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Chapter 7: Q44E (page 281)
How many different 3d states are there? What physical property (as opposed to quantum number) distinguishes them, and what different values may this property assume?
There are five states in the case of 3d states.
The angular momentum could be 0 or or .
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Imagine two classical charges of -q, each bound to a central charge of. +q One -q charge is in a circular orbit of radius R about its +q charge. The other oscillates in an extreme ellipse, essentially a straight line from it鈥檚 +q charge out to a maximum distance .The two orbits have the same energy. (a) Show that. (b) Considering the time spent at each orbit radius, in which orbit is the -q charge farther from its +q charge on average?
Question: Consider a cubic 3D infinite well of side length of L. There are 15 identical particles of mass m in the well, but for whatever reason, no more than two particles can have the same wave function. (a) What is the lowest possible total energy? (b) In this minimum total energy state, at what point(s) would the highest energy particle most likely be found? (Knowing no more than its energy, the highest energy particle might be in any of multiple wave functions open to it and with equal probability.)
Exercise 81 obtained formulas for hydrogen like atoms in which the nucleus is not assumed infinite, as in the chapter, but is of mass , whileis the mass of the orbiting negative charge. In positronium, an electron orbits a single positive charge, as in hydrogen, but one whose mass is the same as that of the electron -- a positron. Obtain numerical values of the ground state energy and 鈥淏ohr radius鈥� of positronium.
In Appendix G. the operator for the square of the angular momentum is shown to be
Use this to rewrite equation (7-19) as
In general, we might say that the wavelengths allowed a bound particle are those of a typical standing wave, , where is the length of its home. Given that , we would have , and the kinetic energy, , would thus be . These are actually the correct infinite well energies, for the argumentis perfectly valid when the potential energy is 0 (inside the well) and is strictly constant. But it is a pretty good guide to how the energies should go in other cases. The length allowed the wave should be roughly the region classically allowed to the particle, which depends on the 鈥渉eight鈥� of the total energy E relative to the potential energy (cf. Figure 4). The 鈥渨all鈥� is the classical turning point, where there is nokinetic energy left: . Treating it as essentially a one-dimensional (radial) problem, apply these arguments to the hydrogen atom potential energy (10). Find the location of the classical turning point in terms of E , use twice this distance for (the electron can be on both on sides of the origin), and from this obtain an expression for the expected average kinetic energies in terms of E . For the average potential, use its value at half the distance from the origin to the turning point, again in terms of . Then write out the expected average total energy and solve for E . What do you obtain
for the quantized energies?
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