Chapter 40
Introduction to
Quantum Physics
Need for Quantum Physics

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Problems remained from classical mechanics that
relativity didn’t explain
Attempts to apply the laws of classical physics to
explain the behavior of matter on the atomic scale
were consistently unsuccessful
Problems included:


Blackbody radiation
 The electromagnetic radiation emitted by a heated object
Photoelectric effect
 Emission of electrons by an illuminated metal
Quantum Mechanics
Revolution



Between 1900 and 1930, another revolution
took place in physics
A new theory called quantum mechanics was
successful in explaining the behavior of
particles of microscopic size
The first explanation using quantum theory
was introduced by Max Planck

Many other physicists were involved in other
subsequent developments
Blackbody Radiation

An object at any temperature is known to
emit thermal radiation


Characteristics depend on the temperature and
surface properties
The thermal radiation consists of a continuous
distribution of wavelengths from all portions of the
em spectrum
Blackbody Radiation, cont.


At room temperature, the wavelengths of the
thermal radiation are mainly in the infrared region
As the surface temperature increases, the
wavelength changes


It will glow red and eventually white
The basic problem was in understanding the
observed distribution in the radiation emitted by a
black body

Classical physics didn’t adequately describe the observed
distribution
Blackbody Radiation, final


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
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
Blackbody Experiment Results

The total power of the emitted radiation
increases with temperature


Stefan’s law (from Chapter 20):
 = sAeT4
The peak of the wavelength distribution shifts
to shorter wavelengths as the temperature
increases


Wien’s displacement law
lmaxT = 2.898 x 10-3 m.K
Intensity of Blackbody
Radiation, Summary


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
Active Figure 40.3


Use the active figure to
adjust the temperature
of the blackbody
Study the emitted
radiation
PLAY
ACTIVE FIGURE
Rayleigh-Jeans Law

An early classical attempt to explain
blackbody radiation was the Rayleigh-Jeans
law
2πck BT
I  λ,T  
λ4

At long wavelengths, the law matched
experimental results fairly well
Rayleigh-Jeans Law, cont.


At short wavelengths, there
was a major disagreement
between the RayleighJeans law and experiment
This mismatch became
known as the ultraviolet
catastrophe

You would have infinite
energy as the wavelength
approaches zero
Max Planck




1858 – 1847
German physicist
Introduced the concept
of “quantum of action”
In 1918 he was
awarded the Nobel
Prize for the discovery
of the quantized nature
of energy
Planck’s Theory of Blackbody
Radiation


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
In 1900 Planck developed a theory of
blackbody radiation that leads to an equation
for the intensity of the radiation
This equation is in complete agreement with
experimental observations
He assumed the cavity radiation came from
atomic oscillations in the cavity walls
Planck made two assumptions about the
nature of the oscillators in the cavity walls
Planck’s Assumption, 1

The energy of an oscillator can have only
certain discrete values En

En = nhƒ


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
n is a positive integer called the quantum number
ƒ is the frequency of oscillation
h is Planck’s constant
This says the energy is quantized
Each discrete energy value corresponds to a
different quantum state
Planck’s Assumption, 2

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
An oscillator emits or absorbs energy only when it
changes quantum states
The energy carried by the quantum of radiation is
E=hƒ
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
e E kBT where E is the energy of the state
Planck’s
Model,
Graph
Active Figure 40.7


Use the active figure to
investigate the energy
levels
Observe the emission
of radiation of different
wavelengths
PLAY
ACTIVE FIGURE
Planck’s Wavelength
Distribution Function

Planck generated a theoretical expression for
the wavelength distribution
2πhc 2
I  λ,T   5 hc λk T
B
λ e
 1


h = 6.626 x 10-34 J.s
h is a fundamental constant of nature
Planck’s Wavelength
Distribution Function, cont.


At long wavelengths, Planck’s equation
reduces to the Rayleigh-Jeans expression
At short wavelengths, it predicts an
exponential decrease in intensity with
decreasing wavelength

This is in agreement with experimental results
Photoelectric Effect

The photoelectric effect occurs when light
incident on certain metallic surfaces causes
electrons to be emitted from those surfaces

The emitted electrons are called photoelectrons
Photoelectric Effect Apparatus



When the tube is kept in the
dark, the ammeter reads
zero
When plate E is illuminated
by light having an
appropriate wavelength, a
current is detected by the
ammeter
The current arises from
photoelectrons emitted from
the negative plate and
collected at the positive
plate
Active Figure 40.9


Use the active figure to
vary frequency or place
voltage
Observe the motion of
the electrons
PLAY
ACTIVE FIGURE
Photoelectric Effect, Results




At large values of DV, the
current reaches a maximum
value
 All the electrons emitted at
E are collected at C
The maximum current
increases as the intensity of
the incident light increases
When DV is negative, the
current drops
When DV is equal to or more
negative than DVs, the
current is zero
Active Figure 40.10


Use the active figure to
change the voltage
range
Observe the current
curve for different
intensities of radiation
Photoelectric Effect Feature 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 Feature 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 Feature 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 Feature 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
Photoelectric Effect Features,
Summary

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
The experimental results contradict all four
classical predictions
Einstein extended Planck’s concept of
quantization to electromagnetic waves
All electromagnetic radiation can be
considered a stream of quanta, now called
photons
A photon of incident light gives all its energy
hƒ to a single electron in the metal
Photoelectric Effect, Work
Function


Electrons ejected from the surface of the
metal and not making collisions with other
metal atoms before escaping possess the
maximum kinetic energy Kmax
Kmax = hƒ – φ


φ is called the work function
The work function represents the minimum energy
with which an electron is bound in the metal
Some Work
Function
Values
Photon Model Explanation of
the Photoelectric Effect

Dependence of photoelectron kinetic energy
on light intensity



Kmax is independent of light intensity
K depends on the light frequency and the work
function
Time interval between incidence of light and
ejection of the photoelectron

Each photon can have enough energy to eject an
electron immediately
Photon Model Explanation of
the Photoelectric Effect, cont.

Dependence of ejection of electrons on light
frequency


There is a failure to observe photoelectric effect
below a certain cutoff frequency, which indicates
the photon must have more energy than the work
function in order to eject an electron
Without enough energy, an electron cannot be
ejected, regardless of the light intensity
Photon Model Explanation of
the Photoelectric Effect, final

Dependence of photoelectron kinetic energy
on light frequency


Since Kmax = hƒ – φ
As the frequency increases, the kinetic energy will
increase


Once the energy of the work function is exceeded
There is a linear relationship between the kinetic
energy and the frequency
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
Cutoff Frequency and
Wavelength


The cutoff frequency is related to the work
function through ƒc = φ / h
The cutoff frequency corresponds to a cutoff
wavelength
c hc
λc 

ƒc φ

Wavelengths greater than lc incident on a
material having a work function φ do not
result in the emission of photoelectrons
Arthur Holly Compton





1892 – 1962
American physicist
Director of the lab at
the University of
Chicago
Discovered the
Compton Effect
Shared the Nobel Prize
in 1927
The Compton Effect,
Introduction



Compton and Debye extended with Einstein’s
idea of photon momentum
The two groups of experimenters
accumulated evidence of the inadequacy of
the classical wave theory
The classical wave theory of light failed to
explain the scattering of x-rays from electrons
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

There should be a range of frequencies for the
scattered electrons
Compton Effect, Observations

Compton’s experiments
showed that, at any
given angle, only one
frequency of radiation is
observed
Compton Effect, Explanation



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 photonelectron 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
h
λ'  λo 
1 cos θ 
mec

This is called the
Compton shift equation
Compton Wavelength

The factor h/mec in the equation is called the
Compton wavelength and is
h
λC 
 0.002 43 nm
mec

The unshifted wavelength, λo, is caused by xrays scattered from the electrons that are
tightly bound to the target atoms
Photons and Waves Revisited

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
Some experiments are best explained by the
photon model
Some are best explained by the wave model
We must accept both models and admit that
the true nature of light is not describable in
terms of any single classical model
Also, the particle model and the wave model
of light complement each other
Louis de Broglie




1892 – 1987
French physicist
Originally studied
history
Was awarded the
Nobel Prize in 1929 for
his prediction of the
wave nature of
electrons
Wave Properties of Particles


Louis de Broglie postulated that because
photons have both wave and particle
characteristics, perhaps all forms of matter
have both properties
The de Broglie wavelength of a particle is
h
h
λ 
p mu
Frequency of a Particle

In an analogy with photons, de Broglie
postulated that a particle would also have a
frequency associated with it
E
ƒ
h

These equations present the dual nature of
matter


Particle nature, p and E
Wave nature, λ and ƒ
Davisson-Germer Experiment



If particles have a wave nature, then under
the correct conditions, they should exhibit
diffraction effects
Davisson and Germer measured the
wavelength of electrons
This provided experimental confirmation of
the matter waves proposed by de Broglie
Complementarity


The principle of complementarity states
that the wave and particle models of either
matter or radiation complement each other
Neither model can be used exclusively to
describe matter or radiation adequately
Electron Microscope



The electron microscope
relies on the wave
characteristics of electrons
The electron microscope
has a high resolving power
because it has a very short
wavelength
Typically, the wavelengths
of the electrons are about
100 times shorter than that
of visible light
Quantum Particle



The quantum particle is a new model that is
a result of the recognition of the dual nature
Entities have both particle and wave
characteristics
We must choose one appropriate behavior in
order to understand a particular phenomenon
Ideal Particle vs. Ideal Wave

An ideal particle has zero size


An ideal wave has a single frequency and is
infinitely long


Therefore, it is localized in space
Therefore,it is unlocalized in space
A localized entity can be built from infinitely
long waves
Particle as a Wave Packet




Multiple waves are superimposed so that one of its
crests is at x = 0
The result is that all the waves add constructively at
x=0
There is destructive interference at every point
except x = 0
The small region of constructive interference is
called a wave packet

The wave packet can be identified as a particle
Active Figure 40.19



Use the active figure to
choose the number of
waves to add together
Observe the resulting
wave packet
The wave packet
represents a particle
PLAY
ACTIVE FIGURE
Wave Envelope


The blue line represents the envelope function
This envelope can travel through space with a
different speed than the individual waves
Active Figure 40.20

Use the active figure to
observe the movement
of the waves and of the
wave envelope
PLAY
ACTIVE FIGURE
Speeds Associated with Wave
Packet

The phase speed of a wave in a wave packet is
given by
v phase  ω


k
This is the rate of advance of a crest on a single wave
The group speed is given by
v g  dω

dk
This is the speed of the wave packet itself
Speeds, cont.

The group speed can also be expressed in
terms of energy and momentum
2

dE d p 
1
vg 

 2p   u


dp dp  2m  2m

This indicates that the group speed of the
wave packet is identical to the speed of the
particle that it is modeled to represent
Electron Diffraction, Set-Up
Electron Diffraction,
Experiment



Parallel beams of mono-energetic electrons
that are incident on a double slit
The slit widths are small compared to the
electron wavelength
An electron detector is positioned far from the
slits at a distance much greater than the slit
separation
Electron Diffraction, cont.



If the detector collects
electrons for a long
enough time, a typical
wave interference pattern
is produced
This is distinct evidence
that electrons are
interfering, a wave-like
behavior
The interference pattern
becomes clearer as the
number of electrons
reaching the screen
increases
Active Figure 40.22


Use the active figure to
observe the
development of the
interference pattern
Observe the destruction
of the pattern when you
keep track of which slit
an electron goes
through
Please replace with
active figure 40.22
PLAY
ACTIVE FIGURE
Electron Diffraction, Equations

A maximum occurs when d sin θ  mλ


This is the same equation that was used for light
This shows the dual nature of the electron


The electrons are detected as particles at a
localized spot at some instant of time
The probability of arrival at that spot is determined
by finding the intensity of two interfering waves
Electron Diffraction Explained


An electron interacts with both slits
simultaneously
If an attempt is made to determine
experimentally which slit the electron goes
through, the act of measuring destroys the
interference pattern


It is impossible to determine which slit the electron
goes through
In effect, the electron goes through both slits

The wave components of the electron are present
at both slits at the same time
Werner Heisenberg




1901 – 1976
German physicist
Developed matrix
mechanics
Many contributions
include:



Uncertainty principle
 Rec’d Nobel Prize in 1932
Prediction of two forms of
molecular hydrogen
Theoretical models of the
nucleus
The Uncertainty Principle,
Introduction


In classical mechanics, it is possible, in
principle, to make measurements with
arbitrarily small uncertainty
Quantum theory predicts that it is
fundamentally impossible to make
simultaneous measurements of a particle’s
position and momentum with infinite accuracy
Heisenberg Uncertainty
Principle, Statement

The Heisenberg uncertainty principle
states: if a measurement of the position of a
particle is made with uncertainty Dx and a
simultaneous measurement of its x
component of momentum is made with
uncertainty Dpx, the product of the two
uncertainties can never be smaller than /2
DxDpx 
2
Heisenberg Uncertainty
Principle, Explained



It is physically impossible to measure
simultaneously the exact position and exact
momentum of a particle
The inescapable uncertainties do not arise
from imperfections in practical measuring
instruments
The uncertainties arise from the quantum
structure of matter
Heisenberg Uncertainty
Principle, Another Form

Another form of the uncertainty principle can
be expressed in terms of energy and time
DE Dt 

2
This suggests that energy conservation can
appear to be violated by an amount DE as
long as it is only for a short time interval Dt