Chapter 28 Quantum Theory – Lecture 23

28.1 Particles, Waves, and Particles-Waves

28.2 Photons

28.3 Wavelike Properties Classical Particles

28.4 Electron Spin

28.5 Meaning of the Wave Function

28.6 Tunneling

28.7 Detection of Photons by the Eye

28.8 The Nature of Quanta: A Few Puzzles

Quantum Regime

• Macroscopic world explanations fail at the atomic-scale world

• Newtonian mechanics

• Maxwell’s equations describing electromagnetism

• The atomic-scale world is referred to as the quantum regime

• Quantum refers to a very small increment or parcel of energy

Introduction

Waves vs. Particles

• In the world of Newton

and Maxwell, energy

can be carried by

particles and waves

• Waves produce an

interference pattern

when passed through a

double slit

• Classical particles

(bullets) will pass

through one of the slits

and no interference

pattern will be formed Section 28.1

Particles and Waves, Classical

• Waves exhibit inference; particles do not

• Particles often deliver their energy (Joules) in

discrete amounts

• Energy arrives in discrete parcels

• Each parcel corresponds to the kinetic energy carried

by a single bullet

• The energy carried or delivered by a wave is not

discrete

• The energy carried by a wave is described by its

intensity (W/m2 = Js-1/m2)

• The amount of energy absorbed depends on the

intensity and the absorption time

Section 28.1

Interference with Electrons

• The separation between waves and particles is not found in

the quantum regime

• Electrons are used in a double slit experiment

• The blue lines show the probability of the electrons striking

particular locations, which has the same form as the variation

of light intensity in the double-slit interference experiment

Section 28.1

Interference with Electrons, cont.

• The experiment shows that electrons undergo

constructive interference at certain locations on the

screen

• At other locations, the electrons undergo destructive

interference

• The probability for an electron to reach those location is

very small or zero

• The experiment also shows aspects of particle-like

behavior since the electrons arrive one at a time at the

screen

Section 28.1

Particles and Waves, Quantum

• All objects, including light and electrons, can exhibit

interference

• All objects, including light and electrons, carry

energy in discrete amounts

• These discrete “parcels” are called quanta

Section 28.1

Work Function

• In the 1880s, studies of

what happens when

light is shone onto a

metal gave some

results that could not be

explained with the wave

theory of light

• The work function, Wc

is the minimum energy

required to remove a

single electron from a

piece of metal

Section 28.2

Work Function, cont.

• A metal contains electrons that are free to move

around within the metal

• The electrons are still bound to the metal and need

energy to be removed from the atom

• This energy is the work function

• The value of the work function is different for

different metals

• If V is the electric potential at which electrons begin

to jump across the vacuum gap, the work function is

Wc = eV

Section 28.2

Work Functions of Metals

Section 28.2

1 eV = 1.6 x 10-19 J

Photoelectric Effect

• Another way to extract

electrons from a metal

is by shining light onto it

• Light striking a metal is

absorbed by the

electrons

• If an electron absorbs

an amount of light

energy greater than Wc,

it is ejected off the metal

• This is called the

photoelectric effect

Section 28.2

Photoelectric Effect, cont.

• No electrons are

emitted unless the

light’s frequency is

greater than a critical

value ƒc

• When the frequency is

above ƒc, the kinetic

energy of the emitted

electrons varies linearly

with the frequency

Section 28.2

( )f c

Photoelectric Effect, Difficulties - 1

• Trying to explain the photoelectric effect with the

classical wave theory of light presented two

difficulties (1) and (2)

• (1) Experiments showed that the critical frequency is

independent of the intensity of the light

• Classically, the energy is proportional to the intensity

• It should always be possible

to eject electrons by increasing

the intensity to a sufficiently high

value

• Below the critical frequency,

there are no ejected electrons no

matter how great the light intensity Section 28.2

Photoelectric Effect, Difficulties - 2

• Difficulties (1) and (2) with classical explanation, cont.

• (2) The kinetic energy of an ejected electron is

independent of the light intensity

• Classical theory predicts increasing the intensity will

cause the ejected electrons to have a higher kinetic

energy

• Experiments actually show the electron kinetic energy

depends on the light’s frequency

• Classical wave theory of light was not able explain the

photoelectric effect experiments

Section 28.2

Photoelectric Effect, Explanation - 1

• The absorption of light by an electron is just like a

collision between two particles, a photon and an

electron

• The photon carries an energy that is absorbed by the

electron

• If this energy is less than the work function, the

electron is not able to escape from the metal

• The energy of a single photon depends on frequency

but not on the light intensity

Section 28.2

Photoelectric Effect, Explanation - 2

• The kinetic energy of the ejected electrons

depends on light frequency but not intensity

• The critical frequency corresponds to photons whose

energy is equal to the work function

h ƒc = Wc

• This photon is just ejected and would have no kinetic

energy

• If the photon has a higher energy, the difference goes

into kinetic energy of the ejected electron

KEelectron = h ƒ - h ƒc = h ƒ - Wc

• This linear relationship is what was found

experimentally

Section 28.2

Photons

• Einstein proposed that light carries energy in

discrete quanta, now called photons

• Each photon carries a parcel of energy

• h is a constant of nature called Planck’s constant

• h = 6.626 x 10-34 J ∙ s = 4.14 x 10-15 eV ∙ s

• A beam of light should be thought of as a collection

of photons

• Each photon has an energy dependent on its

frequency

• If the intensity of monochromatic light is increased,

the number of photons is increased, but the

energy carried by each photon does not change

Section 28.2

photon

hcE hf

Momentum of a Photon

• A light wave with energy E also carries a certain

momentum

• “Particles” of light called photons carry a discrete

amount of both energy and momentum

• Photons have two properties that are different than

classical particles

• Photons do not have any mass

• Photons exhibit interference effects

Section 28.2

.(23.12)photon

Ep Eq

c

photon

E hf hp

c c

Blackbody Radiation

• Blackbody radiation is emitted over a range of

wavelengths

• To the eye, the color of the cavity is determined by

the wavelength at which the radiation intensity is

largest

Section 28.2

Blackbody Radiation, Classical

• The blackbody intensity curve has the same shape for a

wide variety of objects

• Electromagnetic waves form standing waves as they

reflect back and forth inside the oven’s cavity

• The frequencies of the standing waves follow the

pattern ƒn = n ƒ where n = 1, 2, 3, …

• There is no limit to the value of n, so the frequency can

be infinitely large

• But as the frequency increases, so does the energy

• Classical theory (Rayleigh-Jeans Law) predicts that the

blackbody intensity should become infinite as the

frequency approaches infinity (Ultraviolet Catastrophe !)

Section 28.2

Blackbody Radiation, Quantum

• The disagreement between the classical predictions

and experimental observations was called the

“ultraviolet catastrophe”(Rayleigh-Jeans Law)

• Planck proposed solving the problem by assuming

the energy in a blackbody cavity must come in

discrete parcels

• Each parcel would have energy E = h ƒn

• His theory (Planck’s Radiation Law) fit the

experimental results, but gave no reason why it

worked

• Planck’s work is generally considered to be the

beginning of quantum theory Section 28.2

Blackbody Radiation - Summary

• Based on the quantized-

energy hypothesis (E=hf),

Planck obtained Plank’s

radiation law:

2

/5

2( )

( 1)Bhc k T

hc

e

( )

• Rayleigh-Jean Law

(Ultraviolet Catastrophe!)

where is the spectral emittance in SI units of W m-3

kB =1.38 x 10-23 J/K (Boltzmann constant)

4

2( ) 0Bck T

as

Particle-Wave Nature of Light

• Some phenomena can only be understood in terms

of the particle nature of light

• Photoelectric effect

• Blackbody radiation

• Light also has wave properties at the same time

• Interference

• Light has both wave-like and particle-like properties

Section 28.2

Wave-like Properties of Particles

• The notion that the properties of both classical

waves and classical particles are present at the

same time is also called wave-particle duality

• The possibility that all particles are capable of wave-

like properties was first proposed by Prince Louis de

Broglie

• De Broglie suggested in his PhD thesis (two pages

long) in 1923 that if a particle has a momentum p, its

wavelength is

• Nobel Prize in Physics in 1929

Section 28.3

h

p

Electron Interference

• To test de Broglie’s

hypothesis, an experiment

was designed to observe

interference involving

classical particles

• The experiment showed

conclusively that electrons

have wavelike properties

• The calculated wavelength

was in good agreement

with de Broglie’s theory

Section 28.3

h

p

Wavelengths of Macroscopic Particles

• From de Broglie’s equation and using the classical

expression for kinetic energy, we have

• As the mass of the particle (object) increases, its

wavelength decreases

• In principle, you could observe interference with

baseballs

• Has not yet been observed

Section 28.3

22 21 1

( ) ,2 2 2

2 ( )

pKE mv mv p mv

m m

p m KE

2 ( )

h h

p m KE

Electron Spin

• Electrons have another

quantum property that

involves their magnetic

behavior

• An electron has a

magnetic moment, a

property associated with

electron spin

• Classically, the

electron’s magnetic

moment can be thought

of as spinning ball of

charge Section 28.4

Electron Spin, cont.

• The spinning ball of

charge acts like a

collection of current

loops

• This produces a

magnetic field

• It acts like a small bar

magnet

• Therefore, it is attracted

to or repelled from the

poles of other magnets

Section 28.4

Stern-Gerlach Experiment (1920 in Germany)

The Stern-Gerlach Apparatus

Observed results

The magnetic field between the two magnetic

pole pieces is indicated by the field lines.

Electron Spin, Direction

• When a beam of electrons passes near one end of a bar magnet, there are two directions of deflection observed

• Two orientations for the electron magnetic moment are possible

• Classical theory predicts the moment may point in any direction

Section 28.4

Electron Spin, Direction, cont.

• Classically, the electrons should deflect over a range

of angles

• Observing only two directions of deflection indicates

there are only two possible orientations for the

magnetic moment

• The electron magnetic moment is quantized with

only two possible values

• Quantization of the electron’s magnetic moment

applies to both direction and magnitude

• All electrons under all circumstances act as identical

bar magnets

Section 28.4

Quantization of Electron Spin

• Classical explanation of electron spin

• Circulating charge acts as a current loop

• The current loops produce a magnetic field

• This result is called the spin magnetic moment

• You can also say the electron has spin angular

moment

• The classical ideas do not explain the two directions

after the beam of electrons pass the magnet

• Quantum explanation

• Only spin up or spin down are possible

• Other quantum particles also have spin angular

momentum and a resulting magnetic moment Section 28.4