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Quantum Mechanics


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

Chapter 41
  • Quantum Mechanics

Quantum Mechanics
  • The theory of quantum mechanics was developed in
    the 1920s
  • By Erwin Schrödinger, Werner Heisenberg and
  • Enables use to understand various phenomena
  • 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 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

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

Wave Function, cont.
  • The wave function is often complex-valued
  • The absolute square ?2 ?? is always real and
  • ? 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 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
  • 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

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
  • 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
  • Think of the system wave function instead of the
    particle wave function

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
  • The probability of finding the particle in the
    arbitrary interval a x b is
  • and is the area under the curve

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

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
  • The average position is called the expectation
    value of x and is defined as

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

Summary of Mathematical Features of a Wave
  • ?(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

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

Please replace with fig. 41.3 a
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 for the Particle in a Box
  • Since the walls are impenetrable, there is zero
    probability of finding the particle outside the
  • ?(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

Wave Function of a Particle in a Box
  • The wave function can be expressed as a real,
    sinusoidal function
  • Applying the boundary conditions and using the de
    Broglie wavelength

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

Wave Function of the Particle in a Box, cont.
  • Only certain wavelengths for the particle are
  • ?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
  • l 2 L / n
  • Therefore, the momentum values are also

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

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

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

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
  • American physicist
  • Best known as one of the creators of quantum
  • His approach was shown to be equivalent to
  • 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

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

Schrödinger Equation, final
  • ?(x) must be continuous
  • 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
  • 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
  • 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
  • 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

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

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

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

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

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

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

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

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

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

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

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

  • The potential energy has a constant value U in
    the region of width L and zero in all other
  • This a called a square barrier
  • U is the called the barrier height

Tunneling, cont.
  • Classically, the particle is reflected by the
  • 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

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
  • 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
  • T R 1
  • The particle must be either transmitted or
  • T ? e-2CL and can be nonzero
  • 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
  • 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

More Applications of Tunneling Resonant
Tunneling Device
  • The gallium arsenide in the center is a quantum
  • 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
  • The deformation of the potential barrier results
    in an energy level in the quantum dot
  • The resonance of energies gives the device its

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

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 m?2x2
  • Its total energy is
  • E K U ½ kA2 ½ m?2A2

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

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,

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
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