Transcript chapter41

Chapter 41
Quantum Mechanics
Quantum Mechanics

The theory of quantum mechanics was
developed in the 1920s


By Erwin Schrödinger, Werner Heisenberg and
others
Enables use to understand various
phenomena involving

Atoms, molecules, nuclei and solids
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
Probability – A Wave
Interpretation
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From the point of view of a wave, the
intensity of electromagnetic radiation is
proportional to the square of the electric field
amplitude, E
2
I E
Combining the points of view gives
Probability
 E2
V
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
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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 ψ
Wave Function
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The complete wave function ψ 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  r3   rj ,t   rj e iωt


 

rj is the position of the jth particle in the system
ω = 2πƒ is the angular frequency

i  1

Wave Function, cont.
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The wave function is often complex-valued
The absolute square |ψ|2 = ψ*ψ is always
real and positive
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ψ* is the complete conjugate of ψ
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 Interpretation –
Single Particle
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 cannot be measured
||2 is real and can be measured
||2 is also called the probability density
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The relative probability per unit volume that the particle will
be found at any given point in the volume
If dV is a small volume element surrounding some
point, the probability of finding the particle in that
volume element is
P(x, y, z) dV = | |2 dV
Wave Function, General
Comments, Final
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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
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This Schrödinger wave equation represents a key
element in quantum mechanics
Wave Function of a Free
Particle
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The wave function of a free particle moving along
the x-axis can be written as ψ(x) = Aeikx
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A is the constant amplitude
k = 2π/λ is the angular wave number of the wave
representing the particle
Although the wave function is often associated with
the particle, it is more properly determined by the
particle and its interaction with its environment

Think of the system wave function instead of the particle
wave function
Wave Function of a Free
Particle, cont.

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In general, the probability of
finding the particle in a
volume dV is |ψ|2 dV
With one-dimensional
analysis, this becomes |ψ|2
dx
The probability of finding the
particle in the arbitrary
interval a  x  b is
b
Pab   ψ dx
2
a
and is the area under the
curve
Wave Function of a Free
Particle, Final
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Because the particle must be somewhere
along the x axis, the sum of all the
probabilities over all values of x must be 1

Pab   ψ dx  1
2

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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
Expectation Values
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Measurable quantities of a particle can be
derived from ψ
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Remember, ψ is not a measurable quantity
Once the wave function is known, it is possible
to calculate the average position you would
expect to find the particle after many
measurements
The average position is called the expectation
value of x and is defined as

x   ψ * xψdx

Expectation Values, cont.

The expectation value of any function of x
can also be found

f  x    ψ * f  x  ψdx

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The expectation values are analogous to
weighted averages
Summary of Mathematical
Features of a Wave Function
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ψ(x) may be a complex function or a real
function, depending on the system
ψ(x) must be defined at all points in space
and be single-valued
ψ(x) must be normalized
ψ(x) must be continuous in space

There must be no discontinuous jumps in the
value of the wave function at any point
Particle in a Box

A particle is confined to
a one-dimensional
region of space
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The “box” is onedimensional
The particle is bouncing
elastically back and
forth between two
impenetrable walls
separated by L
Please replace
with fig. 41.3 a
Potential Energy for a Particle
in a Box
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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 for the Particle
in a Box

Since the walls are impenetrable, there is
zero probability of finding the particle outside
the box
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ψ(x) = 0 for x < 0 and x > L
The wave function must also be 0 at the walls
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The function must be continuous
ψ(0) = 0 and ψ(L) = 0
Wave Function of a Particle in
a Box – Mathematical

The wave function can be expressed as a
real, sinusoidal function
 2πx 
ψ (x )  A sin 

λ



Applying the boundary conditions and using
the de Broglie wavelength
 nπx 
ψ(x )  A sin 

L


Graphical Representations for
a Particle in a Box
Active Figure 41.4

Use the active figure to
measure the probability
of a particle being
between two points for
three quantum states
PLAY
ACTIVE FIGURE
Wave Function of the Particle
in a Box, cont.
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Only certain wavelengths for the particle are
allowed
|ψ|2 is zero at the boundaries
|ψ|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
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Remember the wavelengths are restricted to
specific values
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l=2L/n
Therefore, the momentum values are also
restricted
h nh
p 
λ 2L
Energy of a Particle in a Box

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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
Energy Level Diagram –
Particle in a Box
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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
Active Figure 41.5
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Use the active figure to
vary

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
The length of the box
The mass of the particle
Observe the effects on
the energy level
diagram
PLAY
ACTIVE FIGURE
Boundary Conditions
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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
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1887 – 1961
American physicist
Best known as one of the
creators of quantum
mechanics
His approach was shown to
be equivalent to
Heisenberg’s
Also worked with:
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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
2
d 2ψ

 Uψ  Eψ
2
2m dx

This is called the time-independent
Schrödinger equation
Schrödinger Equation, cont.
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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
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ψ(x) must be continuous
dψ/dx must also be continuous for finite
values of the potential energy
Solutions of the Schrödinger
Equation
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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
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Classical physics failed to explain this behavior
When quantum mechanics is applied to
macroscopic objects, the results agree with
classical physics
Potential Wells
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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
d 2ψ
2mE
2mE
2
  2 ψ  k ψ where k 
2
dx

The most general solution to the equation is
ψ(x) = A sin kx + B cos kx

A and B are constants determined by the
boundary and normalization conditions
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
 nπx 
ψn (x )  A sin 

L



These match the original results for the particle in
a box
Finite Potential Well
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A finite potential well is
pictured
The energy is zero when
the particle is 0 < x < L
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In region II
The energy has a finite
value outside this region
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Regions I and III
Classical vs. Quantum
Interpretation
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According to Classical Mechanics
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If the total energy E of the system is less than U, the
particle is permanently bound in the potential well
If the particle were outside the well, its kinetic energy would
be negative
 An impossibility
According to Quantum Mechanics
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A finite probability exists that the particle can be found
outside the well even if E < U
The uncertainty principle allows the particle to be outside
the well as long as the apparent violation of conservation of
energy does not exist in any measurable way
Finite Potential Well –
Region II
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U=0
The allowed wave functions are sinusoidal
The boundary conditions no longer require
that ψ be zero at the ends of the well
The general solution will be
ψII(x) = F sin kx + G cos kx

where F and G are constants
Finite Potential Well – Regions
I and III

The Schrödinger equation for these regions
may be written as
d 2ψ 2m U  E 

ψ
2
2
dx

The general solution of this equation is
ψ  Ae
Cx

 Be
Cx
A and B are constants
Finite Potential Well – Regions
I and III, cont.

In region I, B = 0


In region III, A = 0
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This is necessary to avoid an infinite value for ψ
for large negative values of x
This is necessary to avoid an infinite value for ψ
for large positive values of x
The solutions of the wave equation become
 I  Ae
Cx
for x  0 and
 III  Be
Cx
for x  L
Finite Potential Well –
Graphical Results for ψ
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
The wave functions for
various states are shown
Outside the potential well,
classical physics forbids the
presence of the particle
Quantum mechanics shows
the wave function decays
exponentially to approach
zero
Finite Potential Well –
Graphical Results for ψ2
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The probability
densities for the lowest
three states are shown
The functions are
smooth at the
boundaries
Active Figure 41.7
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
Use the active figure to
adjust the length of the
box
See the effect on the
quantized states
PLAY
ACTIVE FIGURE
Finite Potential Well –
Determining the Constants
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The constants in the equations can be
determined by the boundary conditions and
the normalization condition
The boundary conditions are
ψI  ψII
ψII  ψIII
dψI dψII
and

at x  0
dx
dx
dψII dψIII
and

at x  L
dx
dx
Application – Nanotechnology
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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
Tunneling
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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

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Regions II and III would be forbidden
According to quantum mechanics, all regions
are accessible to the particle
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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

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
The particle must be either transmitted or reflected
T  e-2CL and can be nonzero
Tunneling is observed and provides evidence of
the principles of quantum mechanics
Applications of Tunneling

Alpha decay

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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

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
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”
Scanning Tunneling
Microscope


The STM allows highly
detailed images of
surfaces with
resolutions comparable
to the size of a single
atom
At right is the surface of
graphite “viewed” with
the STM
Scanning Tunneling
Microscope, final

The STM is very sensitive to the distance
from the tip to the surface


This is the thickness of the barrier
STM has one very serious limitation



Its operation is dependent on the electrical
conductivity of the sample and the tip
Most materials are not electrically conductive at
their surfaces
The atomic force microscope overcomes this
limitation
More Applications of Tunneling –
Resonant Tunneling Device


The gallium arsenide in the center is a quantum dot
It is located between two barriers formed by the thin
extensions of aluminum arsenide
Resonance Tunneling Devices,
cont



Figure b shows the potential
barriers and the energy
levels in the quantum dot
The electron with the
energy shown encounters
the first barrier, it has no
energy levels available on
the right side of the barrier
This greatly reduces the
probability of tunneling
Resonance Tunneling Devices,
final



Applying a voltage
decreases the potential
with position
The deformation of the
potential barrier results
in an energy level in the
quantum dot
The resonance of
energies gives the
device its name
Active Figure 41.11

Use the active figure to
vary the voltage
PLAY
ACTIVE FIGURE
Resonant Tunneling Transistor




This adds a gate
electrode at the top of
the resonant tunneling
device over the
quantum dot
It is now a resonant
tunneling transistor
There is no resonance
Applying a small
voltage reestablishes
resonance
Simple Harmonic Oscillator



Reconsider black body radiation as vibrating
charges acting as simple harmonic oscillators
The potential energy is
U = ½ kx2 = ½ mω2x2
Its total energy is
E = K + U = ½ kA2 = ½ mω2A2
Simple Harmonic Oscillator, 2

The Schrödinger equation for this problem is
d 2ψ 1
2 2


mω
x ψ  Eψ
2
2m dx
2
2

The solution of this equation gives the wave
function of the ground state as
 mω 2  x2
ψ  Be
Simple Harmonic Oscillator, 3

The remaining solutions that describe the
excited states all include the exponential
function
e


Cx 2
The energy levels of the oscillator are
quantized
The energy for an arbitrary quantum number
n is En = (n + ½)w where n = 0, 1, 2,…
Energy Level Diagrams –
Simple Harmonic Oscillator



The separation between
adjacent levels are equal
and equal to DE = w
The energy levels are
equally spaced
The state n = 0 corresponds
to the ground state


The energy is Eo = ½ hω
Agrees with Planck’s
original equations