Quantum Physics

1
Chapter 27

• Quantum Physics

2
Need for Quantum Physics

• Problems remained that classical mechanics
  couldnt explain
• Blackbody radiation electromagnetic radiation
  emitted by a heated object
• Photoelectric effect emission of electrons by
  an illuminated metal
• Spectral lines emission of sharp spectral lines
  by gas atoms in an electric discharge tube

3
Blackbody Radiation

• An object at any temperature emits
  electromagnetic radiation, sometimes called
  thermal radiation
• Stefans Law describes the total power radiated
• The spectrum of the radiation depends on the
  temperature and properties of the object
• As the temperature increases, the total amount of
  energy increases and the peak of the distribution
  shifts to shorter wavelengths

4
Wiens Displacement Law

• The wavelength of the peak of the blackbody
  distribution was found to follow Weins
  displacement law
• ?max T 0.2898 x 10-2 m K
• ?max wavelength at which the curves peak
• T absolute temperature of the object emitting
  the radiation

5
The Ultraviolet Catastrophe

• Classical theory did not match the experimental
  data
• At long wavelengths, the match is good
• At short wavelengths, classical theory predicted
  infinite energy
• At short wavelengths, experiment showed no or
  little energy
• This contradiction is called the ultraviolet
  catastrophe

6
Plancks Resolution

• Planck hypothesized that the blackbody radiation
  was produced by resonators, submicroscopic
  charged oscillators
• The resonators could only have discrete energies
• En n h
• n quantum number, frequency of vibration, h
  Plancks constant, 6.626 x 10-34 J s
• Key point is quantized energy states

7
Photoelectric Effect

• Photoelectric effect (first discovered by Hertz)
  when light is incident on certain metallic
  surfaces, electrons are emitted from the surface
  photoelectrons
• The successful explanation of the effect was
  given by Einstein
• Electrons collected at C (maintained at a
  positive potential) and passing through the
  ammeter are a current in the circuit

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Photoelectric Effect

• The current increases with intensity, but reaches
  a saturation level for large DVs
• No current flows for voltages less than or equal
  to DVs, the stopping potential, independent of
  the radiation intensity
• The maximum kinetic energy of the photoelectrons
  is related to the stopping potential KEmax e
  DVs

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Features Not Explained by Classical Physics

• No electrons are emitted if the incident light
  frequency is below some cutoff frequency that is
  characteristic of the material being illuminated
• The maximum kinetic energy of the photoelectrons
  is independent of the light intensity
• The maximum kinetic energy of the photoelectrons
  increases with increasing light frequency
• Electrons are emitted from the surface almost
  instantaneously, even at low intensities

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Einsteins Explanation
KEmax h f

• Einstein extended Plancks idea of quantization
  to electromagnetic radiation
• A tiny packet of light energy a photon is
  emitted when a quantized oscillator jumps from
  one energy level to the next lower one
• The photons energy is E h
• Each photon can give all its energy to an
  electron in the metal
• The maximum kinetic energy of the liberated
  photoelectron is KEmax h f
• f is called the work function of the metal

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Einsteins Explanation
KEmax h f

• The effect is not observed below a certain cutoff
  frequency since the photon energy must be greater
  than or equal to the work function
• Without this, electrons are not emitted,
  regardless of the intensity of the light
• The maximum KE depends only on the frequency and
  the work function, not on the intensity
• The maximum KE increases with increasing
  frequency
• The effect is instantaneous since there is a
  one-to-one interaction between the photon and the
  electron

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Verification of Einsteins Theory
KEmax h f

• Experimental observations of a linear
  relationship between KE and frequency confirm
  Einsteins theory
• The x-intercept is the cutoff frequency
• The cutoff wavelength is related to the work
  function
• Wavelengths greater than lC incident on a
  material with a work function f dont result in
  the emission of photoelectrons

c ?
13
Chapter 27Problem 16

• An isolated copper sphere of radius 5.00 cm,
  initially uncharged, is illuminated by
  ultraviolet light of wavelength 200 nm. What
  charge will the photoelectric effect induce on
  the sphere? The work function for copper is 4.70
  eV.

14
X-Rays

• X-rays (discovered and named by Roentgen)
  electromagnetic radiation with short typically
  about 0.1 nm wavelengths
• X-rays have the ability to penetrate most
  materials with relative ease
• X-rays are produced when high-speed electrons are
  suddenly slowed down

15
Production of X-rays

• X-rays can be produced by electrons striking a
  metal target
• A current in the filament causes electrons to be
  emitted
• These freed electrons are accelerated toward a
  dense metal target (the target is held at a
  higher potential than the filament)

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X-ray Spectrum

• The x-ray spectrum has two distinct components
• 1) Bremsstrahlung a continuous broad spectrum,
  which depends on voltage applied to the tube
• 2) The sharp, intense lines, which depend on the
  nature of the target material

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Bremsstrahlung

• An electron passes near a target nucleus and is
  deflected from its path by its attraction to the
  nucleus
• This produces an acceleration of the electron and
  hence emission of electromagnetic radiation
• If the electron loses all of its energy in the
  collision, the initial energy of the electron is
  completely transformed into a photon
• The wavelength then is

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Bremsstrahlung

• Not all radiation produced is at this wavelength
• Many electrons undergo more than one collision
  before being stopped
• This results in the continuous spectrum produced

19
Diffraction of X-rays by Crystals

• For diffraction to occur, the spacing between the
  lines must be approximately equal to the
  wavelength of the radiation to be measured
• The regular array of atoms in a crystal can act
  as a three-dimensional grating for diffracting
  X-rays

20
Diffraction of X-rays by Crystals

• A beam of X-rays is incident on the crystal
• The diffracted radiation is very intense in the
  directions that correspond to constructive
  interference from waves reflected from the layers
  of the crystal

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Diffraction of X-rays by Crystals

• The diffraction pattern is detected by
  photographic film
• The array of spots is called a Laue pattern
• The crystal structure is determined by analyzing
  the positions and intensities of the various spots

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Braggs Law

• The beam reflected from the lower surface travels
  farther than the one reflected from the upper
  surface
• If the path difference equals some integral
  multiple of the wavelength, constructive
  interference occurs
• Braggs Law gives the conditions for constructive
  interference
• 2 d sin ? m ?
• m 1, 2, 3

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The Compton Effect

• Compton directed a beam of x-rays toward a block
  of graphite and found that the scattered x-rays
  had a slightly longer wavelength (lower energy)
  that the incident x-rays
• The change in wavelength (energy) the Compton
  shift depends on the angle at which the x-rays
  are scattered

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The Compton Effect

• Compton assumed the photons acted like other
  particles in collisions with electrons
• Energy and momentum were conserved
• The shift in wavelength is given by

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The Compton Effect

• The Compton shift depends on the scattering angle
  and not on the wavelength
• h/mec 0.002 43 nm (very small compared to
  visible light) is called the Compton wavelength

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Chapter 27Problem 33

• A 0.45-nm x-ray photon is deflected through a 23
  angle after scattering from a free electron. (a)
  What is the kinetic energy of the recoiling
  electron? (b) What is its speed?

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Photons and Electromagnetic Waves

• Light (as well as all other electromagnetic
  radiation) has a dual nature. It exhibits both
  wave and particle characteristics
• The photoelectric effect and Compton scattering
  offer evidence for the particle nature of light
  when light and matter interact, light behaves as
  if it were composed of particles
• On the other hand, interference and diffraction
  offer evidence of the wave nature of light

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Wave Properties of Particles

• In 1924, Louis de Broglie postulated that because
  photons have wave and particle characteristics,
  perhaps all forms of matter have both properties
• Furthermore, the frequency and wavelength of
  matter waves can be determined
• The de Broglie wavelength of a particle is
• The frequency of matter waves is

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Wave Properties of Particles

• The de Broglie equations show the dual nature of
  matter
• Each contains matter concepts (energy and
  momentum) and wave concepts (wavelength and
  frequency)
• The de Broglie wavelength of a particle is
• The frequency of matter waves is

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The Davisson-Germer Experiment

• Davisson and Germer scattered low-energy
  electrons from a nickel target and followed this
  with extensive diffraction measurements from
  various materials
• The wavelength of the electrons calculated from
  the diffraction data agreed with the expected de
  Broglie wavelength
• This confirmed the wave nature of
• electrons
• Other experimenters confirmed the
• wave nature of other particles

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Chapter 27Problem 40

• A monoenergetic beam of electrons is incident on
  a single slit of width 0.500 nm. A diffraction
  pattern is formed on a screen 20.0 cm from the
  slit. If the distance between successive minima
  of the diffraction pattern is 2.10 cm, what is
  the energy of the incident electrons?

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The Wave Function

• In 1926 Schrödinger proposed a wave equation that
  describes the manner in which matter waves change
  in space and time
• Schrödingers wave equation is a key element in
  quantum mechanics
• Schrödingers wave equation is generally solved
  for the wave function, ?, which depends on the
• particles position and the time
• The value of ?2 at some location at
• a given time is proportional to the
• probability of finding the particle
• at that location at that time

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The Uncertainty Principle

• When measurements are made, the experimenter is
  always faced with experimental uncertainties in
  the measurements
• Classical mechanics offers no fundamental barrier
  to ultimate refinements in measurements and would
  allow for measurements with arbitrarily small
  uncertainties
• Quantum mechanics predicts that a barrier to
  measurements with ultimately small uncertainties
  does exist

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The Uncertainty Principle

• In 1927 Heisenberg introduced the uncertainty
  principle If a measurement of position of a
  particle is made with precision ?x and a
  simultaneous measurement of linear momentum is
  made with precision ?px, then the product of the
  two uncertainties can never be smaller than h/4?
• Mathematically,
• It is physically impossible to measure
• simultaneously the exact position and the
• exact linear momentum of a particle

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The Uncertainty Principle

• Another form of the principle deals with energy
  and time
• Energy of a particle can not be measured with
  complete precision in a short interval of time Dt

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The Uncertainty Principle

• A thought experiment for viewing an electron with
  a powerful microscope
• In order to see the electron, at least one photon
  must bounce off it
• During this interaction, momentum is transferred
  from the photon to the electron
• Therefore, the light
• that allows you to
• accurately locate
• the electron changes
• the momentum of the electron

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The Uncertainty Principle
38
Chapter 27Problem 46

• (a) Show that the kinetic energy of a
  nonrelativistic particle can be written in terms
  of its momentum as KE p2/2m. (b) Use the
  results of (a) to find the minimum kinetic energy
  of a proton confined within a nucleus having a
  diameter of 1.0 10-15 m.

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Answers to Even Numbered Problems Chapter 27
Problem 12 5.4 eV
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Answers to Even Numbered Problems Chapter 27
Problem 20 (a) 8.29 10-11 m (b) 1.24
10-11 m
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Answers to Even Numbered Problems Chapter 27
Problem 24 6.7
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Answers to Even Numbered Problems Chapter 27
Problem 28 1.8 keV, 9.7 10-25 kgm/s
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• Answers to Even Numbered Problems
• Chapter 27
• Problem 34
• 1.98 10-11 m
• 1.98 10-14 m

44
Answers to Even Numbered Problems Chapter 27
Problem 52 191 MeV