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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
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