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## Quantum Mechanics

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### Chapter 41 Quantum Mechanics Quantum Mechanics The theory of quantum mechanics was developed in the 1920s By Erwin Schr dinger, Werner Heisenberg and others Enables ... – PowerPoint PPT presentation

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Title: Quantum Mechanics

1
Chapter 41
• Quantum Mechanics

2
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

3
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

4
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

5
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 ?

6
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
• rj is the position of the jth particle in the
system
• ? 2pƒ is the angular frequency

7
Wave Function, cont.
• 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

8
Wave Function Interpretation Single Particle
• Y cannot be measured
• Y2 is real and can be measured
• Y2 is also called the probability density
• 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 Y 2 dV

9
• 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

10
Wave Function of a Free Particle
• The wave function of a free particle moving along
the x-axis can be written as ?(x) Aeikx
• A is the constant amplitude
• k 2p/? 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

11
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 a x b is
• and is the area under the curve

12
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

13
Expectation Values
• Measurable quantities of a particle can be
derived from ?
• Remember, ? 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

14
Expectation Values, cont.
• The expectation value of any function of x can
also be found
• The expectation values are analogous to weighted
averages

15
Summary of Mathematical Features of a Wave
Function
• ?(x) may be a complex function or a real
function, depending on the system
• ?(x) must be defined at all points in space and
be single-valued
• ?(x) must be normalized
• ?(x) must be continuous in space
• There must be no discontinuous jumps in the value
of the wave function at any point

16
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

Please replace with fig. 41.3 a
17
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

18
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 lt 0 and x gt L
• The wave function must also be 0 at the walls
• The function must be continuous
• ?(0) 0 and ?(L) 0

19
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

20
Graphical Representations for a Particle in a Box
21
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
22
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

23
Momentum of the Particle in a Box
• Remember the wavelengths are restricted to
specific values
• l 2 L / n
• Therefore, the momentum values are also
restricted

24
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

25
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

26
Active Figure 41.5
• Use the active figure to vary
• The length of the box
• The mass of the particle
• Observe the effects on the energy level diagram

PLAY ACTIVE FIGURE
27
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

28
Erwin Schrödinger
• 1887 1961
• American physicist
• Best known as one of the creators of quantum
mechanics
• His approach was shown to be equivalent to
Heisenbergs
• Also worked with
• statistical mechanics
• color vision
• general relativity

29
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

30
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

31
Schrödinger Equation, final
• ?(x) must be continuous
• d?/dx must also be continuous for finite values
of the potential energy

32
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

33
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

34
Schrödinger Equation Applied to a Particle in a
Box
• In the region 0 lt x lt 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

35
Schrödinger Equation Applied to a Particle in a
Box, cont.
• Solving for the allowed energies gives
• The allowed wave functions are given by
• These match the original results for the particle
in a box

36
Finite Potential Well
• A finite potential well is pictured
• The energy is zero when the particle is 0 lt x lt L
• In region II
• The energy has a finite value outside this region
• Regions I and III

37
Classical vs. Quantum Interpretation
• According to Classical Mechanics
• 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
• A finite probability exists that the particle can
be found outside the well even if E lt 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

38
Finite Potential Well Region II
• U 0
• The allowed wave functions are sinusoidal
• The boundary conditions no longer require that ?
be zero at the ends of the well
• The general solution will be
• ?II(x) F sin kx G cos kx
• where F and G are constants

39
Finite Potential Well Regions I and III
• The Schrödinger equation for these regions may be
written as
• The general solution of this equation is
• A and B are constants

40
Finite Potential Well Regions I and III, cont.
• In region I, B 0
• This is necessary to avoid an infinite value for
? for large negative values of x
• In region III, A 0
• This is necessary to avoid an infinite value for
? for large positive values of x
• The solutions of the wave equation become

41
Finite Potential Well Graphical Results for ?
• 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

42
Finite Potential Well Graphical Results for ?2
• The probability densities for the lowest three
states are shown
• The functions are smooth at the boundaries

43
Active Figure 41.7
• Use the active figure to adjust the length of the
box
• See the effect on the quantized states

PLAY ACTIVE FIGURE
44
Finite Potential Well Determining the Constants
• The constants in the equations can be determined
by the boundary conditions and the normalization
condition
• The boundary conditions are

45
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

46
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

47
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

48
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

49
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 nonzero
• Tunneling is observed and provides evidence of
the principles of quantum mechanics

50
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

51
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

52
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

53
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

54
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

55
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

56
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

57
Active Figure 41.11
• Use the active figure to vary the voltage

PLAY ACTIVE FIGURE
58
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

59
Simple Harmonic Oscillator
• Reconsider black body radiation as vibrating
charges acting as simple harmonic oscillators
• The potential energy is
• U ½ kx2 ½ m?2x2
• Its total energy is
• E K U ½ kA2 ½ m?2A2

60
Simple Harmonic Oscillator, 2
• The Schrödinger equation for this problem is
• The solution of this equation gives the wave
function of the ground state as

61
Simple Harmonic Oscillator, 3
• The remaining solutions that describe the excited
states all include the exponential function
• 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,

62
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 h?
• Agrees with Plancks original equations