Chapter 7: A6CQ (page 278)
A gas can be too cold to absorb Balmer series lines. Is this also true for the Panchen series? (See Figure 7.5.) for the Lyman series? Explain.
It is true for Paschen series but not true for Lyman series.
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We have noted that for a given energy, as lincreases, the motion is more like a circle at a constant radius, with the rotational energy increasing as the radial energy correspondingly decreases. But is the radial kinetic energy 0 for the largest lvalues? Calculate the ratio of expectation values, radial energy to rotational energy, for thestate. Use the operators
Which we deduce from equation (7-30).
Show that a transition wherecorresponds to a dipole moment in the xy-plane, while corresponds to a moment along the z-axis. (You need to consider only the -parts of therole="math" localid="1659783155213" and , which are of the form ):
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 โheightโ of the total energy E relative to the potential energy (cf. Figure 4). The โwallโ 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?
Classically, an orbiting charged particle radiates electromagnetic energy, and for an electron in atomic dimensions, it would lead to collapse in considerably less than the wink of an eye.
(a) By equating the centripetal and Coulomb forces, show that for a classical charge -e of mass m held in a circular orbit by its attraction to a fixed charge +e, the following relationship holds
.
(b) Electromagnetism tells us that a charge whose acceleration is a radiates power . Show that this can also be expressed in terms of the orbit radius as . Then calculate the energy lost per orbit in terms of r by multiplying the power by the period and using the formula from part (a) to eliminate .
(c) In such a classical orbit, the total mechanical energy is half the potential energy, or . Calculate the change in energy per change in r : . From this and the energy lost per obit from part (b), determine the change in per orbit and evaluate it for a typical orbit radius of . Would the electron's radius change much in a single orbit?
(d) Argue that dividing by P and multiplying by dr gives the time required to change r by dr . Then, sum these times for all radii from to a final radius of 0. Evaluate your result for . (One limitation of this estimate is that the electron would eventually be moving relativistically).
Taking thestates as representative, explain the relationship between the complexity numbers of nodes and antinodes-of hydrogen's standing waves in the radial direction and their complexity in the angular direction at a given value of n. Is it a direct or inverse relationship, and why?
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