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Chapter 7: Q49E (page 282)
Question: Verify the correctness of the normalization constant of the radial wave function given in Table 7.4 as
Answer
It has been proved that the normalization for the case 2pstateis correct.
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Consider a cubic 3D infinite well.
(a) How many different wave functions have the same energy as the one for which ?
(b) Into how many different energy levels would this level split if the length of one side were increased by ?
(c) Make a scale diagram, similar to Figure 3, illustrating the energy splitting of the previously degenerate wave functions.
(d) Is there any degeneracy left? If so, how might it be 鈥渄estroyed鈥�?
What is a quantum number, and how does it arise?
Verify that the solution given in equation (7.6) satisfy differential equations (7.5) as well as the required boundary conditions.
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.
Spectral lines are fuzzy due to two effects: Doppler broadening and the uncertainty principle. The relative variation in wavelength due to the first effect (see Exercise 2.57) is given by
Where T is the temperature of the sample and m is the mass of the particles emitting the light. The variation due to the second effect (see Exercise 4.72) is given by
Where, is the typical transition time
(a) Suppose the hydrogen in a star has a temperature of . Compare the broadening of these two effects for the first line in the Balmer series (i.e., ). Assume a transition time of 10-8s. Which effect is more important?
(b) Under what condition(s) might the other effect predominate?
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