unit12_qm_sup - PHYS 302 Unit 12 Quantum Mechanics...

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Unformatted text preview: PHYS 302: Unit 12 Quantum Mechanics Supplement – Athabasca University Unit 12: Spatial Combination of Waves After completing this Unit 12 supplement, you should be able to 1) compare interference patterns of quantum mechanical waves to those of light. In Unit 8, neither the lecture nor the assigned textbook sections mentioned diffraction and interference, but the supplementary materials on quantum mechanics (notes and Davisson’s Nobel lecture) used diffraction and interference as proof of the wave nature associated with particles. Now that we have studied these phenomena in detail, you can see why this was such strong evidence for the wave properties of electrons. This wave nature also extends to heavier particles, but unless the velocities of the particles are very low, the de Broglie wavelengths are shorter for heavy particles than for electrons. Neutrons can have their speeds reduced to very low values, and both electron and neutron diffraction have important applications. Optical microscopy, for example, is limited by the Rayleigh criterion, based on a wavelength of light of approximately 500 nm. Because electrons and neutrons can have much shorter wavelengths, they can be used for microscopy with much finer resolution of detail. Electron Diffraction Pattern In the book Great Physicists (Oxford University Press, 2004), William Cropper states that if the cathode ray experiments of the late 1800s had been done with more monoenergetic electrons, they might have shown evidence of diffraction. Just as different wavelengths of light overlap in a spectrum from a grating, so will different energies of electrons overlap in an electron diffraction pattern, effectively blurring it out. This is what took place in the early experiments: a blur of many energies of electrons mixed together. Cropper cites Max Jammer, author of The Conceptual Development of Quantum Mechanics (McGraw-Hill, 1966) as speculating what course physics might have followed if electrons had been considered to be waves when they were first discovered. We must bear in mind that in early experiments, the type of “radiation” being detected was often unclear and that X-rays were initially distinguished from electrons mainly by their lack of response to magnetic fields. A charged particle in a magnetic field follows a curved trajectory, whereas light follows a straight line. An electron has charge of -1.60217646×10-19 C (coulombs), while a photon has no charge. However, if the de Broglie wavelengths are the same, both electrons and X-rays have similar diffraction patterns. These patterns are completely explained by the diffraction formulas we have now seen, provided that we can correctly specify the centres from which the diffracting waves effectively originate. Watch Video 5: Quantum Mechanics - Chapter 5 from the Cassiopeia Project. If all particles sharing a state are identical, then a pair of particles must have a combined wave function. The two wave functions can add or subtract. Apparently, electrons and other particles called fermions must have antisymmetric wave functions. If two particles are in the same state, these wave functions add to equal zero; that is, a pair of identical electrons would interfere with each other, giving no electrons. 1 PHYS 302: Unit 12 Quantum Mechanics Supplement – Athabasca University A property called spin, which allows for the identification of electrons, has the value ½ for electrons. If you have studied chemistry, you will have learned that electrons come in pairs in each energy level. Once two electrons of opposite spin have occupied a level, there is no possibility for more electrons to be in that level—hence electrons come in pairs. The entire structure of the universe depends on this fact. Electrons must build up more complex structures, resulting in the chemical elements. How many electrons are required to make a chemical element is determined by the charge on the nucleus: enough electrons will come in to make the atom neutral, filling in higher and higher energy levels with their corresponding wave functions (in principle, we can solve for this using Schrödinger’s Equation). The photon itself has spin 1 and is a boson. Such particles can exist in the same state in unlimited numbers. This allows LASER (Light Amplification by Stimulated Emission of Radiation) to exist with unlimited intensity (in principle). Basically, if more than two electrons are in the same state, they have destructive interference of their wave functions. On the other hand, photons in the same state will exist and interfere constructively; they can exist in unlimited numbers in such a state. Chapter 6 (Cassiopeia Project) deals with topics that we don’t address in PHYS 302, but feel free to watch it if you are interested. 2 ...
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