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Chapter 7: Q38E (page 281)
A particle orbiting due to an attractive central force has angular momentum What z-components of angular momentum is it possible to detect?
The possible z-components of angular momentum are,
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An electron is in anI = 3state of the hydrogen atom, what possible angles might the angular momentum vector make with the z-axis.
A mathematical solution of the azimuthal equation (7-22) is , which applies when is negative, (a) Show that this simply cannot meet itself smoothly when it finishes a round trip about the z-axis. The simplest approach is to consider and . (b) If were , equation (7-22) would say simply that the second derivative of is . Argue than this too leads to physically unacceptable solution, except in the special case of being constant, which is covered by the , case of solutions (7-24).
Exercise 80 discusses the idea of reduced mass. When two objects move under the influence of their mutual force alone, we can treat the relative motion as a one particle system of mass . Among other things, this allows us to account for the fact that the nucleus in a hydrogen like atom isn鈥檛 perfectly stationary, but in fact also orbits the centre of mass. Suppose that due to Coulomb attraction, an object of mass and charge orbits an object of mass and charge +Ze . By appropriate substitution into formulas given in the chapter, show that (a) the allowed energies are , where is the hydrogen ground state, and (b) the 鈥淏ohr Radius鈥� for this system is ,where is the hydrogen Bohr radius.
Residents of flatworld-a two-dimensional world far, far away-have it easy. Although quantum mechanics of course applies in their world, the equations they must solve to understand atomic energy levels involve only two dimensions. In particular, the Schrodinger equation for the one-electron flatrogen atom is
(a) Separate variables by trying a solution of the form , then dividing by . Show that the equation can be written
Here, is the separation constant.
(b) To be physically acceptable, must be continuous, which, since it involves rotation about an axis, means that it must be periodic. What must be the sign of C ?
(c) Show that a complex exponential is an acceptable solution for .
(d) Imposing the periodicity condition find allowed values of .
(e) What property is quantized according of C .
(f) Obtain the radial equation.
(g) Given that , show that a function of the form is a solution but only if C certain one of it, allowed values.
(h) Determine the value of a , and thus find the ground-state energy and wave function of flatrogen atom.
To conserve momentum, an atom emitting a photon must recoil, meaning that not all of the energy made available in the downward jump goes to the photon. (a) Find a hydrogen atom's recoil energy when it emits a photon in a n = 2 to n = 1 transition. (Note: The calculation is easiest to carry out if it is assumed that the photon carries essentially all the transition energy, which thus determines its momentum. The result justifies the assumption.) (b) What fraction of the transition energy is the recoil energy?
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