Particle Wave Duality

What is a particle? What is a wave?

Problems with Classical Physics

•Nature of Light?

•Discrete Spectra?

•Blackbody Radiation?

•Photoelectric Effect?

•Compton Effect?

•Model of Atom?

Double Slit is VERY IMPORTANT because it is evidence

of waves. Only waves interfere like this.

Thomas Young 1804

sind m

REVIEW! Derive Fringe Equations

• For bright fringes

• For dark fringes

bright ( 0 1 2 ), ,λL

y m md

dark

1( 0 1 2 )

2, ,

λLy m m

d

Double Slit for Electrons

shows Wave Interference!

Key to Quantum Theory!

James Clerk Maxwell 1860s

Light is an electromagnetic wave.

The medium is the Ether.

8

0

13.0 10 /

o

c x m s

The Electromagnetic Spectrum

Michelson-Morely

Experiment

1887 The speed of light is independent of the motion and

is always c. The speed of the Ether wind is zero.

OR….

Lorentz Contraction

The apparatus shrinks by a factor :

2 21 / v c

Special Relativity 1905

2 2 2 2 2

0( ) ( ) ( ) E pc m c pc

2 E mc p mu

If m = 0 (photon)

/p E cphoton momentum:

2

0 E mc

2 2( )u u

pc muc m c mc Ec c

22 2

0( ) E pc E

p E

2 2 22 2 2 2 2 2 2 2 2 2 2

0 02 22

2

(1 ) ( )

1

mc u uE mc E mc E E E E E E

c cu

c

Why Continuous vs Discrete?

This is a continuous spectrum of colors: all colors are present.

This is a discrete spectrum of colors: only a few are present.

Kirkoff’s Rules

Kirkoff’s Rules for Spectra: 1859

Bunsen

German physicist who developed the spectroscope and the science of

emission spectroscopy with Bunsen.

Kirkoff

* Rule 1 : A hot and opaque solid, liquid or highly compressed gas emits a continuous spectrum.

* Rule 2 : A hot, transparent gas produces an emission spectrum with bright lines.

* Rule 3 : If a continuous spectrum passes through a gas at a lower temperature, the transparent

cooler gas generates dark absorption lines.

Compare absorption lines in a source with emission

lines found in the laboratory!

Kirchhoff deduced that elements were present in the atmosphere of the Sun

and were absorbing their characteristic wavelengths, producing the absorption

lines in the solar spectrum. He published in 1861 the first atlas of the solar

spectrum, obtained with a prism ; however, these wavelengths were not very

precise : the dispersion of the prism was not linear at all.

Anders Jonas Ångström 1869 Ångström measured the wavelengths on the

four visible lines of the hydrogen spectrum,

obtained with a diffraction grating, whose

dispersion is linear, and replaced

Kirchhoff's arbitrary scale by the

wavelengths, expressed in the metric

system, using a small unit (10-10 m) with

which his name was to be associated.

Line color Wavelength

red 6562.852 Å

blue-green 4861.33 Å

violet 4340.47 Å

violet 4101.74 Å

Balmer Series: 1885 Johann Balmer found an empirical equation that correctly

predicted the four visible emission lines of hydrogen

H 2 2

1 1 1

2R

λ n

RH is the Rydberg constant

RH = 1.097 373 2 x 107 m-1

n is an integer, n = 3, 4, 5,…

The spectral lines correspond to different

values of n

Johannes Robert Rydberg generalized

it in 1888 for all transitions:

Hα is red, λ = 656.3 nm

Hβ is green, λ = 486.1 nm

Hγ is blue, λ = 434.1 nm

Hδ is violet, λ = 410.2 nm

Why this shape? Why the drop?

When an object it heated it will

glow first in the infrared, then the

visible. Most solid materials break

down before they emit UV and

higher frequency EM waves.

Frequency ~ Temperature

Long

Short

All objects radiate energy continuously

in the form of electromagnetic waves

due to thermal vibrations of their

molecules.

A good absorber reflects little and appears Black

A good absorber is also a good emitter.

Blackbody Radiation • A black body is an ideal system that

absorbs all radiation incident on it

• The electromagnetic radiation emitted by a

black body is called blackbody radiation

c

Blackbody Approximation

• A good approximation of a

black body is a small hole

leading to the inside of a

hollow object

• The hole acts as a perfect

absorber

• The nature of the radiation

leaving the cavity through

the hole depends only on the

temperature of the cavity

Stefan’s Law: 1879

Rate of radiation of a Black Body

• P = σAeT 4 – P is the rate of energy transfer, in Watts

– σ = 5.6696 x 10-8 W/m2 . K4

– A is the surface area of the object

– e is a constant called the emissivity • e varies from 0 to 1

• The emissivity is also equal to the absorptivity

– T is the temperature in Kelvins

– With his law Stefan determined the temperature of the Sun’s surface and he calculated a value of 5430C. This was the first sensible value for the temperature of the Sun.

– Boltzmann was his student and derived Stefan’s Law from Thermodynamics in 1884 and extended it to grey bodies.

Jožef Stefan

(1835–1893)

Maxwell-Boltzmann Distribution: 1877

• The observed speed distribution of gas molecules in thermal equilibrium is shown at right

• NV is called the Maxwell-Boltzmann speed distribution function

• The distribution of speeds in N gas molecules is

• The probability of finding the molecule in a particular energy state varies exponentially as the negative of the energy divided by kBT

2

3 / 2

/ 22

B

42

Bmv k ToV

mN N v e

k T

nV (E ) = noe –E /kBT

Ludwig Boltzmann

1844 – 1906

• Temperature ~ Ave KE of each particle

• Particles have different speeds

• Gas Particles are in constant RANDOM motion

• Equipartition of Energy: Average KE of

each particle is: 3/2 kT

• Pressure is due to momentum transfer

Speed ‘Distribution’ at

CONSTANT Temperature

is given by the

Maxwell Speed Distribution

23/ 2 1/ 2 rmskT KE mv

k =1.38 x 10-23 J/K Boltzmann’s Constant

4P e T A

Radiant heat makes it impossible to stand close to a hot lava

flow. Calculate the rate of heat loss by radiation from 1.00

m2 of 1200C fresh lava into 30.0C surroundings, assuming

lava’s emissivity is 1.

The net heat transfer by radiation is: 4 4

2 1( )P e A T T

4 4

2 1( )P e A T T

8 4 2 4 41(5.67 10 / )1 ((303.15 ) (1473.15 ) )x J smK m K K

266P kW

Blackbody Experiment Results • The total power of the radiation emitted from the

surface increases with temperature

– Stefan’s law: P = AeT4

– P is the power and is the Stefan-Boltzmann constant:

= 5.670 x 10-8 W / m2 . K4 (0<e < 1, for a blackbody, e = 1)

• The peak of the wavelength distribution shifts to

shorter wavelengths as the temperature increases

– Wien’s displacement law

(T must be in kelvin):

Finding peak wavelengths

Finding peak wavelengths

The heating effect of a medium such as glass or the Earth’s

atmosphere that is transparent to short wavelengths but opaque

to longer wavelengths: Short get in, longer are trapped!

Intensity of Blackbody Radiation

• The intensity increases with increasing temperature

• The amount of radiation emitted increases with increasing temperature

– The area under the curve

• The peak wavelength decreases with increasing temperature

• Combining gives the Rayleigh-Jeans law:

I = P/A = T4

I , ~4

1λ T

λ

Problems with the Wein’s World

• At short wavelengths, there

was a major disagreement

between the Rayleigh-Jeans

law and experiment

• This mismatch became

known as the ultraviolet

catastrophe

– You would have infinite

energy as the wavelength

approaches zero

I , ~4

1λ T

λ

Max Planck: Father of Quantum

• Introduced the concept of “quantum of action” in 1900 to solve the black body mystery

• In 1918 he was awarded the Nobel Prize for the discovery of the quantized nature of energy

The possible frequencies and energy states of a wave on

a string are quantized.

2

vf n

l

Strings are Quantized

Planck’s Two Assumptions

• The energy of an oscillator can have only certain discrete values En= nhƒ

– This says the energy is quantized

– Each discrete energy value corresponds to a different quantum state

• The oscillators emit or absorb energy when making a transition from one quantum state to another

– The entire energy difference between the initial and final states in the transition is emitted or absorbed as a single quantum of radiation

Energy-Level Diagram

• An energy-level diagram

shows the quantized energy

levels and allowed

transitions

• Energy is on the vertical axis

• Horizontal lines represent

the allowed energy levels

• The double-headed arrows

indicate allowed transitions

More About Planck’s Model

• The average energy of a wave is the average

energy difference between levels of the oscillator,

weighted according to the probability of the wave

being emitted

• This weighting is described by the Boltzmann

distribution law and gives the probability of a state

being occupied as being proportional to

BE k Te

Planck’s Wavelength

Distribution Function

• Planck generated a theoretical expression

for the wavelength distribution

– h = 6.626 x 10-34 J.s

– h is a fundamental constant of nature

2

5

2

1I ,

Bhc λk T

πhcλ T

λ e

Planck’s

Model,

Graphs

Intensity of Blackbody Radiation P40.61

The total power per unit area radiated by a black body at a temperature T is the area under the I(λ, T)-versus-λ curve, as shown in Figure 40.3. (a) Show that this power per unit area is

where I(λ, T) is given by Planck’s radiation law and σ is a constant independent of T.

This result is Stefan’s law.

To carry out the integration, you should make

the change of variable x = hc/λkT and use the

fact that

4

0λ λ, TdTI

0

43

151

xe

dxx

2

5

2

1I ,

Bhc λk T

πhcλ T

λ e

34

, n= 0,1,2,3,...

6.626 10

E nhf

h x Js

Atomic Energy is quantized.

It comes in chunks of Planck’s constant, h.

Max Planck NEVER liked the idea

of quantized energy states.

In classical physics particles have

continuous energy states….to say they

have discrete energy states would mean

that you can only drive at 10mph and

20mph but not at 15mph or at any speed

in between 10 and 20 mph! In classical

physics only special bound states have

discrete or quantum energy states….

Why doesn’t Planck like Quantum?????

The Photoelectric Effect • In 1886 Hertz noticed, in the

course of his investigations, that a

negatively charged electroscope

could be discharged by shining

ultraviolet light on it.

• In 1899, Thomson showed that the

emitted charges were electrons.

• The emission of electrons from a

substance due to light striking its

surface came to be called the

photoelectric effect.

• The emitted electrons are often

called photoelectrons to indicate

their origin, but they are identical

in every respect to all other

electrons.

The Problem with Waves:

Increasing the intensity of a low frequency

light beam doesn’t eject electrons. This

didn’t agree with wave picture of light

which predicts that the energy of waves

add so that if you increase the intensity of

low frequency light (bright red light)

eventually electrons would be ejected –

but they don’t! There is a cut off

frequency, below which no electrons will

be ejected no matter how bright the beam!

Also there is no time delay in the ejection

of electrons as the waves build up!

The PROBLEM with the

Photoelectric Effect

The Problem with Waves:

Increasing the intensity of a low frequency

light beam doesn’t eject electrons. This

didn’t agree with wave picture of light

which predicts that the energy of waves

add so that if you increase the intensity of

low frequency light (bright red light)

eventually electrons would be ejected –

but they don’t! There is a cut off

frequency, below which no electrons will

be ejected no matter how bright the beam!

Also there is no time delay in the ejection

of electrons as the waves build up!

The Photoelectric Effect Proof that Light is a Particle

Characteristics of the

Photoelectric Effect 1. The current I is directly proportional to the light intensity.

2. Photoelectrons are emitted only if the light frequency f

exceeds a threshold frequency f0.

3. The value of the threshold frequency f0 depends on the

type of metal from which the cathode is made.

4. If the potential difference ΔV is positive, the current does

not change as ΔV is increased. If ΔV is made negative, the

current decreases until, at ΔV = −Vstop the current reaches

zero. The value of Vstop is called the stopping potential.

5. The value of Vstop is the same for both weak light and

intense light. A more intense light causes a larger current,

but in both cases the current ceases when ΔV = −Vstop.

The Photoelectric Effect

Photoelectric Effect Problem 1

• Dependence of photoelectron kinetic energy on light intensity

– Classical Prediction • Electrons should absorb energy continually from the

electromagnetic waves

• As the light intensity incident on the metal is increased, the electrons should be ejected with more kinetic energy

– Experimental Result • The maximum kinetic energy is independent of light intensity

• The maximum kinetic energy is proportional to the stopping potential (DVs)

Photoelectric Effect Problem 2

• Time interval between incidence of light and

ejection of photoelectrons

– Classical Prediction

• At low light intensities, a measurable time interval should pass

between the instant the light is turned on and the time an

electron is ejected from the metal

• This time interval is required for the electron to absorb the

incident radiation before it acquires enough energy to escape

from the metal

– Experimental Result

• Electrons are emitted almost instantaneously, even at very low

light intensities

Photoelectric Effect Problem 3

• Dependence of ejection of electrons on light

frequency

– Classical Prediction

• Electrons should be ejected at any frequency as long as the light

intensity is high enough

– Experimental Result

• No electrons are emitted if the incident light falls below some

cutoff frequency, ƒc

• The cutoff frequency is characteristic of the material being

illuminated

• No electrons are ejected below the cutoff frequency regardless

of intensity

Photoelectric Effect Problem 4

• Dependence of photoelectron kinetic energy on

light frequency

– Classical Prediction

• There should be no relationship between the frequency of the

light and the electric kinetic energy

• The kinetic energy should be related to the intensity of the light

– Experimental Result

• The maximum kinetic energy of the photoelectrons increases

with increasing light frequency

Einstein’s Postulates: Light Quanta

Einstein framed three postulates about light quanta and

their interaction with matter:

1. Light of frequency f consists of discrete quanta, each

of energy E = hf, where h is Planck’s constant

h = 6.63 × 10−34 J s. Each photon travels at the

speed of light c = 3.00 × 108 m/s.

2. Light quanta are emitted or absorbed on an all-or-

nothing basis. A substance can emit 1 or 2 or 3

quanta, but not 1.5. Similarly, an electron in a metal

can absorb only an integer number of quanta.

3. A light quantum, when absorbed by a metal, delivers

its entire energy to one electron.

Light is quantized in chunks of Planck’s constant. Electrons will not be ejected in the Photoelectric Effect unless every

photon has the right energy. One photon is completely absorbed by

each electron ejected from the metal. As you increase the intensity of

the beam, more electrons are ejected, but their energy stays the same.

Photons

E hf

EX 39.2 The energy of a light quantum

Einstein’s Explanation of the

Photoelectric Effect An electron that has just absorbed a quantum of light

energy has Eelec = hf. (The electron’s thermal energy at

room temperature is so much less than that we can neglect

it.) This electron can escape from the metal, becoming a

photoelectron, if

In other words, there is a threshold frequency

for the ejection of photoelectrons because each light

quantum delivers all of its energy to one electron.

Einstein’s Explanation of the

Photoelectric Effect • A more intense light delivers a larger number of light

quanta to the surface. These quanta eject a larger number

of photoelectrons and cause a larger current.

• There is a distribution of kinetic energies, because

different photoelectrons require different amounts of

energy to escape, but the maximum kinetic energy is

The stopping potential Vstop is directly proportional to Kmax.

Einstein’s theory predicts that the stopping potential is

related to the light frequency by

max 0hf KE E

Photon

Energy

Max KE of

ejected electron

Work to eject

(Work Function)

or Binding Energy

Cutoff Frequency • The lines show the linear

relationship between K and ƒ The slope of each line is h

• The x-intercept is the cutoff frequency. This is the frequency below which no photoelectrons are emitted

• The cutoff frequency is related to the work function through ƒc = φ / h

• The cutoff frequency corresponds to a cutoff wavelength

ƒc

c

c hcλ

φ

Work Function

The Photoelectric Effect What is the maximum velocity of electrons ejected from

a material by 80nm photons, if they are bound to the

material by 4.73eV? Ignore relatavistic effects.

MaxK BE BEhc

hf

1 2

18

2 6

31

2 1.729 10 J1 2KEK 1.95 10 m s

2 9.11 10 kgmv v

m

34 8 19

9

6.63 10 J s 3.00 10 m s 1.60 10 J4.73 eV

80.0 10 m 1 eV

181.7295 10 J

(SR: ) 2( 1)K mc

Maximum photoelectron speed

Compton Effect, Classical

Predictions • According to the

classical theory, EM

waves incident on

electrons should:

– have radiation pressure

that should cause the

electrons to accelerate

– set the electrons

oscillating

Compton Effect, Observations

• Compton’s

experiments showed

that, at any given

angle, only one

frequency of radiation

is observed

Compton Effect, Explianed

• The results could be explained by treating the photons as point-like particles having energy hƒ

• Assume the energy and momentum of the isolated system of the colliding photon-electron are conserved

• This scattering phenomena is known as the Compton effect

Compton Shift Equation

• The graphs show the scattered

x-ray for various angles

• The shifted peak, λ’ is caused

by the scattering of free

electrons

– This is called the Compton shift

equation

1' coso

e

hλ λ θ

m c

Arthur Holly Compton

• 1892 - 1962

• Director of the lab at

the University of

Chicago

• Discovered the

Compton Effect, 1923

• Shared the Nobel

Prize in 1927

hp

The phenomenon in which an X-ray photon is scattered from an

electron, the scattered photon having a smaller frequency than the

incident photon is called The Compton Effect.

hcE hf pc

2 E mc p mv

E pcDivide:

incident scattered electronp p p

The photon transfers momentum, acts like a particle.

The Compton wavelength of a particle is

equivalent to the wavelength of a photon whose

energy is the same as the rest mass of the particle.

It gives the limits of measuring the position of a

particle using traditional QM and not QED.

The compton wavelength of the elctron is:

1' coso

e

hλ λ θ

m c

. 122 43 10e

hx m

m c

Compton Wavelength

The incident X-ray photon has an energy of 3.98 keV and is scattered by an angle of 140.0 degrees.

a) What is the wavelength of incident X-ray?

b) What is the wavelength of the scattered X-ray?

c) What is the energy of the scattered X-ray?

d) What is the kinetic energy of the recoil electron?

e) What is the de Broglie wavelength of the recoil electron?

82.998 10 /c x m s319.109 10em x kg

346.626 10h x Js

191.602 10 /x J eV

,

Joseph John Thomson

“Plum Pudding” Model 1904

• Received Nobel Prize in

1906

• Usually considered the

discoverer of the electron

• Worked with the

deflection of cathode rays

in an electric field

• His model of the atom

– A volume of positive

charge

– Electrons embedded

throughout the volume

1911: Rutherford’s

Planetary Model of the

Atom

(Couldn’t explain the stability or spectra of atoms.)

•A beam of positively charged alpha

particles hit and are scattered from a

thin foil target.

•Large deflections could not be

explained by Thomson’s pudding

model.

1911: Rutherford’s Planetary

Model of the Atom

(Couldn’t explain the stability or spectra of atoms.)

•A beam of positively charged alpha

particles hit and are scattered from a

thin foil target.

•Large deflections could not be

explained by Thomson’s model.

Classical Physics at the Limit

WHY IS MATTER (ATOMS)

STABLE?

Electrons exist in quantized orbitals with energies given by

multiples of Planck’s constant. Light is emitted or absorbed when

an electron makes a transition between energy levels. The energy of

the photon is equal to the difference in the energy levels:

i fE E E hf

34

, n= 0,1,2,3,...

6.626 10

E nhf

h x Js

Light Absorption & Emission i fE E E hf

1. Electrons in an atom can occupy only certain discrete quantized

states or orbits.

2. Electrons are in stationary states: they don’t accelerate and they

don’t radiate.

3. Electrons radiate only when making a transition from one

orbital to another, either emitting or absorbing a photon.

Bohr’s Assumptions

Postulate: The angular momentum of an electron is

always quantized and cannot be zero:

2

( 1,2,3,....)

hL n

n

: = ( 1,2,3,....)2

hFrom L n mvr n

Bohr’s Derivation of the Energy for Hydrogen:

E K U

F is centripetal:

Conservation of E:

Sub back into E:

From Angular

Momentum:

(1)

Sub r back into (1):

(2)

Sub into (2):

Why is it negative?

2 2 2

0 0

2 2

4 2

e e ekq q q

vnh hn hn

Bohr Line Spectra of Hydrogen

2

2 2

1 1 1( )

f i

RZn n

7 11.097 10R x m

Balmer: Visible

Lyman: UV

Paschen: IR

Bohr’s Theory derived the spectra equations that Balmer,

Lyman and Paschen had previously found experimentally!

2

0 13.6E Z eV

1. Bohr model does not explain why electrons don’t radiate in orbit.

2. Bohr model does not explain splitting of spectral lines.

3. Bohr model does not explain multi-electron atoms.

4. Bohr model does not explain ionization energies of elements.

1. Bohr model does not explain angular momentum postulate:

The angular momentum of an electron is

always quantized and cannot be zero*:

2

( 1,2,3,....)

hL n

n

*If L=0 then the electron travels linearly and does not ‘orbit’….

But if it is orbiting then it should radiate and the atom would be unstable…eek gads!

WHAT A MESS!

E hf hp

c c

If photons can be particles, then

why can’t electrons be waves?

e

h

p

Electrons are

STANDING

WAVES in

atomic orbitals.

deBroglie Wavelength:

2 nr n

2

( 1,2,3,....)

e n

hL m vr n

n

2e n

hm vr n

1924: de Broglie Waves

Explains Bohr’s postulate of angular

momentum quantization:

h

p 2 nr n

2 n

e

h hr n n

p m v

Electrons are STANDING WAVES in atomic orbitals.

h

p

1924: de Broglie Waves

2 nr n

De Broglie Wavelength

346.626 10h x J s

h

mv

Lynda’s De Broglie Wavelength

346.626 10

(75 )2 /

x J s

kg m s

364.4 10x m

Too small to notice or to interact with anything!

Particle-Wave: Light

A gamma ray photon has a momentum of 8.00x10-21 kg m/s.

What is its wavelength? What is its energy in Mev?

h

p

663 10

800 10829 10

34

21

14.

..

J s

kg m s m

21 88.00 10 kg m s 3.00 10 m sE pc

12

13

1 MeV2.40 10 J 15.0 MeV

1.60 10 J

Electron De Broglie Wavelength

for electron v = .1c

34

31 7

6.626 10

(9.1 10 )(3 10 / )

x J s

x kg x m s

/h mv

112.4 10x m

Limits of Vision

112.4 10e x m

Electron

Waves

Electron Microscope

Electron microscope picture of a fly.

The resolving power of an optical lens depends on the wavelength of

the light used. An electron-microscope exploits the wave-like

properties of particles to reveal details that would be impossible to see

with visible light.

Electron Microscope

Stem Cells

The fossilized shell of

a microscopic ocean

animal is magnified

392 times its actual

size.

Salmonella Bacteria

Double Slit for Electrons

shows Wave Interference

Double Slit for Electrons

A modified oscilloscope is used to

perform an electron interference

experiment. Electrons are incident on

a pair of narrow slits 0.060 0 μm

apart. The bright bands in the

interference pattern are separated by

0.400 mm on a screen 20.0 cm from

the slits. Determine the potential

difference through which the

electrons were accelerated to give

this pattern.

Particle-Wave Duality

Interference pattern builds one

electron at a time.

Electrons act like

waves going through

the slits but arrive at

the detector like a

particle.

Trying to see what slit an

electron goes through destroys

the interference pattern.

Feynman version of the

Uncertainty Principle

It is impossible to design an apparatus

to determine which hole the electron

passes through, that will not at the

same time disturb the electrons enough

to destroy the interference pattern.

-Richard Feynman

Next Time….

Electron waves are probability waves

in the ocean of uncertainty. - Richard Feynman: