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Chapter 8: Q3CQ (page 338)
Summarize the connection between angular momentum quantization and the stem-Gerlach experiment.
The quantized magnetic moment of the electron manifests as a splitting of the atomic beam.
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Here we consider adding two electrons to two "atoms," represented as finite wells. and investigate when the exclusion principle must be taken into account. In the accompanying figure, diagram (a) shows the four lowest-energy wave functions for a double finite well that represents atoms close together. To yield the lowest energy. the first electron added to this system must have wave function and is shared equally between the atoms. The second would al so have function and be equally shared. but it would have to be opposite spin. A third would have function B. Now consider atoms far a part diagram(b) shows, the bumps do not extend much beyond the atoms - they don't overlap-and functions and approach equal energy, as do functions and . Wave functionsandin diagram (b) describe essentially identical shapes in the right well. while being opposite in the left well. Because they are of equal energy. sums or differences ofandare now a valid alternative. An electron in a sum or difference would have the same energy as in either alone, so it would be just as "happy" inrole="math" localid="1659956864834" , or- B. Argue that in this spread-out situation, electrons can be put in one atom without violating the exclusion principle. no matter what states electrons occupy in the other atom.
Using a beam of electrons accelerated in an X-ray tube, we wish to knock an electron out of the shell of given element in a target. Section \(7.8\) gives the energies in a hydrogen like atom as . Assume that for fairly high Z , aK-shell electron can be treated as orbiting the nucleus alone.
(a) A typical accelerating potential in an X-ray tube is . In roughly how high aZcould a hole in the K -shell be produced?
(b) Could a hole be produced in elements of higher Z?
In its ground state, nitrogen's 2p electrons interact to produce . Given Hund's rule, how might the orbit at angular momenta of these three electrons combine?
Question: In classical electromagnetism, the simplest magnetic dipole is a circular current loop, which behaves in a magnetic field just as an electric dipole does in an electric field. Both experience torques and thus have orientation energies -p.Eand(a) The designation "orientation energy" can be misleading. Of the four cases shown in Figure 8.4 in which would work have to be done to move the dipole horizontally without reorienting it? Briefly explain. (b) In the magnetic case, using B and u for the magnitudes of the field and the dipole moment, respectively, how much work would be required to move the dipole a distance dx to the left? (c) Having shown that a rate of change of the "orientation energy'' can give a force, now consider equation (8-4). Assuming that B and are general, writein component form. Then, noting thatis not a function of position, take the negative gradient. (d) Now referring to the specific magnetic field pictured in Figure 8.3 which term of your part (c) result can be discarded immediately? (e) Assuming thatandvary periodically at a high rate due to precession about the z-axis what else may be discarded as averaging to 0? (f) Finally, argue that what you have left reduces to equation (8-5).
Question: Concisely, why is the table periodic?
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