Class 21 - Early Quantum Mechanics and the Wave Nature of Matter€¦ · · 2017-03-22Class 21...

Class 21

Physics 106

LearningOutcomes

QuantumMechanicsOverview

BlackbodyRadiation

PhotoelectricEffect

X RaysCompton Scattering

Some ProblemsPhotons

Photoelectric Effect

Compton Scattering

Matter Waves

The SchrödingerEquation

Wave Functions

UncertaintyPrinciple

Tunneling

Some Philosophy

Class 21Early Quantum Mechanics and the Wave Nature of

Matter

Physics 106

Winter 2018

Press CTRL-L to view as a slide show.

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Last Time

Last time we discussed:I Optical systemsI Midterm 2

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Learning Outcomes

Today we will discuss:I Quick Overview of Quantum MechanicsI Blackbody RadiationI Photoelectric EffectI X-ray diffractionI Compton scattering

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Quantum Mechanics −An Overview

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

April 27, 1900

In a speech before the Royal Institution, Lord Kelvinproposed that we basically understood all of physics,except for "two dark clouds on the hoirizon."These were

I The null result of the Michelson-Morley experimentI The inability of classical physics to explain blackbody

radiation

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Need for Quantum Physics

Blackbody RadiationI The electromagnetic radiation emitted by a heated

objectPhotoelectric Effect

I Emission of electrons by an illuminated metalSpectral Lines

I Emission of sharp spectral lines by gas atoms in anelectric discharge tube

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Blackbody Radiation

Light is emitted from hot objects inbundles (quanta) of energy

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Blackbody Radiation

I An object at anytemperature emitselectromagneticradiation

I The spectrum ofthe radiationdepends on thetemperature andproperties of theobject

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Blackbody Radiation Graph

I As T increases,the total energyemitted increases

I As T increases,the peak of thedistribution shiftsto shorterwavelengths

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Wien’s Law

I Based onthermodynamicsand EM Theory

I Gave goodagreement atshort wavelengths

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Wien’s Law

I But it wasn’t verygood at longwavelength

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Rayleigh-Jeans Law

I Also based onthermodynamicsand EM Theory

I Gave goodagreement at longwavelength

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Rayleigh-Jeans Law

I But it didn’t agreeat all at shortwavelength

I This was calledthe "ultravioletcatastrophe"

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Blackbody Radiation

I Classical physicsgave two laws thatdidn’t agree

I . . . and neitheragreed with all thedata

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Max Planck

I 1858 -1947I Introduced a

"quantum ofaction," h

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Planck’s Resolution

I Oscillators in matter could only emit light at discreteenergies

I En = nhfI n is called the quantum numberI f is the frequency of oscillationI h is Planck’s constant

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Planck’s Constant

There are many versions

I h = 6.626× 10−34 Js (SI)I h = 4.136× 10−15 eV s (eV)

I hc = 1240 eV nmI ~ = h/2π ("hbar")

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Today’s Interpretation

I Light from atoms in the blackbody is emitted asphotons from atoms

I Each photon of light carries energy

E = hf

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Conclusion 1

I At very small scales, measurements of manyphysical quantities can only have discrete values.These include

I Energy of bound electronsI Angular momentumI Angular momentum component along one axisI Spin

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy


Light is absorbed by matter inquanta of energy called photons

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy


I Light is absorbed by matter in quanta of energy→ photons

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy


I Light can knock electrons off metallic surfacesI The effect was first discovered by HertzI The successful explanation of the effect was given by

Albert Einstein in 1905

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy


I When light strikesE , photoelectronsare emitted

I Electrons strike Cand form a currentin the circuit

I A voltage isplaced on C. IfV > 0, it attractselectrons. IfV < 0, it repelselectrons.

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy


I By adjusting thevoltage, we canmeasure thekinetic energy ofthephotoelectrons

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Photoelectric Current/Voltage Graph

I When V > 0, thecurrent quicklyreaches asaturation level

I No current flowsfor voltages lessthan or equal to−∆Vs, thestopping potential

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Einstein’s Explanation

I Light is emitted only in packets of energy calledphotons with

E = hf

I The maximum kinetic energy of the liberatedphotoelectron is

KEmax = hf − Φ

where Φ is the work function of the metalI The work function is the minimum energy

needed to knock an elctron off the metal surface

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Cutoff Frequency

I The cutoff frequency is related to the work function Φ

Ec = hfc = Φ

I If the energy of the photon equals the work function,there is just enough energy to remove the electronfrom the metal surface with no extra kinetic energy.

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Cutoff Wavelength

I A cutoff wavelength can be defined:

Ec = hfc =hcλc

I Wavelengths greater than λc incident on a materialwith a work function Φ don’t result in the emission ofphotoelectrons

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Conclusion 2

I We can think of light as particles, each havingenergy and momentum:

I E = hfI p = h/λ

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Consequences of Conclusion 2

I Understanding light in terms photons is helpful inunderstanding many phenomena including

I Photoelectric effectI Compton scattering

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

X Rays

X rays can behave as waves

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Diffraction of X Rays by Crystals

I For diffraction to occur, the spacing between thelines must be approximately equal to the wavelengthof the radiation to be measured

I The regular array of atoms in a crystal can act as athree-dimensional grating for diffracting x rays

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Schematic for X-ray Diffraction

I A beam of X-rayswith a continuousrange of wavelengthsis incident on thecrystal

I In directions wherethere is constructiveinterference fromwaves reflected fromthe layers of thecrystal, the diffractedradiation is veryintense

I The diffractionpattern is detectedby photographic film

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Compton Scattering

X rays can behave as particles −high energy photons

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Arthur Holly Compton

I 1892 -1962I Discovered the

Compton effectI Worked with

cosmic raysI Director of the lab

at U of Chicago

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

The Compton Effect

I Compton directed a beam of x rays toward a block ofgraphite

I He found that the scattered x rays had a slightly lessenergy than the incident x rays

I The amount of energy reduction depended on theangle at which the x rays were scattered

I The change in wavelength is called the Compton shift

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Compton Scattering

I Comptonassumed thephotons acted likeother particles incollisions

I Energy andmomentum wereconserved

I The shift inwavelength is

∆λ =h

mec(1−cos θ)

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Photons and Electromagnetic Waves

I Light has a dual nature. It exhibits both wave andparticle characteristics

I This applies to all electromagnetic radiationI The photoelectric effect and Compton scattering

offer evidence for the particle nature of lightI Interference and diffraction offer evidence of the

wave nature of light

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Some Problems

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Wavelength

I What is the energy of photon that has the wavelengthof 1 nm, the size of a typical atom?

E = hf =hcλ

=1240eVnm

1nm= 1240eV

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Work Function

The work function of a given metal is 2.3 eV. What is themaximum wavelength of light (minimum photon energy)that will produce photoelectons?

Φ = hf =hcλ

= 2.3eV

λ =hcφ

=1240eVnm

2.3eV= 539nm

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Compton Scattering

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Compton Scattering

A 4.4 MeV gamma ray Compton scatters through anangle of 20◦ (measured with respect to its incidentdirection). What is its energy after scattering?

∆λ = λ− λ0 =h

mec(1− cos θ)

First, we need the relation between E and λ.

E = hf =hcλ

⇒ λ =hcE

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Compton Scattering


∆λ = λ− λ0 =h

mec(1− cos θ) λ =

hcE

hcE− hc

E0=

hmec

(1− cos θ)

1E− 1

E0=

1mec2 (1− cos θ)

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Compton Scattering


1E

=1E0

+1

0.511MeV(1− cos θ)

1E

=1

4.4MeV+

10.511MeV

(1− cos 20◦)

E = 2.90MeV

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

de Broglie and MatterWaves

Electrons as well as photons sharein wave-particle duality

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Photons and Electromagnetic Waves

I Light has a dual nature. It exhibits both wave andparticle characteristics

I The photoelectric effect and Compton scatteringoffer evidence for the particle nature of light

I Interference and diffraction offer evidence of thewave nature of light

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Photon Momentum

I If a photon has energy, it also has momentum.

I Relativity tells us

p =hλ

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Photon Momentum



p =hλ

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Photon Momentum



p =hλ

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Wave Properties of Particles

I In 1924, Louis de Broglie postulated that becausephotons have wave and particle characteristics,perhaps all forms of matter have both properties

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

de Broglie Wavelength and Frequency

I Furthermore, he

postulated that

both light and

matter obeyed the

relationships:

E = hf

p =hλ

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

The Davisson-Germer Experiment

I Davisson and Germer reasoned that if x raysdiffracted from crystals, electron waves shoulddiffract as well

I Electron diffraction experiments showed that λ = h/pas de Broglie predicted

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

The Schrödinger Equation

A guess about how matter wavesshould behave

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Measurement Theory

I Physics had always assumed that position, velocity,etc., could be measured accurately

I In the early 20th century, physicists worried that atatomic levels, measurements would change systems

I They looked for mathematics that gave differentresults when two quantities were measured indifferent order

I If you measure A then B, you get different resultsthan when you measure B then A

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Measurement Theory

I Erwin Schrödinger suggested using derivatives to

represent measurement of physical quantities:

ddx

(xf (x)) 6= xddx

f (x)

I This led to "wave mechanics"

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Measurement Theory

I Werner Heisenberg suggested using matrices to

represent measurement of physical quantities:[a bc d

] [e fg h

]6=

[e fg h

] [a bc d

]I This led to "matrix mechanics"

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Measurement Theory

I Wave mechanics and matrix mechanics were latershown to be identical

I They came to be known as "quantum mechanics"

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

The Wave Function

I In 1926 Schrödinger proposed a wave equation thatdescribes the manner in which matter waves changein space and time

I Schrödinger’s wave equation is a key element inquantum mechanics

I Schrödinger’s wave equation is generally solved forthe wave function, ψ

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

The Foundations of Quantum Mechanics

Postulate 1:We can represent a system (like an electron in an atom)by a function called a wave function.

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

The Wave Function

I Has both wave and particle characteristics built intoit.

I The wave function of a particle moving in onedimension in a box with very hard walls:

ψ(x , t) = A sin2πxλ

e−i 2πf t

wave function wavelength: p = h/λ frequency: E = hf

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

The Wave Function




e−i 2πf t


Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

The Wave Function




e−i 2πf t


Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

The Wave Function




e−i 2πf t


Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

The Wave Function




e−i 2πf t


Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

The Wave Function

I What’s with the e−i 2πft?

e−i 2πft = cos(−2πft) + i sin(2πft)

I This form makes calculations more convenient, butstill represents a sine (or cosine) wave.

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

The Wave Function

I Only wave functions with certain wavelengths areallowed.

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

The Wave Function

I When we square the wave function, we get theprobability distribution.

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

The concept of a probability distribution

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Wave function = Probability wave

I The wave function gives us the probability of findingthe particle at a given location.

I Wave functions add like waves − they interfere anddiffract

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy


Postulate 2:We can represent things we can measure (energy,momentum, spin) with operators.

An operator is anything that does something to somethingelse.The operators in quantum mechanics do something towave functions.

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Two Operators

I Momentum operatorConstant × slope of ( )

I Position operatorposition × ( )

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy


Postulate 3:Sometimes an operator satisfies the equation:operator (ψ) = number × ψIf this is true, the number is the value you obtain whenyou measure the quantity indicated by the operator.

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Huh????

Postulate 3:Ifmomentum operator (ψ) = 3 × ψthe momentum is 3and ψ is the special function that describes a particle withmomentum = 3

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

So what?

Energy = kinetic energy +potential energy= p2/(2m) + P.E.

If

(momentum operator)2

2mψ+(potential energy)×ψ = E×ψ

Thenψ is the special wave function that describes a particle

with energy E .

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Schrödinger’s Equation

− ~2

2m∇2ψ(~r) + V (~r)ψ(~r) = Eψ(~r)

I Constructed by ErwinSchrödinger in 1926

I It describes atoms veryaccurately!

I It was a guess − it hadno right to work and wedon’t really understandit yet ...

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Wave Functions

Understanding and dealing withmatter waves

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

The Wave Function

I So, what does the wave function mean?I That was the subject of much debate in the 1920s!I We settled on the "Copenhagen Interpretation,"

proposed by Neils Bohr and his associates.

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

The Wave Function

I The value of |ψ|2 at some location at a given time isthe probability density (the probability per unitvolume) of finding the particle at that location and atthat time

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

The Wave Function

I The value of |ψ|2 at some location at a given time isthe probability density (the probability per unitvolume) of finding the particle at that location and atthat time

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

The Wave Function

I Atomic orbitalsare the wavefunctions forelectrons inatoms.

http://www.geo.arizona.edu/xtal/geos306/d-orbitals.gif

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Conclusion 3

I We can describe all we know about a particle interms of the wave function.

I The absolute square of the wave function is theprobability density.

I The measurement process can be described by"operators"

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

The Heisenberg UncertaintyPrinciple

One consequence of dealing withmatter waves

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

The Uncertainty Principle

I When measurements are made, the experimenter isalways faced with experimental uncertainties in themeasurements

I Classical mechanics allows for measurements witharbitrarily small uncertainties

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

The Uncertainty Principle and Waves

I Two special wave functions:

I All other wave functions are somewhere in between.

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy


I Quantum mechanics predicts that a barrier tomeasurements with ultimately small uncertaintiesdoes exist

I In 1927 Heisenberg introduced the uncertaintyprinciple

I If a measurement of position of a particle is madewith precision ∆x and a simultaneous measurementof linear momentum is made with precision ∆px ,then the product of the two uncertainties can neverbe smaller than h/4π

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy


I Mathematically,

∆x∆px ≥h

4π

I It is physically impossible to measure simultaneouslythe exact position and the exact linear momentum ofa particle

I Another form of the principle deals with energy and

time:

∆E∆t ≥ h4π

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Uncertainty Principle Applied to anElectron

I View the electron as a particleI Its position and velocity cannot both be known

precisely at the same timeI Its energy can be uncertain for a period given by

∆E = h/(4π∆t)

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Conclusion 4

I Measuring a quantity changes the wave function.I Whenever the measurement of one quantity affects

another quantity, the two quantities can not bemeasured accurately together.

I In these cases, the quantities satisfy an uncertaintyprinciple relationship.

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Tunneling

An application of quantummechanics

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Tunneling

I Consider a positively charged plate with a small holein it.

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Tunneling

I A proton is moving in the x direction along the axis ofthe hole.

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Tunneling

I We can make a graph of the potential energy of theproton vs position

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Tunneling

I We can make a graph of the potential energy of theproton vs position

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Tunneling

I Let the total energy of the proton be E .

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Tunneling

I The kinetic energy is the difference between E andU.

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Tunneling

I As the proton approaches the plate, it slows down.I After if passes through the hole, it speeds up again.U

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Tunneling

I If the total energy is less than U, the electron cannever get through the hole...

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Tunneling

I ... unless the electron is a wave!I Then a small wave is found on the far side of the

plate.

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Scanning Tunneling Microscope (STM)

I A conducting probewith a sharp tip isbrought near thesurface

I Electrons can tunnelacross the workfunction potentialenergy

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Scanning Tunneling Microscope

I By applying a voltage between the surface and thetip, the electrons can be made to tunnel preferentiallyfrom surface to tip

I The larger the distance to the tip, the more difficult itis for electrons to tunnel through the barrier.

I The STM allows measurements of the height ofsurface features within 0.001 nm

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

STM Result, Example

I This is a "quantumcorral" of 48 ironatoms on a coppersurface

I The diameter of thering is 143 nm

I Obtained with a lowtemperature STM

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

Some Philosophy

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

A Thought Experiment

I Shoot many individual electrons toward two narrowslits (really at a crystal lattice). What do you see onthe screen behind?

I An interference pattern is seen, but can one electrongo through both slits?

I Try an experiment - put a pick-up coil around one slit– What happens?

I The interference pattern goes away - but theexperiment changes the system.

Class 21

Physics 106

LearningOutcomes


BlackbodyRadiation

PhotoelectricEffect




Compton Scattering

Matter Waves


Wave Functions


Tunneling

Some Philosophy

The Bottom Line

I The electron is observed at creation and detection.I Between these times, the electron seems to do all

possible things the electron could do.I The probability of the different outcomes is the same

as if the electron were a wave that experiencedinterference.

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Class 21 - Early Quantum Mechanics and the Wave Nature of Matter€¦ · · 2017-03-22Class 21...

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