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
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

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

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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

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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
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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

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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

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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

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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

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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

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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

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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
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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

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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

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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

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Schrödinger Equation, final

• ?(x) must be continuous
• d?/dx must also be continuous for finite values
  of the potential energy

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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

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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

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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

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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

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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

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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

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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

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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

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Finite Potential Well Graphical Results for ?2

• The probability densities for the lowest three
  states are shown
• The functions are smooth at the boundaries

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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
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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

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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

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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

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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

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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

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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

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Active Figure 41.11

• Use the active figure to vary the voltage

PLAY ACTIVE FIGURE
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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

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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,

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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