chapter 28 quantum physics. simplification models particle model allowed us to ignore unnecessary...
TRANSCRIPT
Chapter 28
Quantum Physics
Simplification Models Particle Model
Allowed us to ignore unnecessary details of an object when studying its behavior
Systems and rigid objects Extension of particle model
Wave Model Two new models
Quantum particle Quantum particle under boundary conditions
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 nature of the radiation leaving the cavity through the hole depends only on the temperature of the cavity walls
Blackbody Experiment Results The total power of the emitted radiation
increases with temperature Stefan’s Law P = A e T4
For a blackbody, e = 1 The peak of the wavelength distribution shifts
to shorter wavelengths as the temperature increases Wien’s displacement law max T = 2.898 x 10-3 m.K
Stefan’s Law – Details P = Ae T4
P is the power is the Stefan-Boltzmann constant
= 5.670 x 10-8 W / m2 . K4
Was studied in Chapter 17
Wien’s Displacement Law
max T = 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
Ultraviolet Catastrophe At short wavelengths,
there was a major disagreement between classical theory and experimental results for black body radiation
This mismatch became known as the ultraviolet catastrophe You would have infinite
energy as the wavelength approaches zero
Max Planck 1858 – 1947 He 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 structural model
for blackbody radiation that leads to an equation in agreement with the experimental results
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 = n h ƒ 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 only in
discrete units They do this 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
Correspondence Principle Quantum results must blend smoothly with
classical results when the quantum number becomes large Quantum effects are not seen on an everyday
basis since the energy change is too small a fraction of the total energy
Quantum effects are important and become measurable only on the submicroscopic level of atoms and molecules
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 The effect was first discovered by Hertz
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 (E) and collected at the positive plate (C)
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 current goes to zero at the same negative voltage
for all intensity curves
Photoelectric Effect Feature 2 Time interval between incidence of light and
ejection of photoelectrons Classical Prediction
For very weak light, 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 Less than 10-9 s
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 – 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 The intensity will change the number of photoelectrons
being emitted, but not the energy of an individual electron
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 – 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 absolute value of the
y-intercept is the work function
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 = / h The cutoff frequency corresponds to a cutoff
wavelength
Wavelengths greater than c incident on a material having a work function do not result in the emission of photoelectrons
ƒcc
c hc
Applications of the Photoelectric Effect Detector in the light meter of a camera Phototube
Used in burglar alarms and soundtrack of motion picture films
Largely replaced by semiconductor devices Photomultiplier tubes
Used in nuclear detectors and astronomy
Arthur Holly Compton 1892 - 1962 Director at the lab of
the University of Chicago
Discovered the Compton Effect
Shared the Nobel Prize in 1927
The Compton Effect, Introduction Compton and coworkers dealt with Einstein’s
idea of photon momentum Einstein proposed a photon with energy E carries
a momentum of E/c = hƒ / c Compton and others 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,
electromagnetic waves of frequency ƒo incident on electrons should Accelerate in the direction of propagation of the x-
rays by radiation pressure Oscillate at the apparent frequency of the radiation
since the oscillating electric field should set the electrons in motion
Overall, the scattered wave frequency at a given angle should be a distribution of Doppler-shifted values
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 and momentum hƒ / c
Assume the energy and momentum of the isolated system of the colliding photon-electron are conserved Adopted a particle model for a well-known wave
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, ', is caused by the scattering of free electrons
This is called the Compton shift equation
' 1 cosoe
h
m c
Compton Wavelength The unshifted wavelength, o, is caused
by x-rays scattered from the electrons that are tightly bound to the target atoms
The shifted peak, ', is caused by x-rays scattered from free electrons in the target
The Compton wavelength is 0.00243
e
hnm
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
Light has a dual nature in that it exhibits both wave and particle characteristics
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 h
p mv
Frequency of a Particle In an analogy with photons, de Broglie
postulated that particles would also have a frequency associated with them
These equations present the dual nature of matter particle nature, m and v wave nature, and ƒ
ƒE
h
Davisson-Germer Experiment If particles have a wave nature, then
under the correct conditions, they should exhibit diffraction effects
Davission and Germer measured the wavelength of electrons
This provided experimental confirmation of the matter waves proposed by de Broglie
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
simplification model that is a result of the recognition of the dual nature of light and of material particles
In this model, entities have both particle and wave characteristics
We much choose one appropriate behavior in order to understand a particular phenomenon
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
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
phasev k
gdv 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 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
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 sin2 2 x
hd or
p d
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 (brown 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
2px x
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
2tE
Probability – A Particle Interpretation From the particle point of view, the
probability per unit volume of finding a photon in a given region of space at an instant of time is proportional to the number N of photons per unit volume at that time and to the intensity
Probability
V
NI
V
Probability – A Wave Interpretation From the point of view of a wave, the
intensity of electromagnetic radiation is proportional to the square of the electric field amplitude, E
Combining the points of view gives
2Probability
VE
2I E
Probability – Interpretation Summary For electromagnetic radiation, the probability
per unit volume of finding a particle associated with this radiation is proportional to the square of the amplitude of the associated em wave The particle is the photon
The amplitude of the wave associated with the particle is called the probability amplitude or the wave function The symbol is
Wave Function The complete wave function for a
system depends on the positions of all the particles in the system and on time The function can be written as (r1, r2, … rj…., t) = (rj)e-it
rj is the position of the jth particle in the system = 2 ƒ is the angular frequency 1i
Wave Function, con’t The wave function is often complex-valued The absolute square ||2 = is always real
and positive * is the complete conjugate of It is proportional to the probability per unit volume
of finding a particle at a given point at some instant
The wave function contains within it all the information that can be known about the particle
Wave Function, General Comments, Final The probabilistic interpretation of the
wave function was first suggested by Max Born
Erwin Schrödinger proposed a wave equation that describes the manner in which the wave function changes in space and time This Schrödinger Wave Equation represents a
key element in quantum mechanics
Wave Function of a Free Particle The wave function of a free particle moving
along the x-axis can be written as (x) = Aeikx
k = 2 is the angular wave number of the wave representing the particle
A is the constant amplitude If represents a single particle, ||2 is the
relative probability per unit volume that the particle will be found at any given point in the volume ||2 is called the probability density
Wave Function of a Free Particle, Cont In general, the probability
of finding the particle in a volume dV is ||2 dV
With one-dimensional analysis, this becomes ||2 dx
The probability of finding the particle in the arbitrary interval axb is
and is the area under the curve
b
a
2
ab dxP
Wave Function of a Free Particle, Final Because the particle must be
somewhere along the x axis, the sum of all the probabilities over all values of x must be 1
Any wave function satisfying this equation is said to be normalized
Normalization is simply a statement that the particle exists at some point in space
1dxP 2
ab
Expectation Values is not a measurable quantity Measurable quantities of a particle can
be derived from The average position is called the
expectation value of x and is defined as
dxx*x
Expectation Values, cont The expectation value of any function of
x can also be found
The expectation values are analogous to averages
dxxf*xf
Particle in a Box A particle is confined to a
one-dimensional region of space The “box” is one-
dimensional The particle is bouncing
elastically back and forth between two impenetrable walls separated by L
Classically, the particle’s momentum and kinetic energy remain constant
Wave Function for the Particle in a Box Since the walls are impenetrable, there
is zero probability of finding the particle outside the box (x) = 0 for x < 0 and x > L
The wave function must also be 0 at the walls The function must be continuous (0) = 0 and (L) = 0
Potential Energy for a Particle in a Box As long as the particle is
inside the box, the potential energy does not depend on its location We can choose this
energy value to be zero The energy is infinitely
large if the particle is outside the box This ensures that the
wave function is zero outside the box
Wave Function of a Particle in a Box – Mathematical The wave function can be expressed as
a real, sinusoidal function
Applying the boundary conditions and using the de Broglie wavelength
2( ) sin
xx A
( ) sinn x
x AL
Graphical Representations for a Particle in a Box
Wave Function of the Particle in a Box, cont Only certain wavelengths for the particle
are allowed ||2 is zero at the boundaries ||2 is zero at other locations as well,
depending on the values of n The number of zero points increases by
one each time the quantum number increases by one
Momentum of the Particle in a Box Remember the wavelengths are
restricted to specific values Therefore, the momentum values are
also restricted
2
h nhp
L
Energy of a Particle in a Box We chose the potential energy of the
particle to be zero inside the box Therefore, the energy of the particle is
just its kinetic energy
The energy of the particle is quantized
22
21, 2, 3
8n
hE n n
mL
Energy Level Diagram – Particle in a Box
The lowest allowed energy corresponds to the ground state
En = n2E1 are called excited states
E = 0 is not an allowed state The particle can never be at
rest The lowest energy the
particle can have, E = 1, is called the zero-point energy
Boundary Conditions Boundary conditions are applied to determine
the allowed states of the system In the model of a particle under boundary
conditions, an interaction of a particle with its environment represents one or more boundary conditions and, if the interaction restricts the particle to a finite region of space, results in quantization of the energy of the system
In general, boundary conditions are related to the coordinates describing the problem
Erwin Schrödinger 1887 – 1961 Best known as one of
the creators of quantum mechanics
His approach was shown to be equivalent to Heisenberg’s
Also worked with statistical mechanics color vision general relativity
Schrödinger Equation The Schrödinger equation as it applies
to a particle of mass m confined to moving along the x axis and interacting with its environment through a potential energy function U(x) is
This is called the time-independent Schrödinger equation
2 2
22
h dU E
m dx
Schrödinger Equation, cont Both for a free particle and a particle in
a box, the first term in the Schrödinger equation reduces to the kinetic energy of the particle multiplied by the wave function
Solutions to the Schrödinger equation in different regions must join smoothly at the boundaries
Schrödinger Equation, final (x) must be continuous (x) must approach zero as x
approaches ± This is needed so that (x) obeys the
normalization condition d / dx must also be continuous for
finite values of the potential energy
Solutions of the Schrödinger Equation Solutions of the Schrödinger equation may be
very difficult The Schrödinger equation has been
extremely successful in explaining the behavior of atomic and nuclear systems Classical physics failed to explain this behavior
When quantum mechanics is applied to macroscopic objects, the results agree with classical physics
Potential Wells A potential well is a graphical
representation of energy The well is the upward-facing region of
the curve in a potential energy diagram The particle in a box is sometimes said
to be in a square well Due to the shape of the potential energy
diagram
Schrödinger Equation Applied to a Particle in a Box In the region 0 < x < L, where U = 0, the
Schrödinger equation can be expressed in the form
The most general solution to the equation is (x) = A sin kx + B cos kx A and B are constants determined by the
boundary and normalization conditions
22
2 2
2d mEk
dx
Schrödinger Equation Applied to a Particle in a Box, cont. Solving for the allowed energies gives
The allowed wave functions are given by
The second expression is the normalized wave function These match the original results for the particle in a box
22
28n
hE n
mL
2( ) sin sin
n x n xx A
L L L
Application – Nanotechnology Nanotechnology refers to the design and
application of devices having dimensions ranging from 1 to 100 nm
Nanotechnology uses the idea of trapping particles in potential wells
One area of nanotechnology of interest to researchers is the quantum dot A quantum dot is a small region that is grown in a
silicon crystal that acts as a potential well Storage of binary information using quantum dots is
being researched
Quantum Corral Corrals and other
structures are used to confine surface electron waves
This corral is a ring of 48 iron atoms on a copper surface
The ring has a diameter of 143 nm
Tunneling The potential energy
has a constant value U in the region of width L and zero in all other regions
This a called a square barrier
U is the called the barrier height
Tunneling, cont Classically, the particle is reflected by the
barrier Regions II and III would be forbidden
According to quantum mechanics, all regions are accessible to the particle The probability of the particle being in a classically
forbidden region is low, but not zero According to the Uncertainty Principle, the particle
can be inside the barrier as long as the time interval is short and consistent with the Principle
Tunneling, final The curve in the diagram represents a full
solution to the Schrödinger equation Movement of the particle to the far side of the
barrier is called tunneling or barrier penetration
The probability of tunneling can be described with a transmission coefficient, T, and a reflection coefficient, R
Tunneling Coefficients The transmission coefficient represents the
probability that the particle penetrates to the other side of the barrier
The reflection coefficient represents the probability that the particle is reflected by the barrier
T + R = 1 The particle must be either transmitted or reflected T e-2CL and can be non zero
Tunneling is observed and provides evidence of the principles of quantum mechanics
Applications of Tunneling Alpha decay
In order for the alpha particle to escape from the nucleus, it must penetrate a barrier whose energy is several times greater than the energy of the nucleus-alpha particle system
Nuclear fusion Protons can tunnel through the barrier
caused by their mutual electrostatic repulsion
More Applications of Tunneling – Scanning Tunneling Microscope An electrically conducting probe with a very
sharp edge is brought near the surface to be studied
The empty space between the tip and the surface represents the “barrier”
The tip and the surface are two walls of the “potential well”
The vertical motion of the probe follows the contour of the specimen’s surface and therefore an image of the surface is obtained
Cosmic Temperature In the 1940’s, a structural model of the
universe was developed which predicted the existence of thermal radiation from the Big Bang The radiation would now have a
wavelength distribution consistent with a black body
The temperature would be a few kelvins
Cosmic Temperature, cont In 1965 two workers at Bell Labs found
a consistent “noise” in the radiation they were measuring They were detecting the background
radiation from the Big Bang It was detected by their system regardless
of direction It was consistent with a back body at about
3 K
Cosmic Temperature, Final Measurements at
many wavelengths were needed
The brown curve is the theoretical curve
The blue dots represent measurements from COBE and Bell Labs