3Units of Chapter 30Blackbody Radiation and Planck’s Hypothesis of Quantized EnergyPhotons and the Photoelectric EffectThe Mass and Momentum of a PhotonPhoton Scattering and the Compton Effect
4Units of Chapter 30The de Broglie Hypothesis and Wave-Particle DualityThe Heisenberg Uncertainty PrincipleQuantum Tunneling
530-1 Blackbody Radiation and Planck’s Hypothesis of Quantized Energy An ideal blackbody absorbs all the light that is incident upon it.
630-1 Blackbody Radiation and Planck’s Hypothesis of Quantized Energy An ideal blackbody is also an ideal radiator. If we measure the intensity of the electromagnetic radiation emitted by an ideal blackbody, we find:
730-1 Blackbody Radiation and Planck’s Hypothesis of Quantized Energy This illustrates a remarkable experimental finding:The distribution of energy in blackbody radiation is independent of the material from which the blackbody is constructed — it depends only on the temperature, T.The peak frequency is given by:
830-1 Blackbody Radiation and Planck’s Hypothesis of Quantized Energy The peak wavelength increases linearly with the temperature. This means that the temperature of a blackbody can be determined by its color.Classical physics calculations were completely unable to produce this temperature dependence, leading to something called the “ultraviolet catastrophe.”
930-1 Blackbody Radiation and Planck’s Hypothesis of Quantized Energy Classical predictions were that the intensity increased rapidly with frequency, hence the ultraviolet catastrophe.
1030-1 Blackbody Radiation and Planck’s Hypothesis of Quantized Energy Planck discovered that he could reproduce the experimental curve by assuming that the radiation in a blackbody came in quantized energy packets, depending on the frequency:The constant h in this equation is known as Planck’s constant:
1130-1 Blackbody Radiation and Planck’s Hypothesis of Quantized Energy Planck’s constant is a very tiny number; this means that the quantization of the energy of blackbody radiation is imperceptible in most macroscopic situations. It was, however, a most unsatisfactory solution, as it appeared to make no sense.
1230-2 Photons and the Photoelectric Effect Einstein suggested that the quantization of light was real; that light came in small packets, now called photons, of energy:
1330-2 Photons and the Photoelectric Effect Therefore, a more intense beam of light will contain more photons, but the energy of each photon does not change.
1430-2 Photons and the Photoelectric Effect The photoelectric effect occurs when a beam of light strikes a metal, and electrons are ejected.Each metal has a minimum amount of energy required to eject an electron, called the work function, W0. If the electron is given an energy E by the beam of light, its maximum kinetic energy is:
1530-2 Photons and the Photoelectric Effect This diagram shows the basic layout of a photoelectric effect experiment.
1630-2 Photons and the Photoelectric Effect Classical predictions:Any beam of light of any color can eject electrons if it is intense enough.The maximum kinetic energy of an ejected electron should increase as the intensity increases.Observations:Light must have a certain minimum frequency in order to eject electrons.More intensity results in more electrons of the same energy.
1730-2 Photons and the Photoelectric Effect Explanations:Each photon’s energy is determined by its frequency. If it is less than the work function, electrons will not be ejected, no matter how intense the beam.
1830-2 Photons and the Photoelectric Effect 2. A more intense beam means more photons, and therefore more ejected electrons.
1930-3 The Mass and Momentum of a Photon Photons always travel at the speed of light (of course!). What does this tell us about their mass and momentum?The total energy can be written:Since the left side of the equation must be zero for a photon, it follows that the right side must be zero as well.
2030-3 The Mass and Momentum of a Photon The momentum of a photon can be written:Dividing the momentum by the energy and substituting, we find:
2130-3 The Mass and Momentum of a Photon Finally, we can write the momentum of a photon in the following way:
2230-4 Photon Scattering and the Compton Effect The Compton effect occurs when a photon scatters off an atomic electron.
2330-4 Photon Scattering and the Compton Effect In order for energy to be conserved, the energy of the scattered photon plus the energy of the electron must equal the energy of the incoming photon. This means the wavelength of the outgoing photon is longer than the wavelength of the incoming one:
2430-5 The de Broglie Hypothesis and Wave-Particle Duality In 1923, de Broglie proposed that, as waves can exhibit particle-like behavior, particles should exhibit wave-like behavior as well.He proposed that the same relationship between wavelength and momentum should apply to massive particles as well as photons:
2530-5 The de Broglie Hypothesis and Wave-Particle Duality The correctness of this assumption has been verified many times over. One way is by observing diffraction. We already know that X-rays can diffract from crystal planes:
2630-5 The de Broglie Hypothesis and Wave-Particle Duality The same patterns can be observed using either particles or X-rays.
2730-5 The de Broglie Hypothesis and Wave-Particle Duality Indeed, we can even perform Young’s two-slit experiment with particles of the appropriate wavelength and find the same diffraction pattern.
2830-5 The de Broglie Hypothesis and Wave-Particle Duality This is even true if we have a particle beam so weak that only one particle is present at a time – we still see the diffraction pattern produced by constructive and destructive interference.Also, as the diffraction pattern builds, we cannot predict where any particular particle will land, although we can predict the final appearance of the pattern.
2930-5 The de Broglie Hypothesis and Wave-Particle Duality These images show the gradual creation of an electron diffraction pattern.
3030-6 The Heisenberg Uncertainty Principle The uncertainty just mentioned – that we cannot know where any individual electron will hit the screen – is inherent in quantum physics, and is due to the wavelike properties of matter.
3130-6 The Heisenberg Uncertainty Principle The width of the central maximum is given by:Therefore, it would be possible to have a narrower central peak by using light of a shorter wavelength. However, from the de Broglie relation, as the wavelength goes down, the momentum goes up:
3230-6 The Heisenberg Uncertainty Principle When the electrons diffract through the slit, they acquire a y-component of momentum that they had not had before. This leads to the uncertainty principle:If we know the position of a particle with greater precision, its momentum is more uncertain; if we know the momentum of a particle with greater precision, its position is more uncertain.
3330-6 The Heisenberg Uncertainty Principle Mathematically,
3430-6 The Heisenberg Uncertainty Principle The uncertainty principle can be cast in terms of energy and time rather than position and momentum:The effects of the uncertainty principle are generally not noticeable in macroscopic situations due to the smallness of Planck’s constant, h.
3530-7 Quantum TunnelingWaves can “tunnel” through narrow gaps of material that they otherwise would not be able to traverse. As the gap widens, the intensity of the transmitted wave decreases exponentially.
3630-7 Quantum TunnelingGiven their wavelike properties, it is not surprising that particles can tunnel as well. A practical application is the scanning tunneling microscope, which can image single atoms using the tunneling of electrons.
37Summary of Chapter 30An ideal blackbody absorbs all light incident on it. The distribution of energy within it as a function of frequency depends only on its temperature.Frequency of maximum radiation:Planck’s hypothesis:
38Summary of Chapter 30 Light is composed of photons, each with energy: In terms of wavelength:Photoelectric effect: photons eject electrons from metal surface.Minimum energy: work function, W0Minimum frequency:
39Summary of Chapter 30 Photons have zero rest mass. Photon momentum, frequency, and wavelength:Compton effect: a photon scatters off an atomic electron, and exits with a longer wavelength:
40Summary of Chapter 30de Broglie hypothesis: particles have wavelengths, depending on their momentum:Both X-rays and electrons can be diffracted by crystals.Light and matter display both wavelike and particle-like properties.
41Summary of Chapter 30The position and momentum of waves and particles cannot both be determined simultaneously with arbitrary precision:Nor can the energy and time:Particles can “tunnel” through a region that classically would be forbidden to them.