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Chapter 40
Introduction to
Quantum Physics
Need for Quantum Physics 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 mechanics 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):
P = AeT4
The peak of the wavelength distribution shifts to shorter wavelengths as the temperature increases Wien’s displacement law maxT = 2.898 x 10-3 m.K
Stefan’s Law – Details P = AeT4
P is the power is the Stefan-Boltzmann constant
= 5.670 x 10-8 W / m2 . K4
Stefan’s law can be written in terms of intensity I = P/A = T4
For a blackbody, where e = 1
Wien’s Displacement Law
maxT = 2.898 x 10-3 m.K max is the wavelength at which the curve
peaks T is the absolute temperature
The wavelength is inversely proportional to the absolute temperature As the temperature increases, the peak is
“displaced” to shorter wavelengths
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
(SLIDESHOW MODE ONLY)
Rayleigh-Jeans Law An early classical attempt to explain
blackbody radiation was the Rayleigh-Jeans law
At long wavelengths, the law matched experimental results fairly well
4
2I , Bπck T
λ Tλ
Rayleigh-Jeans Law, cont. 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
Max Planck 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 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 n is a positive integer called the quantum number h is Planck’s constant ƒ is the frequency of oscillation
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
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
where E is the energy of the state BE k Te
Planck’s Model, Graphs
Active Figure 40.7
(SLIDESHOW MODE ONLY)
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 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
(SLIDESHOW MODE ONLY)
Photoelectric Effect, Results At large values of V, 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 V is negative, the current drops
When V is equal to or more negative than Vs, the current is zero
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 (Vs)
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 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
Wavelengths greater than c incident on a material having a work function φ do not result in the emission of photoelectrons
ƒcc
c hcλ
φ
Arthur Holly Compton 1892 - 1962 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 dealt 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
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 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' cosoe
hλ λ θ
m c
Compton Wavelength
The unshifted wavelength, λo, is caused by x-rays scattered from the electrons that are tightly bound to the target atoms
The Compton wavelength is
0002 43 nm.e
h
m c
Photons and Waves Revisited 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 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 hλ
p mv
Frequency of a Particle In an analogy with photons, de Broglie
postulated that a particle would also have a frequency associated with it
These equations present the dual nature of matter particle nature, m and v wave nature, λ and ƒ
ƒE
h
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
depends 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
Active Figure 40.20
(SLIDESHOW MODE ONLY)
Ideal Particle vs. Ideal Wave An ideal particle has zero size
Therefore, it is localized in space An ideal wave has a single frequency
and is infinitely long 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
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 40.21
(SLIDESHOW MODE ONLY)
Speeds Associated with Wave Packet
The phase speed of a wave in a wave packet is given by
This is the rate of advance of a crest on a single wave
The group speed is given by
This is the speed of the wave packet itself
phaseωv k
gdωv dk
Speeds, cont. The group speed can also be expressed
in terms of energy and momentum
This indicates that the group speed of the wave packet is identical to the speed of the particle that it is modeled to represent
2 1
22 2g
dE d pv p u
dp dp m m
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 40.23
(SLIDESHOW MODE ONLY)
Electron Diffraction, Equations A minimum occurs when
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
sin or sin 2 2
λ hd θ θ θ
pd
Electron Diffraction, Closed Slits If one slit is closed, the
maximum is centered around the opening
Closing the other slit produces another maximum centered around that opening
The total effect is the blue line It is completely different from
the interference pattern (red curve)
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 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 x and a simultaneous measurement of its x component of momentum is made with uncertainty p, the product of the two uncertainties can never be smaller than
2xx p
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
This suggests that energy conservation can appear to be violated by an amount E as long as it is only for a short time interval t
2E t