bei48482_FM

ra 0 (that is, to be between r0 and ra 0 ). Verify this by

calculating the relevant probabilities.

23. Unsöld’s theoremstates that for any value of the orbital
    quantum number l, the probability densities summed over all
    possible states from mllto mllyield a constant
    independent of angles or ; that is,
    l
    mll

 ^2  ^2 constant

This theorem means that every closed subshell atom or ion

(Sec. 7.6) has a spherically symmetric distribution of electric

charge. Verify Unsöld’s theorem for l0, l1, and l 2

with the help of Table 6.1.

6.9 Selection Rules

24. A hydrogen atom is in the 4pstate. To what state or states can
    it go by radiating a photon in an allowed transition?


25. With the help of the wave functions listed in Table 6.1 verify
    that l1 for n 2 Sn1 transitions in the hydrogen
    atom.


26. The selection rule for transitions between states in a harmonic
    oscillator is n1. (a) Justify this rule on classical grounds.
    (b) Verify from the relevant wave functions that the n 1 S
    n3 transition in a harmonic oscillator is forbidden whereas
    the n 1 Sn0 and n 1 Sn2 transitions are allowed.


27. Verify that the n 3 Sn1 transition for the particle in a
    box of Sec. 5.8 is forbidden whereas the n 3 Sn2 and
    n 2 Sn1 transitions are allowed.

6.10 Zeeman Effect

28. In the Bohr model of the hydrogen atom, what is the magni-
    tude of the orbital magnetic moment of an electron in the
    nth energy level?


29. Show that the magnetic moment of an electron in a Bohr orbit
    of radius rnis proportional to rn.


30. Example 4.7 considered a muonic atom in which a negative
    muon (m 207 me) replaces the electron in a hydrogen atom.
    What difference, if any, would you expect between the Zeeman
    effect in such atoms and in ordinary hydrogen atoms?


31. Find the minimum magnetic field needed for the Zeeman effect
    to be observed in a spectral line of 400-nm wavelength when a
    spectrometer whose resolution is 0.010 nm is used.


32. The Zeeman components of a 500-nm spectral line are
    0.0116 nm apart when the magnetic field is 1.00 T. Find the
    ratio e mfor the electron from these data.

6.6 Magnetic Quantum Number

9. Under what circumstances, if any, is Lzequal to L?


10. What are the angles between Land the zaxis for l1?
    For l2?


11. What are the possible values of the magnetic quantum number
    mlof an atomic electron whose orbital quantum number is
    l4?


12. List the sets of quantum numbers possible for an n4 hydro-
    gen atom.


13. Find the percentage difference between Land the maximum
    value of Lzfor an atomic electron in p,d, and fstates.

6.7 Electron Probability Density

14. Under what circumstances is an atomic electron’s probability-
    density distribution spherically symmetric? Why?


15. In Sec. 6.7 it is stated that the most probable value of rfor a 1s
    electron in a hydrogen atom is the Bohr radius a 0. Verify this.


16. At the end of Sec. 6.7 it is stated that the most probable value
    of rfor a 2pelectron in a hydrogen atom is 4a 0 , which is the
    same as the radius of the n2 Bohr orbit. Verify this.


17. Find the most probable value of rfor a 3delectron in a hydro-
    gen atom.


18. According to Fig. 6.11, P drhas twomaxima for a 2selectron.
    Find the values of rat which these maxima occur.


19. How much more likely is the electron in a ground-state hydro-
    gen atom to be at the distance a 0 from the nucleus than at the
    distance 2a 0?


20. In Section 6.7 it is stated that the average value of rfor a 1s
    electron in a hydrogen atom is 1.5a 0. Verify this statement by
    calculating the expectation value r  r||^2 dV.


21. The probability of finding an atomic electron whose radial wave
    function is R(r) outside a sphere of radius r 0 centered on the
    nucleus is
    
    r 0
    R(r)^2 r^2 dr
    (a) Calculate the probability of finding a 1selectron in a hydro-
    gen atom at a distance greater than a 0 from the nucleus.
    (b) When a 1selectron in a hydrogen atom is 2a 0 from the nu-
    cleus, all its energy is potential energy. According to classical
    physics, the electron therefore cannot ever exceed the distance
    2 a 0 from the nucleus. Find the probability r> 2a 0 for a 1s
    electron in a hydrogen atom.


22. According to Fig. 6.11, a 2selectron in a hydrogen atom is
    more likely than a 2pelectron to be closer to the nucleus than

Exercises 227

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