Transcript chapter28 - Serveftp.com

Chapter 28
Quantum Physics
Fig. 28-CO,2p. 937
28.1 Simplification Models

Particle Model


Systems and rigid objects



Allowed us to ignore unnecessary details of an
object when studying its behavior
Extension of particle model
Wave Model
Two new models


Quantum particle
Quantum particle under boundary conditions
3
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
4
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
5
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
6
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
Fig 28.1
7
Blackbody Experiment Results

The total power of the emitted radiation
increases with temperature




Stefan’s Law
P = s 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
lmax T = 2.898 x 10-3 m.K
8
Stefan’s Law – Details

P = s Ae T4


P is the power
s is the Stefan-Boltzmann constant


s = 5.670 x 10-8 W / m2 . K4
Was studied in Chapter 17
9
Wien’s Displacement Law

lmax T = 2.898 x 10-3 m.K



lmax 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
10
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
Fig 28.2
11
Active Figure
28.2
• If you can't see the image above, please install Shockwave Flash Player.
• If this active figure can’t auto-play, please click right button, then click play.
NEXT
12
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
Fig 28.3
13
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
14
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
15
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
16
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
17
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
Fig 28.4
18
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
19
20p. 940
Fig. 28-5,
21
22
23
24
25
26
27
28.2 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
28
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)
Fig 28.7
29
Active Figure
28.7
• If you can't see the image above, please install Shockwave Flash Player.
• If this active figure can’t auto-play, please click right button, then click play.
NEXT
30
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
Fig 28.8
31
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
32
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
33
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
34
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
35
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
36
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 – f


f is called the work function
The work function represents the minimum energy
with which an electron is bound in the metal
37
Some Work Function Values
38
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
39
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
40
Photon Model Explanation of
the Photoelectric Effect, final

Dependence of photoelectron kinetic
energy on light frequency


Since Kmax = hƒ – f
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
41
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
Fig 28.9
42
Cutoff Frequency
and Wavelength


The cutoff frequency is related to the work
function through ƒc = f / h
The cutoff frequency corresponds to a cutoff
wavelength
c
hc
lc 

ƒc
f

Wavelengths greater than lc incident on a
material having a work function f do not
result in the emission of photoelectrons
43
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
44
45p. 946
Fig. 28-10,
46
47
28.3 Arthur Holly Compton




1892 - 1962
Director at the lab of
the University of
Chicago
Discovered the
Compton Effect
Shared the Nobel
Prize in 1927
48
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
49
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 xrays 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
50
Compton Effect, Observations

Compton’s
experiments
showed that, at any
given angle, only
one frequency of
radiation is
observed
Fig 28.11
51
52p. 948
Fig. 28-12,
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 photonelectron are conserved


Adopted a particle model for a well-known wave
This scattering phenomena is known as the
Compton Effect
53
Compton Shift Equation


The graphs show the
scattered x-ray for various
angles
The shifted peak, l', is caused
by the scattering of free
electrons
h
l ' lo 
1  cos 
me c

This is called the Compton shift
equation
Fig 28.13
54
Compton Wavelength



The unshifted wavelength, lo, is caused
by x-rays scattered from the electrons
that are tightly bound to the target
atoms
The shifted peak, l', is caused by x-rays
scattered from free electrons in the
target
The Compton wavelength is
h
 0.00243 nm
me c
55
56
57
28.4 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
58
28.5 Louis de Broglie



1892 – 1987
Originally studied
history
Was awarded the
Nobel Prize in 1929
for his prediction of
the wave nature of
electrons
59
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
l 
p mv
60
Frequency of a Particle


In an analogy with photons, de Broglie
postulated that particles would also have a
frequency associated with them
E
ƒ
h
These equations present the dual nature of
matter


particle nature, m and v
wave nature, l and ƒ
61
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
62
63
64
65
66
67
68
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
Fig 28.14
69
70p. 953
Fig. 28-14b,
28.6 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
71
72p. 954
Fig. 28-15,
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
73
74p. 955
Fig. 28-16,
Active Figure
28.16
• If you can't see the image above, please install Shockwave Flash Player.
• If this active figure can’t auto-play, please click right button, then click play.
NEXT
75
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
76
Wave Envelope
Fig 28.17


The blue line represents the envelope
function
This envelope can travel through space with
a different speed than the individual waves
77
Active Figure
28.17
• If you can't see the image above, please install Shockwave Flash Player.
• If this active figure can’t auto-play, please click right button, then click play.
NEXT
78
Speeds Associated
with Wave Packet

The phase speed of a wave in a wave
packet is given by
v phase  


k
This is the rate of advance of a crest on a
single wave
The group speed is given by
v g  d

dk
This is the speed of the wave packet itself
79
Speeds, cont

The group speed can also be expressed
in terms of energy and momentum
dE d  p2 
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
80
28.18 Electron Diffraction, SetUp
Fig 28.18
81
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
82
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 wavelike behavior
The interference pattern becomes
clearer as the number of
electrons reaching the screen
increases
Fig 28.19
83
Active Figure
28.19
• If you can't see the image above, please install Shockwave Flash Player.
• If this active figure can’t auto-play, please click right button, then click play.
NEXT
84
Electron Diffraction, Equations

A minimum occurs when
d sin 

l
2
or
h
sin   
2 px d
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
85
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)
Fig 28.20
86
87p. 958
Fig. 28-20,
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
88
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
89
28.8 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
90
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 Dp, the product of the
two uncertainties can never be smaller
than

DxDp x 
2
91
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
92
Heisenberg Uncertainty
Principle, Another Form


Another form of the Uncertainty
Principle can be expressed in terms of
energy and time

DEDt 
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
93
94
95
28.9 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 N
 I
V
V
96
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
I E

2
Combining the points of view gives
Probability
 E2
V
97
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 Y
98
Wave Function

The complete wave function Y for a
system depends on the positions of all
the particles in the system and on time

The function can be written as
Y(r1, r2, … rj…., t) = y(rj)e-it



rj is the position of the jth particle in the system
 = 2 p ƒ is the angular frequency
i  1
99
Wave Function, cont


The wave function is often complex-valued
The absolute square |y|2 = y*y is always real
and positive



y* is the complete conjugate of y
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
100
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
101
Wave Function
of a Free Particle

The wave function of a free particle moving
along the x-axis can be written as y(x) = Aeikx



k = 2 p / l is the angular wave number of the wave
representing the particle
A is the constant amplitude
If y represents a single particle, |y|2 is the
relative probability per unit volume that the
particle will be found at any given point in the
volume

|y|2 is called the probability density
102
Wave Function
of a Free Particle, Cont



In general, the probability
of finding the particle in a
volume dV is |y|2 dV
With one-dimensional
analysis, this becomes
|y|2 dx
The probability of finding
the particle in the arbitrary
interval axb is
Pab  a y 2 dx
b
and is the area under the
curve
Fig 28.21
103
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

2
Pab   y dx  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
104
Expectation Values



y is not a measurable quantity
Measurable quantities of a particle can
be derived from y
The average position is called the
expectation value of x and is defined
as

x   y * xydx
105
Expectation Values, cont

The expectation value of any function of
x can also be found
f x    y * f x ydx


The expectation values are analogous to
averages
106
28.10 Particle in a Box

A particle is confined to a
one-dimensional region of
space



The “box” is onedimensional
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
Fig 28.22
107
Wave Function
for the Particle in a Box

Since the walls are impenetrable, there
is zero probability of finding the particle
outside the box


y(x) = 0 for x < 0 and x > L
The wave function must also be 0 at the
walls


The function must be continuous
y(0) = 0 and y(L) = 0
108
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
Fig 28.22
109
Wave Function of a Particle
in a Box – Mathematical

The wave function can be expressed as
a real, sinusoidal function
 2p x 
y ( x )  A sin 

 l 

Applying the boundary conditions and
using the de Broglie wavelength
 np x 
y ( x )  A sin 

 L 
110
Graphical Representations
for a Particle in a Box
Fig 28.23
111
Active Figure
28.23
• If you can't see the image above, please install Shockwave Flash Player.
• If this active figure can’t auto-play, please click right button, then click play.
NEXT
112
Wave Function of the
Particle in a Box, cont




Only certain wavelengths for the particle
are allowed
|y|2 is zero at the boundaries
|y|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
113
Momentum of the
Particle in a Box


Remember the wavelengths are
restricted to specific values
Therefore, the momentum values are
also restricted
h
nh
p 
l 2L
114
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
 h2  2
En  
n
2 
 8mL 

n  1, 2, 3
The energy of the particle is quantized
115
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
Fig 28.24
116
Active Figure
28.24
• If you can't see the image above, please install Shockwave Flash Player.
• If this active figure can’t auto-play, please click right button, then click play.
NEXT
117
118
119
120
121
122
28.11 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
123
28.12 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
124
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
h2 d 2y

 Uy  Ey
2
2m dx

This is called the time-independent
Schrödinger equation
125
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
126
Schrödinger Equation, final


y(x) must be continuous
y(x) must approach zero as x
approaches ±


This is needed so that y(x) obeys the
normalization condition
dy / dx must also be continuous for
finite values of the potential energy
127
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
128
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
129
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
d 2y
2mE
2


y


k
y
2
2
dx

The most general solution to the equation is
y(x) = A sin kx + B cos kx

A and B are constants determined by the
boundary and normalization conditions
130
Schrödinger Equation Applied
to a Particle in a Box, cont.

Solving for the allowed energies gives
 h2  2
En  
n
2 
 8mL 

The allowed wave functions are given by
2
 np x 
 np x 
y ( x )  A sin 

sin 


L
L
L






The second expression is the normalized wave
function
These match the original results for the particle in
a box
131
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
132
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
Fig 28.25
133
134
135
28.13 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
Fig 28.26
136
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
137
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
138
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
139
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 nucleusalpha particle system
Nuclear fusion

Protons can tunnel through the barrier
caused by their mutual electrostatic
repulsion
140
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
141
142
143
144
28.14 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
145
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
3K
146
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
Fig 28.27
147