General Physics (PHY 2140)

http://www.physics.wayne.edu/~alan/2140Website/Main.htm

Lecture 15Lecture 15Modern Physics1. Quantum Physics

The Compton EffectPhotons and EM WavesWave Properties of ParticlesWave FunctionsThe Uncertainty Principle

Chapter 27Chapter 27

Reminder: Exam 3 Reminder: Exam 3 Friday, July 6Friday, July 6

1212--13 questions.13 questions.

Show your work for credit.Show your work for credit.

Closed book. Closed book.

You may bring a page of notes.You may bring a page of notes.

Bring a calculator and a pen or pencilBring a calculator and a pen or pencil

Review Problem: A xenon arc lamp is covered with an interference filter that only transmits light of 400-nm wavelength. When the transmitted light strikes a metal surface, a stream of electrons emerges from the metal. If the intensity ofthe light striking the surface is doubled,

1. more electrons are emitted in a given time interval.2. the electrons that are emitted are more energetic.3. both of the above.4. neither of the above.

Lightning ReviewLightning Review

Last lecture:

1.1.

Quantum physicsQuantum physicsBlackbody radiationBlackbody radiationPlanckPlanck’’s hypothesiss hypothesisPhotoelectric effectPhotoelectric effectXX--raysrays

KE hf= −Φ

2max 0.2898 10T m Kλ −= × ⋅

, 1,2,3,...nE nhf n= =

( )minhc

e Vλ =

Δ

The Compton EffectThe Compton EffectCompton directed a beam of xCompton directed a beam of x--rays toward a block of graphiterays toward a block of graphiteHe found that the scattered xHe found that the scattered x--rays had a slightly longer wavelength rays had a slightly longer wavelength that the incident xthat the incident x--raysrays

This means they also had less energyThis means they also had less energyThe amount of energy reduction depended on the angle at which thThe amount of energy reduction depended on the angle at which the e xx--rays were scatteredrays were scatteredThe change in wavelength is called the The change in wavelength is called the Compton shiftCompton shift

Compton ScatteringCompton Scattering

Compton assumed the Compton assumed the photons acted like other photons acted like other particles in collisionsparticles in collisions

Energy and momentum Energy and momentum were conservedwere conserved

The shift in wavelength isThe shift in wavelength is )cos1(cm

he

o θ−=λ−λ=λΔ

Compton wavelength = 0.00243 nm

Compton ScatteringCompton Scattering

The quantity The quantity h/mh/mee

cc

is called the is called the Compton wavelengthCompton wavelengthCompton wavelength = 0.00243 nmCompton wavelength = 0.00243 nmVery small compared to visible lightVery small compared to visible light

The Compton shift depends on the The Compton shift depends on the scattering anglescattering angle

and and not not on on the the wavelengthwavelengthExperiments confirm the results of Compton scattering and Experiments confirm the results of Compton scattering and strongly support the photon conceptstrongly support the photon concept

Problem: Compton scatteringProblem: Compton scattering

A beam of 0.68A beam of 0.68--nm photons (E=1828 nm photons (E=1828 eVeV) undergoes Compton scattering ) undergoes Compton scattering from free electrons. What are the energy and momentum of the phofrom free electrons. What are the energy and momentum of the photons tons that emerge at a 45that emerge at a 45°°

angle with respect to the incident beam? angle with respect to the incident beam?

(1 cos )oe

hm c

λ λ λ θΔ = − = −

Δλ = 0.00243 nm x (1-0.707) = 7.11x10-4

nm

E=hc/λ =hc/0.6807nm = 1826 eV

p = h/λ = h/0.6807nm = 1826 eV/c

QUICK QUIZ 1

An x-ray photon is scattered by an electron. The frequency of the scattered photon relative to that of the incident photon (a) increases, (b) decreases, or (c) remains the same.

(b).

Some energy is transferred to the electron in the scattering process. Therefore, the scattered photon must have less energy (and hence, lower frequency) than the incident photon.

QUICK QUIZ 2

A photon of energy E0

strikes a free electron, with the scattered photon of energy E moving in the direction opposite that of the incident photon. In this Compton effect interaction, the resulting kinetic energy of the electron is (a) E0 , (b) E , (c) E0

−

E , (d) E0 + E , (e) none of the above.

(c).

Conservation of energy requires the kinetic energy given to the electron be equal to the difference between the energy of the incident photon and that of the scattered photon.

27.7 Wave Properties of Particles27.7 Wave Properties of Particles

In 1924, Louis de Broglie postulated that In 1924, Louis de Broglie postulated that because because photons have photons have wave and particle characteristics, perhaps all forms of matter hwave and particle characteristics, perhaps all forms of matter have ave both propertiesboth propertiesFor instance, for a photon:For instance, for a photon:

De Broglie suggested that this formula is true for De Broglie suggested that this formula is true for anyany

particle! Thus, particle! Thus, the frequency and wavelength of matter waves can be determined. the frequency and wavelength of matter waves can be determined. I.e. I.e. de Broglie wavelengthde Broglie wavelength of a particle isof a particle is

hmv

λ=

hcE hfλ

= =E hc hpc cλ λ

= = =thus orhp

λ =

Wave Properties of ParticlesWave Properties of Particles

The frequency of matter waves can also be determinedThe frequency of matter waves can also be determined

De Broglie postulated that all particles satisfy EinsteinDe Broglie postulated that all particles satisfy Einstein’’s s relation relation

Or, in other words,Or, in other words,

ƒ Eh

=

E hf=

The DavissonThe Davisson--Germer ExperimentGermer Experiment

They scattered lowThey scattered low--energy electrons from a nickel targetenergy electrons from a nickel target

They followed this with extensive They followed this with extensive diffraction measurementsdiffraction measurements

from from various materialsvarious materials

The wavelength of the electrons calculated from the diffraction The wavelength of the electrons calculated from the diffraction data data agreed with the expected de Broglie wavelengthagreed with the expected de Broglie wavelength

This confirmed the wave nature of electronsThis confirmed the wave nature of electrons

Other experimenters have confirmed the wave nature of other Other experimenters have confirmed the wave nature of other particlesparticles

Problem: the wavelength of a protonProblem: the wavelength of a proton

Calculate the de Broglie wavelength for a proton (mCalculate the de Broglie wavelength for a proton (mpp

=1.67x10=1.67x10--2727

kg ) kg ) moving with a speed of 1.00 x 10moving with a speed of 1.00 x 1077

m/sm/s..

Calculate the de Broglie wavelength for a proton (mCalculate the de Broglie wavelength for a proton (mpp

=1.67x10=1.67x10--2727

kg ) moving with a kg ) moving with a speed of 1.00 x 10speed of 1.00 x 1077

m/s.m/s.

Given:

v = 1.0 x 107m/s

Find:

λp

= ?

Given the velocity and a mass of the proton we can compute its wavelength

pp

hm v

λ =

Or numerically,

( )( )( )

3414

31 7

6.63 103.97 10

1.67 10 1.00 10ps

J sm

kg m sλ

−−

−

× ⋅= = ×

× ×

QUICK QUIZ 2

A non-relativistic electron and a non-relativistic proton are moving and have the same de Broglie wavelength. Which of the following are also the same for the two particles: (a) speed, (b) kinetic energy, (c) momentum, (d) frequency?

(c).

Two particles with the same de Broglie wavelength will have the

same momentum p = mv. If the electron and proton have the same momentum, they cannot have the same speed because of the difference in their masses. For the same reason, remembering that KE = p2/2m, they cannot have the same kinetic energy. Because the kinetic energy is the only type of energy an

isolated particle can have, and we have argued that the particles have different energies, Equation 27.15 (

f = E/h )

tells us that the particles do not have the same frequency.

pp

hm v

λ =

The Electron MicroscopeThe Electron Microscope

The electron microscope depends The electron microscope depends on the wave characteristics of on the wave characteristics of electronselectronsMicroscopes can only resolve details Microscopes can only resolve details that are slightly smaller than the that are slightly smaller than the wavelength of the radiation used to wavelength of the radiation used to illuminate the objectilluminate the objectThe electrons can be accelerated to The electrons can be accelerated to high energies and have small high energies and have small wavelengthswavelengths

λe

-

≈

5×10-12

m (5 pm) for 50 kV acceleration potential.

27.8 The Wave Function27.8 The Wave Function

In 1926 SchrIn 1926 Schröödinger proposed a dinger proposed a wave equationwave equation

that that describes the manner in which matter waves change in describes the manner in which matter waves change in space and timespace and timeSchrSchröödingerdinger’’s wave equations wave equation

is a key element in is a key element in

quantum mechanicsquantum mechanics

SchrSchröödingerdinger’’s wave equation is generally solved for the s wave equation is generally solved for the wave functionwave function, , ΨΨ

i Ht

ΔΨ= Ψ

Δ

The Wave FunctionThe Wave Function

The wave function depends on the particleThe wave function depends on the particle’’s position and s position and the timethe time

The The value of |value of |ΨΨ||22

at some location at a given time is at some location at a given time is proportional to the probability of finding the particle at proportional to the probability of finding the particle at that location at that timethat location at that time

OrbitalsOrbitals of of Atomic Atomic

HydrogenHydrogen

Computer generated Computer generated figures of atomic figures of atomic orbitalsorbitals (electron wave (electron wave functions) for the functions) for the Hydrogen atom.Hydrogen atom.

27.9 The Uncertainty Principle27.9 The Uncertainty Principle

When measurements are made, the experimenter is When measurements are made, the experimenter is always faced with experimental uncertainties in the always faced with experimental uncertainties in the measurementsmeasurements

Classical mechanics offers no fundamental barrier to Classical mechanics offers no fundamental barrier to ultimate refinements in measurementsultimate refinements in measurementsClassical mechanics would allow for measurements with Classical mechanics would allow for measurements with arbitrarily small uncertaintiesarbitrarily small uncertainties

The Uncertainty PrincipleThe Uncertainty Principle

Quantum mechanics predicts that a barrier to measurements Quantum mechanics predicts that a barrier to measurements with ultimately small uncertainties does existwith ultimately small uncertainties does exist

In 1927 Heisenberg introduced the In 1927 Heisenberg introduced the uncertainty principleuncertainty principle

If a measurement of position of a particle is made with precisioIf a measurement of position of a particle is made with precision n ΔΔx x and a simultaneous measurement of linear momentum is made with and a simultaneous measurement of linear momentum is made with precision precision ΔΔp, then the product of the two uncertainties can never be p, then the product of the two uncertainties can never be smaller than h/4smaller than h/4ππ

The Uncertainty PrincipleThe Uncertainty Principle

Mathematically,Mathematically,

It is physically impossible to measure simultaneously the It is physically impossible to measure simultaneously the exact position and the exact linear momentum of a exact position and the exact linear momentum of a particleparticle

Another form of the principle deals with energy and time: Another form of the principle deals with energy and time:

π≥ΔΔ

4hpx x

π≥ΔΔ

4htE

Thought Experiment Thought Experiment ––

the Uncertainty the Uncertainty PrinciplePrinciple

A thought experiment for viewing an electron with a powerful A thought experiment for viewing an electron with a powerful microscopemicroscopeIn order to see the electron, at least one photon must bounce ofIn order to see the electron, at least one photon must bounce off itf itDuring this interaction, momentum is transferred from the photonDuring this interaction, momentum is transferred from the photon

to to the electronthe electronTherefore, the light that allows you to accurately locate the elTherefore, the light that allows you to accurately locate the electron ectron changes the momentum of the electronchanges the momentum of the electron

Problem: macroscopic uncertaintyProblem: macroscopic uncertainty

A 50.0A 50.0--g ball moves at 30.0 g ball moves at 30.0 m/sm/s. If its speed is measured to an . If its speed is measured to an accuracy of 0.10%, what is the minimum uncertainty in its accuracy of 0.10%, what is the minimum uncertainty in its position?position?

A 50.0A 50.0--g ball moves at 30.0 m/s. If its speed is measured to an accuracg ball moves at 30.0 m/s. If its speed is measured to an accuracy of 0.10%, y of 0.10%, what is the minimum uncertainty in its position?what is the minimum uncertainty in its position?

Given:

v = 30 m/sΔv/v

= 0.10%m = 50.0 g

Find:

δx = ?

Notice that the ball is non-relativistic. Thus, p = mv, and uncertainty in measuring momentum is

( )

( )( )2 3 250.0 10 1.0 10 30 1.5 10

vp m v m vv

kg m s kg m s

δ

− − −

⎛ ⎞Δ = Δ = ⋅⎜ ⎟⎝ ⎠

= × × ⋅ = × ⋅

Thus, uncertainty relation implies

( ) ( )24

323

6.63 10 3.5 104 4 1.5 10

h J sx mp kg m sπ π

−−

−

× ⋅Δ ≥ = = ×

Δ × ⋅

Problem: Macroscopic measurementProblem: Macroscopic measurement

A 0.50-kg block rests on the icy surface of a frozen pond, which we can assume to be frictionless. If the location of the block is measured to a precision of 0.50 cm, what speed must the block acquire because of the measurement process?

4x

hx pπ

Δ Δ ≥Recall: p mv=and

Scanning Tunneling Microscope (STM)Scanning Tunneling Microscope (STM)

Allows highly detailed images with Allows highly detailed images with resolution comparable to the size of resolution comparable to the size of a single atoma single atomA conducting probe with a sharp tip A conducting probe with a sharp tip is brought near the surfaceis brought near the surfaceThe electrons can The electrons can ““tunneltunnel””

across across the barrier of empty spacethe barrier of empty space

Scanning Tunneling Microscope, contScanning Tunneling Microscope, cont

By applying a voltage between the surface and the tip, the electBy applying a voltage between the surface and the tip, the electrons rons can be made to tunnel preferentially from surface to tipcan be made to tunnel preferentially from surface to tipThe tip samples the distribution of electrons just above the surThe tip samples the distribution of electrons just above the surfacefaceThe STM is very sensitive to the distance between the surface anThe STM is very sensitive to the distance between the surface and d the tipthe tip

Allows measurements of the height of surface features within 0.0Allows measurements of the height of surface features within 0.001 nm01 nm

Limitation of the STMLimitation of the STM

There is a serious limitation to the STM since it depends There is a serious limitation to the STM since it depends on the conductivity of the surface and the tipon the conductivity of the surface and the tip

Most materials are not conductive at their surfaceMost materials are not conductive at their surfaceAn An atomic force microscopeatomic force microscope has been developed that overcomes has been developed that overcomes this limitationthis limitationIt measures the force between the tip and the sample surfaceIt measures the force between the tip and the sample surfaceHas comparable sensitivityHas comparable sensitivity

STM ImagesSTM Images

More STM ImagesMore STM Images