapted fromAdapted from

Wavelike properties of Wavelike properties of matter matter

Class 8-1-4: (ThT Q) Class 8-1-4: (ThT Q)


Did you complete at Did you complete at least 70% of Chapter 3: least 70% of Chapter 3:

1-3?1-3?A.A. Yes B. No Yes B. No ReviewReview


In our reference frame the In our reference frame the beam droopsbeam droops

It happens near stars, tooIt happens near stars, too http://www.theory.caltech.edu/peoplhttp://www.theory.caltech.edu/peopl

e/patricia/lclens.htmle/patricia/lclens.html

And helps show dark matter. And helps show dark matter. http://apod.nasa.gov/apod/ap080917http://apod.nasa.gov/apod/ap080917.html.html

http://www.theory.caltech.edu/people/patricia/lclens.html

http://www.theory.caltech.edu/people/patricia/lclens.html

http://apod.nasa.gov/apod/ap080917.html

http://apod.nasa.gov/apod/ap080917.html


Gravitational lensingGravitational lensing

http://www.nature.com/nature/journal/v417/n6892/fig_tab/417905a_F1.html


http://www.utahskies.org/HST/Archives/misc.htmlhttp://www.utahskies.org/HST/Archives/misc.html The Gravitational Lens G2237 + 0305The European Space Agency's Faint Object Camera on board NASA's Hubble Space Telescope has provided astronomers with the most detailed image ever taken of the gravitational lens G2237 + 0305—sometimes referred to as the "Einstein Cross". The photograph shows four images of a very distant quasar which has been multiple-imaged by a relatively nearby galaxy acting as a gravitational lens. The angular separation between the upper and lower images is 1.6 arc seconds.The quasar seen here is at a distance of approximately 8 billion light years, whereas the galaxy at a distance of 400 million light years is 20 times closer. The light from the quasar is bent in its path by the gravitational field of the galaxy. This bending has produced the four bright outer images seen in the photograph. The bright central region of the galaxy is seen as the diffuse central object.

http://www.utahskies.org/HST/Archives/misc.html


Gravitational lensingGravitational lensinghttp://www.astronomy.org.nz/aas/MonthlyMeetings/MeetingOct2002.asphttp://www.astronomy.org.nz/aas/MonthlyMeetings/MeetingOct2002.asp

http://www.physics.brown.edu/physics/demopages/Demo/astro/demo/grvlns8.jpg

http://www.astronomy.org.nz/aas/MonthlyMeetings/MeetingOct2002.asp


de Brogliede Broglie

Photons: Photons: p=h/p=h/λλ

Particle: Particle: λλ=h/p=h/p

Hint: If the particle Hint: If the particle is going slow is going slow (K=½p(K=½p22/m)/m)


Quick Writing AssignmentQuick Writing Assignment

In one minute, write a short, clear, In one minute, write a short, clear, and concise paragraph which and concise paragraph which explains why the Compton effect explains why the Compton effect suggests that light is quantized.suggests that light is quantized.


Puzzles at the Beginning of the Puzzles at the Beginning of the Twentieth CenturyTwentieth Century

Null result of the Michelson-Morley Null result of the Michelson-Morley ExperimentExperiment

Ultraviolet CatastropheUltraviolet Catastrophe Photoelectric EffectPhotoelectric Effect Maxwell’s Equations Spell the Demise of Maxwell’s Equations Spell the Demise of

Atoms!Atoms! Discrete atomic emission linesDiscrete atomic emission lines


Radiating AtomsRadiating Atoms


Puzzles at the Beginning of the Puzzles at the Beginning of the Twentieth CenturyTwentieth Century

Null result of the Michelson-Morley Null result of the Michelson-Morley ExperimentExperiment

Ultraviolet CatastropheUltraviolet Catastrophe Photoelectric EffectPhotoelectric Effect Maxwell’s Equations Spell the Demise of Maxwell’s Equations Spell the Demise of

Atoms!Atoms! Discrete atomic emission linesDiscrete atomic emission lines


The Hydrogen SpectrumThe Hydrogen Spectrum

The Balmer SeriesThe Balmer Series

2

1

4

11

nR


Quiz QuestionQuiz Question

Where did de Broglie get his equation for the wavelength Where did de Broglie get his equation for the wavelength of a massive particle?of a massive particle?

A – From special relativityA – From special relativityB – It is the same equation used for light.B – It is the same equation used for light.C – It is the same equation used for sound waves travelingC – It is the same equation used for sound waves traveling through a medium with mass.through a medium with mass.D – From the principle of least actionD – From the principle of least actionE – He found it on the internet.E – He found it on the internet.


Louis de BroglieLouis de Broglie

If light, which we thought of as a wave, behaves as a particle, then maybe things we think of as particles behave as waves…

photo from http://www.aip.org/history/heisenberg/p08.htm


Energy/Frequency and Energy/Frequency and Momentum/Wavelength Relations for a Momentum/Wavelength Relations for a

Photon Photon

hfE ?p

c

Ep

h

Ef

Remember from 220 ...

h

c

hf

h

p

or

p

h

or


Energy/Frequency and Energy/Frequency and Momentum/Wavelength Relations for Momentum/Wavelength Relations for

an Electron/Proton/Apple Pie/Ford an Electron/Proton/Apple Pie/Ford TaurusTaurus

hfE

h

Ef

h

p

or

p

h

or

Day 9: Vernal equinox Day 9: Vernal equinox 09-22-2008 @9:39 MDT09-22-2008 @9:39 MDT

3.4 Phase and Group Velocities3.4 Phase and Group Velocities p 99p 99–A group of waves need not have the same A group of waves need not have the same velocity as the waves themselvesvelocity as the waves themselves

3.5 Particle Diffraction3.5 Particle Diffraction p 104p 104–An experiment that confirms the existence of An experiment that confirms the existence of de Broglie wavesde Broglie waves

3.6 Particle in a Box3.6 Particle in a Box p 106p 106–Why the energy of a trapped particle is Why the energy of a trapped particle is quantizedquantized

(3.2 Waves of What?(3.2 Waves of What? Waves of Waves of probability)probability)


Public Star Party and Opening Social

When: Friday, Sept. 268 - 11pm

Where: Big Springs Park

What: Telescopes, dark sky, food

Dress warm! Bring your friends.

• Get to know the night sky• Look through BIG telescopes• View spiral structure in galaxies!• See nebulas of all types• Watch the moons of Jupiter move• And a whole lot more (no, really, we’re not just saying this - there really is a whole lot more!)

Directions to Big Springs Park:

Head up Provo Canyon. Take a right at Vivian Park. Go through Vivian Park and up the canyon a little over 3 miles. Look for a sign for the star party.


Did you Did you complete complete at least at least 50% of 50% of

Chapter 3: Chapter 3: 4-6?4-6?

A.A. Yes B. No Yes B. No

ReviewReview


Draw light coneDraw light cone

For electron accelerated to 511 keV For electron accelerated to 511 keV in 20 cmin 20 cm


The Wave Equation: The Wave Equation:

Which satisfy the Wave Which satisfy the Wave equation? ppequation? pp

A.A. y = yy = y00sin(kx- sin(kx- ωωt) t)

B.B. y = yy = y00ee-a(x- -a(x- vt)²vt)²; (Gaussian) ; (Gaussian)

C.C. Any fAny f(kx- (kx- ωωt) t)

D.D. All threeAll three

E.E. Only A and BOnly A and BAdapted fromAdapted from

3.4 Phase and 3.4 Phase and Group VelocitiesGroup Velocities

A group of waves A group of waves need not have the need not have the

same velocity as same velocity as the waves the waves

themselvesthemselves


Consider this experiment:Consider this experiment:

Throwing a pebble in a still pool. Throwing a pebble in a still pool. http://en.wikipedia.org/wiki/Group_velocithttp://en.wikipedia.org/wiki/Group_velocit

yy

Note Note green dotsgreen dots mark the beginning mark the beginning and ends of the group of waves but the and ends of the group of waves but the red dotsred dots mark the top of a given wave mark the top of a given wave

If you only had one group of waves, If you only had one group of waves, what would it be like?what would it be like?

Why does this work this way? Why does this work this way?

http://en.wikipedia.org/wiki/Group_velocity

http://en.wikipedia.org/wiki/Group_velocity


Dispersion Dispersion

Prism: breaks Prism: breaks up white up white light. How?light. How?

Refractive Refractive index n and index n and velocity of velocity of light?light?

n= c/v n= c/v v(v(λλ) = c/n) = c/n((λλ) ) v(v(k) = c/nk) = c/n((k)k)


What does dispersion have to What does dispersion have to do with Matter-waves?do with Matter-waves?

Dispersion means that the velocity Dispersion means that the velocity depends on the wavelength, or k or depends on the wavelength, or k or frequency. frequency.

Look at space-time diagrams. (On Look at space-time diagrams. (On blackboard) blackboard)

Definition of phase and group Definition of phase and group velocities.velocities.

Problem 6-1 (draw the dispersion Problem 6-1 (draw the dispersion relationship)relationship)


(draw the dispersion (draw the dispersion relationship)relationship)

Which one? Which one?


Just a note...Just a note...

cvhchv

hfE unless

nmeV8.1239 and hchc

hfE

For a photon

But...


What Exactly is Waving?What Exactly is Waving? For a photon...For a photon...

– electric and magnetic fieldselectric and magnetic fields– You can measure them if You can measure them if ºº is small is small

enough.enough.– For visible light, you can see that it is a For visible light, you can see that it is a

wave indirectly.wave indirectly. For a massive particleFor a massive particle

– You can’t measure them --- even in theory!You can’t measure them --- even in theory!– They are complex!They are complex!– How do we know that there’s really a How do we know that there’s really a

wave?wave?


How might I verify that my How might I verify that my Ford is a wave? Ford is a wave?


Thought QuestionThought Question Which of the following would be the easiest Which of the following would be the easiest

particle to use if I wanted to see a matter-particle to use if I wanted to see a matter-wave diffraction pattern?wave diffraction pattern?

1: A car moving at 100 mph1: A car moving at 100 mph

2: A car moving at 1 mph2: A car moving at 1 mph

3: A 1 MeV electron3: A 1 MeV electron

4: A 1 keV electron4: A 1 keV electron

5: What was the question?5: What was the question?


Wavelength of a FordWavelength of a Ford

mv

h

p

h

kg 105.1 lb 3336 3mv m/s 10 3

m 104.4 34


Wavelength of a 10 eV Wavelength of a 10 eV ElectronElectron

mv

h

p

h kg 101.9 31m

v m/s 1088.1 7 nm 39.0


3.5 Particle Diffraction3.5 Particle Diffraction

An experiment that An experiment that confirms the confirms the existence of de existence of de Broglie wavesBroglie waves


Davisson and GermerDavisson and Germer

photo from http://faculty.rmwc.edu/tmichalik/davisson.htm



Why did Davison and Why did Davison and Germer heat their nickel Germer heat their nickel target?target?

A – To induce thermal emission of A – To induce thermal emission of electronselectrons

B – To remove oxide contaminationB – To remove oxide contaminationC – To study the thermal expansion C – To study the thermal expansion

coefficient of pure nickelcoefficient of pure nickelD – To produce blackbody radiation D – To produce blackbody radiation E – To keep their graduate students E – To keep their graduate students

from sitting on it (ouch, that’s hot!)from sitting on it (ouch, that’s hot!)


Bragg DiffractionBragg Diffraction



Why so many peaks?


1. Many orders (n=1,2,3,4,...)

2. Many Bragg planes


Scanning the Energy of Scanning the Energy of the Electronsthe Electrons


de Brogliede Broglie

Photons: Photons: p=h/p=h/λλ

Particle: Particle: λλ=h/p=h/p

Hint: If the particle Hint: If the particle is going slow is going slow (K=½p(K=½p22/m)/m)



In one minute, write a short, clear, In one minute, write a short, clear, and concise paragraph which and concise paragraph which explains how the Davisson-Germer explains how the Davisson-Germer experiment shows that electrons are experiment shows that electrons are waves.waves.


Cesium InterferometerCesium Interferometer

Rotation rate (x10-5) rad/sec

-10 -5 0 5 10 15 20

Nor

mal

ized

sig

nal

-1

0

1

/2

/2

2

1

3


Interference of BECInterference of BEC


CC6060 Interference Interference

Interference fringes!

The interfering particle: Buckyballs

Apparatus

http://www.quantum.univie.ac.at/

Recent results from Vienna group of Anton Zielinger:

Not only more mass,but more degrees offreedom too!


3.6 Particle in a Box3.6 Particle in a Box p. 106p. 106Why is the energy of a trapped Why is the energy of a trapped

particle quantized?particle quantized?

Rather talk about this now we should Rather talk about this now we should spend some more time with waves. spend some more time with waves.


(3.2 Waves of What?(3.2 Waves of What?Waves of probabilityWaves of probability))

Extra Credit Activity (2 points) SIM: Davisson-Germer: Electron Diffraction

Run the simulation found at http://phet.colorado.edu/new/simulations/s

ims.php?sim=DavissonGermer_Electron_Diffraction

Use a variety of parameters in the simulation. Write a paragraph describing the Davisson-Germer experiment and what you have observed and learned. You may include fugures or a table if you'd like.

http://phet.colorado.edu/new/simulations/sims.php?sim=DavissonGermer_Electron_Diffraction




2 Important Properties of 2 Important Properties of WavesWaves

Not localizedNot localized

Greater localization in Greater localization in space implies poorer space implies poorer localization in localization in wavelengthwavelength



If If ψψ is is the wavefunction of a the wavefunction of a particle, where am I most particle, where am I most likely to find the particle?likely to find the particle?

A: Where A: Where ψψ is the largest. is the largest.

B: Where B: Where ψψ is the smallest. is the smallest.

C: Where C: Where ψψ22 is the largest. is the largest.

D: Where |D: Where |ψψ||22 is the largest. is the largest.

E: It has nothing to do with E: It has nothing to do with ψψ..


The Wave FunctionThe Wave Function

),,,( tzyx

2|| Related to probability of finding the particle at a given location at a given time.

dV2||

dV2|| is the probability of finding the particle in an infinitesimal volume element dV


The Wave Function in One The Wave Function in One DimensionDimension

),( tx

bx

ax

dx2||

dx2||

is the probability of finding the particle between x and x+dx at time t.



In one minute, write a short, clear, In one minute, write a short, clear, and concise paragraph which and concise paragraph which explains why |explains why |ΨΨ((xx,,tt)|)|22 needs to be needs to be multiplied by multiplied by dVdV to really be to really be meaningful.meaningful.


No you feeble minded rodent, No you feeble minded rodent, why is why is ªª not defined such that not defined such that ªª and not |and not |ªª||22 is proportional to is proportional to

probability of finding the particle?probability of finding the particle?

Are you ponderingAre you ponderingwhat I’m pondering?what I’m pondering?

Yeah, but I don’t thinkYeah, but I don’t thinksnails like yodeling.snails like yodeling.


Thought QuestionThought Question

What tells us about the probability of What tells us about the probability of finding a photon at a particular finding a photon at a particular location?location?

A: The electric fieldA: The electric field

B: The magnetic fieldB: The magnetic field

C: The intensityC: The intensity

D: The wave vectorD: The wave vector

E: The Poynting vectorE: The Poynting vector


What else does the photon’s What else does the photon’s wave function tell us?wave function tell us?

The Poynting vector The Poynting vector (ExB) tells us the (ExB) tells us the direction it is going.direction it is going.

The electric field tells The electric field tells us how it will interfere. us how it will interfere.


What do Quantum Waves What do Quantum Waves Represent?Represent?

Everything about the object’s stateEverything about the object’s state– Position, momentum, angular momentum, Position, momentum, angular momentum,

excitation energy, dipole moment, etc.excitation energy, dipole moment, etc. Does it express the object’s Does it express the object’s

temperature?temperature? What does the amplitude represent? What does the amplitude represent? Are there other ways to represent Are there other ways to represent

the wave function?the wave function?


Quantum Probability – Quantum Probability – Ensemble AverageEnsemble Average

Start with a million copies Start with a million copies of the same wavefunction.of the same wavefunction.

Measure the location of Measure the location of each particle.each particle.

Generate a HistogramGenerate a Histogram– This plot is essentially This plot is essentially ªª**ªª


PP((xx,,tt))//EE22

Photon with well defined wavelength: E=E0sin(kx-! t)

Red=E(x,t)Blue=P(x,t)


PP((xx,,tt))//ªª**ªªElectron in free space with well defined wavelength:

ª=Aei(kx-! t)

Red=Re[ª(x,t)]Green=Im[ª(x,t)]

Blue=P(x,t)


A moving standing wave...A moving standing wave...


Fourier’s TheoremFourier’s Theorem

)cos()sin()( 00

0 xnkBxnkAx nn

n

Any periodic function can be written as a sum of sines and cosines...

dkkkxkAx )(sin)()(

dkekAx ikx)(~

)(


Pure Sine WavePure Sine Wavey=sin(5 x) Power Spectrum


““Shuttered” Sine WaveShuttered” Sine Wavey=sin(5 x)*shutter(x) Power Spectrum


““Thin” GaussianThin” Gaussiany=exp(-(x/0.2)^2) Power Spectrum


““Fat” GaussianFat” Gaussiany=exp(-(x/2)^2) Power Spectrum


Femtosecond Laser PulseFemtosecond Laser PulseEt=0=sin(10 x)*exp(-x^2) Power Spectrum


Quiz Question: Which of the Quiz Question: Which of the following best describes following best describes “dispersion?”“dispersion?”A – When a wave passing through a slit A – When a wave passing through a slit

spreads out sphericallyspreads out sphericallyB – When the phase velocity varies with B – When the phase velocity varies with

wavelengthwavelengthC – When a wave spontaneously C – When a wave spontaneously

changes its momentumchanges its momentumD – When a wave only contains a finite D – When a wave only contains a finite

number of frequenciesnumber of frequenciesE – When the police try to get the crowd E – When the police try to get the crowd

of waves to “move along” at the scene of waves to “move along” at the scene of a crime (“there’s nothing to see of a crime (“there’s nothing to see here”).here”).


Femtosecond Laser PulseFemtosecond Laser PulseEt=0=sin(10 x)*exp(-x^2) Power Spectrum


Thought Question: Thought Question: What will happen What will happen to my laser pulse as it travels to my laser pulse as it travels

through empty space?through empty space?

A: The pulse will get wider.A: The pulse will get wider.B: The pulse will get thinner.B: The pulse will get thinner.C: The pulse will stay the same.C: The pulse will stay the same.D: It depends on the spectrum of D: It depends on the spectrum of the pulse.the pulse.

E: I have no idea.E: I have no idea.


Propagation Of Light PulsePropagation Of Light PulseE(x,t) Power Spectrum


Tracking a Moving PulseTracking a Moving PulseE(x+vt,t) Power Spectrum


Laser Pulse in Dispersive Laser Pulse in Dispersive MediumMedium

Et=0 = sin(10 x)*exp(-x^2) Power Spectrum


Thought Question: Thought Question: What will What will happen to my laser pulse as happen to my laser pulse as it travels through a piece of it travels through a piece of

glass?glass?A: The pulse will get wider.A: The pulse will get wider.B: The pulse will get thinner.B: The pulse will get thinner.C: The pulse will stay the C: The pulse will stay the same.same.

D: It depends on the D: It depends on the spectrum of the pulse.spectrum of the pulse.

E: I have no idea.E: I have no idea.


Time Evolution of Dispersive Time Evolution of Dispersive PulsePulse


Zooming In on a Dispersive Zooming In on a Dispersive PulsePulse

E(x+vpt,t)


Tracking a Dispersive Pulse Tracking a Dispersive Pulse AgainAgain

E(x+vgt,t)



In one minute, write a short, clear, In one minute, write a short, clear, and concise paragraph which and concise paragraph which explains why a pulse of light spreads explains why a pulse of light spreads out (or sometimes shrinks) in length out (or sometimes shrinks) in length as it travels through glass.as it travels through glass.


Thought Question: Thought Question: What happens What happens to the power spectrum of to the power spectrum of the pulse as goes through the pulse as goes through

the glass?the glass? A: It narrowsA: It narrowsB: It broadensB: It broadensC: It stays the same widthC: It stays the same widthD: It depends on the initial D: It depends on the initial spectrum of the pulse.spectrum of the pulse.


Time Evolution of Power Time Evolution of Power Spectrum in Dispersive Spectrum in Dispersive

MediumMedium


Phase and Group VelocityPhase and Group Velocity

)()( kvk

velocityparticle classical torelated ,envelope"" of velocity dk

dvg

ceinterferenfor important ,wiggles"" ofvelocity

kvp


Phase vs. Group Velocity: Pure Phase vs. Group Velocity: Pure Sine WaveSine Wave

)sin( tkx

kv

For a pure sine wave...

Only vp has meaning


Phase vs. Group Velocity: Phase vs. Group Velocity: Sum of Two SinesSum of Two Sines

2

sin2

cos2)sin()sin(baba

ba

2

sin2

cos2 22112211 txktxktxktxk

)sin()sin( 2211 txktxk

txkk

txkk

22sin

22cos2 21212121

wigglesenvelope


Phase vs. Group Velocity: Phase vs. Group Velocity: Sum of Two SinesSum of Two Sines

txkk

txkk

22sin

22cos2 21212121

avgk

kkk

221

wiggles

221

envelopebeat

kkk

avg

avg

kvv

pwiggles

2

21envelopebeat

dk

d

kkvv

21

21genvelopebeat

avg

221

wiggles


Tracking a Free Particle Tracking a Free Particle Wavefunction (with stationary Wavefunction (with stationary

cameras)cameras)


Relativistic vs. Non-relativistic Relativistic vs. Non-relativistic Phase Velocity...Phase Velocity...



In one minute, write a short, clear, In one minute, write a short, clear, and concise paragraph which and concise paragraph which explains why the phase velocity of a explains why the phase velocity of a massive particle is different for massive particle is different for relativistic and non-relativistic relativistic and non-relativistic calculations (even at low velocities).calculations (even at low velocities).


Infinite Square WellInfinite Square Well

Particle in a BoxParticle in a Box p 106p 106– Why the energy of a trapped particle is Why the energy of a trapped particle is

quantizedquantized



In one minute, write a short, clear, In one minute, write a short, clear, and concise paragraph which and concise paragraph which explains how boundary conditions explains how boundary conditions resulted in quantized energies for a resulted in quantized energies for a particle in a box.particle in a box.



How would we have to change the How would we have to change the energy equation for a particle in energy equation for a particle in a box if the potential in the box a box if the potential in the box were not zero but some finite were not zero but some finite positive value positive value UU00?_____ ?_____ A: We would have to add A: We would have to add UU00..B: We would have to subtract B: We would have to subtract UU00..C: We would have to add C: We would have to add nUnU00

D: We would have to subtract D: We would have to subtract nUnU00..E: We wouldn’t have to change E: We wouldn’t have to change anything.anything.


3.6 Particle in a Box3.6 Particle in a Box ::Why is Why is the energy of a trapped particle the energy of a trapped particle

quantized?quantized?7.4. (2 points) Consider an electron in a 1-

dimensional infinitely deep well.(a) If the well is 9 nm wide, what is the

ground-state energy of the electron, in eV?(b) What is the energy of the first excited

state?(c) What is the wavelength of a photon that

will excite the electron from its ground state to it's third excited state (with n = 4)?

(d) If the well is 2 cm wide, for what value of n will the electron have an energy of 1 eV?


3.7 – 3.93.7 – 3.9

3.7 Uncertainty Principle I3.7 Uncertainty Principle IWe cannot know the future because we cannot We cannot know the future because we cannot know the presentknow the present

3.8 Uncertainty Principle II3.8 Uncertainty Principle IIA particle approach gives the same resultA particle approach gives the same result

3.9 Applying the Uncertainty 3.9 Applying the Uncertainty Principle Principle A useful tool, not just a negative A useful tool, not just a negative statementstatement



Which type of wave packet has the Which type of wave packet has the minimum uncertainty product? minimum uncertainty product? (i.e. for which one is (i.e. for which one is ¢¢xx¢¢k k the the smallestsmallest?)?)

A – A square packetA – A square packetB – A triangular packetB – A triangular packetC – A Gaussian packetC – A Gaussian packetD – A Lorentzian packetD – A Lorentzian packetE – I can’t possibly know much about E – I can’t possibly know much about

physics because I understand too physics because I understand too much about social interactions:much about social interactions:

((¢¢PPhh¢¢SSii¸̧1/21/2).).


Uncertainty in a Classical Uncertainty in a Classical WaveWave

2

1 t

2

1 kx


Sine WaveSine WaveWhat is it’s wavelength?

What is it’s location?

What is it’s frequency?

When does it occur?

sin(kx ¡ ! t)


Beats in TimeBeats in TimeWhat is it’s wavelength?



When does it occur?

sin(kx ¡ ! t) + sin(kx ¡ 2wt)


Localization in Localization in Time/FrequencyTime/Frequency

t

2212

beatf

2t

2 t

12

beatf

1

Have we really localized this wave?


Localization in Localization in Position/WavenumberPosition/Wavenumber

What is it’s wavelength?



When does it occur?

sin(kx ¡ ! t) + sin(1:1kx ¡ wt)


Localization in Localization in Time/FrequencyTime/Frequency

k

x

22"" 12 kkk

fbeat

12 kk

""

1

beatf ...


Beats in Both...Beats in Both...


True Localization...True Localization...

What does it take to make a single pulse of wavyness?


Thought Question: Thought Question: Consider two Consider two short pulses of light. Pulse “A” short pulses of light. Pulse “A” has a duration of 10 fs. Pulse has a duration of 10 fs. Pulse “B” has a duration of 50 fs. “B” has a duration of 50 fs. Which one has the smallest Which one has the smallest

¢¢ωω??A : Pulse “A”A : Pulse “A”

B : Pulse “B”B : Pulse “B”

C : They are both the sameC : They are both the same

D : Not enough information has D : Not enough information has been givenbeen given


Time Evolution of Dispersive Time Evolution of Dispersive PulsePulse



In one minute, write a short, clear, In one minute, write a short, clear, and concise paragraph which and concise paragraph which explains how the uncertainty explains how the uncertainty relations allow two waves to have the relations allow two waves to have the same uncertainty in same uncertainty in xx, but different , but different uncertainty in uncertainty in kk..



A Gaussian pulse of light strikes a diffraction grating and A Gaussian pulse of light strikes a diffraction grating and spreads out. I let a tiny piece of the diffracting beam spreads out. I let a tiny piece of the diffracting beam through a pinhole. How will the spectrum of the light through a pinhole. How will the spectrum of the light coming through the pinhole compare to the spectrum of coming through the pinhole compare to the spectrum of the pulse before it hit the grating?the pulse before it hit the grating?

A : It will be narrowerA : It will be narrowerB : It will be widerB : It will be widerC : It will be the sameC : It will be the sameD : Not enough information givenD : Not enough information givenE : I have no ideaE : I have no idea



A Gaussian pulse of light strikes a A Gaussian pulse of light strikes a diffraction grating and spreads out. I diffraction grating and spreads out. I let a tiny piece of the diffracting beam let a tiny piece of the diffracting beam through a pinhole. How will the through a pinhole. How will the duration of the pulse coming through duration of the pulse coming through the pinhole compare to the duration of the pinhole compare to the duration of the pulse before it hit the grating?the pulse before it hit the grating?

A : It will be shorterA : It will be shorterB : It will be longerB : It will be longerC : It will be the sameC : It will be the sameD : Not enough information givenD : Not enough information givenE : I have no ideaE : I have no idea


New_sqr60.avi


““Shuttered” Sine Wave on a Shuttered” Sine Wave on a GrattingGratting

y=sin(5 x)*shutter(x) Power Spectrum


Atom Emitting “Shuttered” Atom Emitting “Shuttered” Sine WaveSine Wave

y=sin(5 x)*shutter(x) Power Spectrum


Something to think aboutSomething to think about

Can’t we think of a pure sine wave as Can’t we think of a pure sine wave as a string of pulses too?a string of pulses too?

So why doesn’t a pure sine wave So why doesn’t a pure sine wave contain a spread of frequencies?contain a spread of frequencies?

COHERENCECOHERENCE


Calcium SpectroscopyCalcium Spectroscopy


Quantum Uncertainty RelationsQuantum Uncertainty Relations

Position – Momentum

Energy – Time

Other Dimensions

Angular Momentum


7.5. (2 points)

(a) How accurately can the position of a proton with v << c be determined without giving it more than 1.00 keV of kinetic energy?

(b) The position and momentum of a 1.00-keV electron are simultaneously determined. If its position is located to within 0.100 nm, what is the percentage of uncertainty in its momentum?



Imagine that I measure the location and momentum of an Imagine that I measure the location and momentum of an electron. I measure the location with a precision of electron. I measure the location with a precision of 1nm. If I make a second measurement one second 1nm. If I make a second measurement one second later, about how well will I be able to predict where I later, about how well will I be able to predict where I will find the electron with the second measurement? will find the electron with the second measurement? (me ~ 1e-30 kg, h/4(me ~ 1e-30 kg, h/4¼¼ ~ 0.5e-34 Js) ~ 0.5e-34 Js)

A : To better than 1 nmA : To better than 1 nm B : To within around 1nmB : To within around 1nmC : To within 1 C : To within 1 μμmm D : To within 1 mmD : To within 1 mmE : Not even to within 1 mmE : Not even to within 1 mm


7.6. (2 points)(a) How much time is needed to measure

the kinetic energy of an electron whose speed is 10.0 m/s with an uncertainty no more than 0.100 percent?

(b) How far will the electron have traveled in this period of time?

(c) Make the same calculation for a 1.00-g insect whose speed is the same.

(d) What do these sets of figures indicate?


Wave-Particle DualityWave-Particle Duality Things act as wave when propagatingThings act as wave when propagating

– or, in other words, we use waves to make or, in other words, we use waves to make predictions as to what we will find when we predictions as to what we will find when we make our measurement.make our measurement.

Things act as waves when we measure Things act as waves when we measure wave-like properties.wave-like properties.

Things act as particles when we measure Things act as particles when we measure particle-like propertiesparticle-like properties

Example: BEC interference --- theorists Example: BEC interference --- theorists confused about “undefined phase”confused about “undefined phase”


Postulates of Quantum Postulates of Quantum MechanicsMechanics

Every physically-realizable system is described Every physically-realizable system is described by a state function by a state function ψψ that contains all that contains all accessible physical information about the accessible physical information about the system in that statesystem in that state

The probability of finding a system within the The probability of finding a system within the volume volume dvdv at time at time tt is equal to | is equal to |ψψ||22dvdv

Every observable is represented by an Every observable is represented by an operator which is used to obtain information operator which is used to obtain information about the observable from the state functionabout the observable from the state function

The time evolution of a state function is The time evolution of a state function is determined by Schrödinger’s Equationdetermined by Schrödinger’s Equation

physics 222

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light years

broglieif light

gravitational lens g2237

adapted fromlouis

gravitational field

nearby galaxy

demise of atoms

momentumwavelength relations