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Goals for Chapter 39 To study the wave nature of electrons To examine the evidence for the nuclear model of the atom (Rutherford scattering) To understand the ideas of atomic energy levels and the Bohr model of the hydrogen atom To learn the fundamental physics of how lasers operate To see how the ideas of photons and atomic energy levels explain the continuous spectrum of light emitted by a blackbody To see how the Heisenberg uncertainty principle applies to the behavior of particles

Particles behaving as waves (another aspect of QM) At the end of the 19th century light was regarded as a wave and matter as a collection of particles. Just as light was found to have particle characteristics (photons), matter proved to have wave characteristics. The wave nature of matter allows us to use electrons to make images (e.g. the viruses shown here on a bacterium). This picture is the output of an “electron microscope”

The Prince of Quantum Mechanics The photoelectric effect and Compton scattering show that light waves also behave as particles. The wave nature of light is revealed by interference - the particle nature by the fact that light is detected as quanta: “photons”. Photons of light have energy and momentum given by: Prince Louis de Broglie (1923) proposed that particles also behave as waves; i.e., for all particles there is a quantum wave with a wavelength given by the same relation: But be careful c=fλ does not work for matter waves.

Application of de Broglie waves 71 pm x-rays passing through aluminum foil; 600 eV electrons passing through the same. Question: By the way, what is 71 pm in MKS units ?

Electron microscopy The wave aspect of electrons means that they can be used to form images, just as light waves can. This is the basic idea of the electron microscope What accelerating voltage is needed to provide electrons with wavelength, 10 pm =0.010 nm in an electron microscope ? Question: The non-relativistic kinetic energy of a point particle K=1/2mv2. How can we rewrite in terms of p, the momentum ?

Electron microscope example (cont’d) How can the accelerating voltage be related to the wavelength ?

Electron microscope example (cont’d) Question: This means that the electrons have energies of 15keV. How does this compare to the rest mass of the electron ? Are the electrons non-relativistic ? Ans: 15 keV<<511 keV, so the electrons are non-relativistic

Electron microscope example (conceptual question) Question: What limits the resolution of an optical microscope ? Ans: the diffraction limit What is the diffraction limit for an electron microscope ? Ans: Compare diffraction limit of10 pm (0.01 nm) to 500 nm In fact, quality of electron optics is a worse limitation for a TEM (transmission electron microscope) than diffraction.

Davisson-Germer Experiment: Electron Diffraction In 1927, Davisson and Germer accidentally discover electron diffraction at Bell Labs

Davisson-Germer Experiment: Electron Diffraction The diffraction maxima occur at Question: How does electron diffraction differ from x-ray (Bragg) diffraction ? Ans: 2d is twice the distance between planes in a crystal in Bragg; here the angle θ is measured wrt the normal.

Rutherford’s discovery of the nucleus at Manchester “It was quite the most incredible event that ever happened to me in my life. It was almost as incredible as if you had fired a 15-inch shell at a piece of tissue paper and it came back and hit you.” “Plum pudding” Graduate students Geiger and Marsden carried out the experiment. Have you heard of Geiger ?

Simulation: Scatter from a large nucleus Question: What is an α particle ?

Simulation: Hard scatter from a compact nucleus Question: Compare to the large nucleus. What is different ?

Rutherford scattering example Question: An alpha particle (charge 2e) is aimed directly at a gold nucleus (charge 79e). What minimum initial kinetic energy must the alpha particle to approach within 5.0 x 10-14m of the center of the gold nucleus before reversing direction. (Assume that the heavy gold nucleus remains at rest). Potential energy at distance of closest approach. Potential at infinity is zero.

Rutherford scattering example (cont’d) Question: An alpha particle (charge 2e) is aimed directly at a gold nucleus (charge 79e). What minimum initial kinetic energy must the alpha particle have to approach within 5.0 x 10-14m of the center of the gold nucleus before reversing direction. (Assume that the heavy gold nucleus remains at rest). Potential energy at distance of closest approach. Potential at infinity is zero.

Breakdown of classical physics (Crisis) Rutherford’s experiment suggested that electrons orbit around the nucleus like a miniature solar system. However, classical physics predicts that an orbiting electron (accelerating charge) would emit electromagnetic radiation and fall into the nucleus. So classical physics could not explain why atoms are stable. There is a ground state energy level Question: What is the solution to this crisis ?