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Chapter 28. Quantum Theory April 8 th , 2013. Quantum Theory. Two things are really different at the atomic and sub-atomic scale: Wave-particle duality Particles are waves, waves are particles Energies are not continuous but discrete. They vary by discrete increments called quanta

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Chapter 28

Quantum Theory

April 8th, 2013

quantum theory
Quantum Theory

Two things are really different at the atomic and sub-atomic scale:

  • Wave-particle duality
    • Particles are waves, waves are particles
  • Energies are not continuous but discrete. They vary by discrete increments called quanta
    • e. g. electron binding energies
  • Young’s double-slit experiment demonstrates the wave-nature of light
  • Wave-particle duality: electron are waves, they diffract like x-ray photons (1956 electron diffraction instrument)
quantum regime
Quantum Regime
  • Macroscopic-world explanations fail at the atomic-scale
    • Newtonian mechanics
    • Maxwell’s equations describing electromagnetism
  • The atomic-scale world is referred to as the quantum regime
  • Quantum refers to a very small increment, or parcel, or packet of energy
  • The discovery and development of quantum theory began in the late 1800s and continued during the early 1900s
waves vs particles
Waves vs. Particles
  • In the world of Newton and Maxwell, energy can be carried by particles and waves
  • Waves produce an interference pattern when passed through a double slit
  • Classical particles (bullets) will pass through one of the slits and no interference pattern will be formed
particles and waves classical
Particles and Waves, Classical
  • Waves exhibit interference; particles do not
  • Particles often deliver their energy in discrete amounts
  • The energy carried or delivered by a wave is not discrete
    • The energy carried by a wave is described by its intensity
    • The amount of energy absorbed depends on the intensity and the absorption time
interference with electrons
Interference with Electrons
  • The separation between waves and particles is not found in the quantum regime
  • Electrons are used in a double slit experiment
  • The blue lines show the probability of the electrons striking particular locations
interference with electrons cont
Interference with Electrons, cont.
  • The probability curve of the electrons has the same form as the variation of light intensity in the double-slit interference experiment
  • The experiment shows that electrons undergo constructive interference at certain locations on the screen
  • At other locations, the electrons undergo destructive interference
    • The probability for an electron to reach those location is very small or zero
  • The experiment also shows aspects of particle-like behavior since the electrons arrive one at a time at the screen
particles waves quanta
Particles, Waves, Quanta
  • All objects, including light and electrons, can exhibit interference
  • All objects, including light and electrons, carry energy in discrete amounts
    • These discrete parcels of energy are called quanta
work function
Work function
  • In the 1880s, Hertz discovered the work function and the photoelectric effect
  • If V is the electric potential at which electrons begin to jump across the vacuum gap, the work function is Wc = eV
  • The work function, Wc is the minimum energy required to remove a single electron from a piece of metal
  • This energy can be delivered either as electric potential or by shining light on the metal

Wc = eV

work function cont
Work Function, cont.
  • A metal contains electrons that are free to move around within the metal
  • The electrons are still bound to the metal and need energy to be removed from the metal
    • This energy is the work function
  • The value of the work function is different for different metals
photoelectric effect
Photoelectric Effect
  • Another way to extract electrons from a metal is by shining light onto it
  • Light striking a metal is absorbed by the electrons
  • If an electron absorbs an amount of light energy greater than Wc, it is ejected off the metal
  • This is called the photoelectric effect
photoelectric effect cont
Photoelectric Effect, cont.
  • No electrons are emitted unless the light’s frequency is greater than a critical value ƒc, the intensity of light does not matter
  • When the frequency is above ƒc, the kinetic energy of the emitted electrons varies linearly with the frequency, not the intensity of light
  • These results could not be explained with the classical wave theory of light
photoelectric effect problems
Photoelectric Effect, Problems
  • Trying to explain the photoelectric effect with the classical wave theory of light presented two difficulties:
    • Experiments showed that the critical frequency is independent of the intensity of the light
      • Classically, the energy is proportional to the intensity
      • It should always be possible to eject electrons by increasing the intensity to a sufficiently high value
      • Yet it is observable that below the critical frequency, there are no ejected electrons no matter how great the light intensity
    • The kinetic energy of an ejected electron is independent of the light intensity
      • Classical theory predicts that increasing the intensity will cause the ejected electrons to have a greater kinetic energy
      • Yet experiments show that the electron kinetic energy depends on the frequency of light, not at all on its intensity
photoelectric effect solution photons
Photoelectric Effect, solution: Photons
  • Einstein proposed that light carries energy in discrete quanta, now called photons
  • Each photon carries a quantum of energy Ephoton = hƒ
    • h is a constant of nature called Planck’s constant
    • h = 6.626 x 10-34 J ∙ s
  • A beam of light should be thought of as a collection of photons
    • Each photon has an energy dependent on its frequency
  • If the intensity of monochromatic light is increased, the number of photons increases, but the energycarried by each photon does not change
  • Vending machine analogy
photoelectric effect solution photons1
Photoelectric Effect, solution: Photons
  • The introduction of photons accounts for all the problems with the classical explanation
    • The absorption of light by an electron is just like a collision between two particles, a photon and an electron
      • The photon carries an energy that is absorbed by the electron
      • If this energy is less than the work function, the electron is not able to escape from the metal
      • The energy of a single photon depends on frequency but not on the light intensity
photoelectric effect solution photons2
Photoelectric Effect, solution: Photons
  • The kinetic energy of the ejected electrons depends on light frequency but not intensity
    • The critical frequency corresponds to photons whose energy is equal to the work function

h ƒc = Wc

    • This electron is ejected with 0 kinetic energy
    • If the photon has a higher frequency, the difference goes into kinetic energy of the ejected electron

KEelectron = h ƒ - h ƒc = h ƒ - Wc

    • This linear relationship is what was observed experimentally

Photoelectric EffectNobel prizeAlbert Einstein 1921

(his discovery was in 1905)

KEelectron = h ƒ - h ƒc = h ƒ– Wc

With his explanation of the photoelectric effect, Einstein introduced the idea that

light is made of particles, now called photons,and that their energies are quantized.

We now say that:

a photon is a quantum of light

momentum of a photon
Momentum of a Photon
  • A light wave with energy E also carries a certain momentum
  • Particles of light called photonscarry a discrete amount of both energy and momentum
  • Photons have two properties that are different from classical particles
    • Photons do not have any mass
    • Photons exhibit interference effects
blackbody radiation
Blackbody Radiation
  • Blackbody radiation is emitted over a range of wavelengths
  • To the eye, the color of the cavity is determined by the wavelength at which the radiation intensity is largest
blackbody radiation classical
Blackbody Radiation, Classical
  • The blackbody intensity curve has the same shape for a wide variety of objects
  • Electromagnetic waves form standing waves as they reflect back and forth inside the oven’s cavity
  • The frequencies of the standing waves follow the pattern ƒn = n ƒ where n = 1, 2, 3, …
  • There is no limit to the value of n, so the frequency can be infinitely large
  • But as the frequency increases, so does the energy
  • Classical theory predicts that the blackbody intensity should become infinite as the frequency approaches infinity. This is nonsense!
blackbody radiation and quanta
Blackbody Radiation and Quanta
  • The disagreement between the classical predictions and experimental observations was called the “ultraviolet catastrophe”
  • Planck proposed solving the problem by assuming the energy in a blackbody cavity must come in discrete quanta
  • Each parcel would have energy E = h ƒn
  • His theory fit the experimental results, but gave no reason why it worked
  • Planck’s work is generally considered to be the beginning of quantum theory
particle wave nature of light
Particle-Wave Nature of Light
  • Some phenomena can only be understood in terms of the particle nature of light
    • Photoelectric effect
    • Blackbody radiation
  • Light also has wave properties at the same time
    • Interference
  • Light has both wave-like and particle-like properties
wave like duality
Wave-like Duality
  • The notion that the properties of both classical waves and classical particles are present at the same time is also called wave-particle duality and it is essential for understanding the micro-scale world
  • All particles at all scales are capable of wave-like properties, as first proposed by Louis de Broglie
  • De Broglie suggested that if a particle has a momentum p, its wavelength is
  • Even baseballs, although those would have a wavelength, albeit a very small one (e.g. 10-34m)!
electrons are waves
Electrons are waves!
  • To test de Broglie’s hypothesis, an experiment was designed by Davisson-Germer to observe interference of electrons
  • The experiment showed conclusively that electrons have wavelike properties: the diffraction pattern they form is identical to that obtained by x-ray photons.
  • The calculated wavelength was in good agreement with de Broglie’s theory
wavelengths of macroscopic particles
Wavelengths of Macroscopic Particles
  • From de Broglie’s equation and using the classical expression for kinetic energy
  • As the mass of the particle (object) increases, its wavelength decreases
  • In principle, you could observe interference with baseballs
    • Has not yet been observed
problem 28 36
Problem 28.36

An electron and a neutron have the same wavelength. What is the ration of (a) their kinetic energies and (b) their momenta? Assume the speeds are low enough that you can ignore relativity.