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量子化学作业
第一章（2016-2017 学期）
1‒5. Before Planck’s theoretical work on blackbody radiation, Wien showed
empirically that (Equation 1.4)
λ max T  2.90 10 3 m K
where λmax is the wavelength at which the blackbody spectrum has its maximum value
at a temperature T. This expression is called the Wien displacement law; derive it from
Planck’s theoretical expression for the blackbody distribution by differentiating
Equation 1.3 with respect to λ. Hint: Set hc/λmaxkBT = x and derive the intermediate
result e‒x + (x/5) = 1. This equation cannot be solved analytically but must be solved
numerically. Solve it by iteration on a hand calculator, and show that x = 4.965 is the
solution.
1-6. At what wavelength does the maximum in the energy-density distribution
function for a blackbody occur if (a) T = 300 K, (b) T = 3000 K, and (c) T = 10 000
K?
1‒9. We can use the Planck distribution to derive the Stefan‒Boltzmann law, which
gives the total energy density emitted by a blackbody as a function of temperature.
Derive the Stefan‒Boltzmann law by integrating the Planck distribution over all

frequencies. Hint: You’ll need to use the integral  dxx3 /(e x  1)   4 / 15.
0
1-11. Calculate the energy of a photon for a wavelength of 100 pm (about one atomic
diameter).
1-13. Calculate the number of photons in a 2.00 mJ light pulse at (a) l.06 µm, (b) 537
nm, and (c) 266 nm.
1-19. Given that the work function of chromium is 4.40 eV, calculate the kinetic
energy of electrons emitted from a chromium surface that is irradiated with ultraviolet
radiation of wavelength 200 nm.
1‒21. Some data for the kinetic energy of ejected electrons as a function of the
wavelength of the incident radiation for the photoelectron effect of sodium metal are
shown below. Plot these data to obtain a straight line, and calculate h from the slope
of the line and the work function ϕ from its intercept with the horizontal axis.
λ/nm
100
200
300
400
500
KE/eV
10.1
3.94
1.88
0.842
0.222
1‒24. Use the Rydberg formula (Equation 1.14) to calculate the wavelengths of the
first three lines of the Lyman series.
1‒26. A ground‒state hydrogen atom absorbs a photon of light that has a wavelength
of 97.2 nm. It then gives off a photon that has a wavelength of 486 nm. What is the
final state of the hydrogen atom?
1-27. Show that the Lyman series occurs between 91.2 nm and 121.6 nm, that the
Balmer series occurs between 364.7 nm and 656.3 nm, and that the Paschen series
occurs between 821.0 nm and 1876 nm. Identify the spectral regions to which these
wavelengths correspond.
1‒34. Using the Bohr theory, calculate the ionization energy (in electron volts and in
kJ·mol‒1) of singly ionized helium.
1-36. Ionizing a hydrogen atom in its electronic ground state requires 2.l79x10-18 J of
energy. The sun's surface has a temperature of ≈ 6000 K and is composed, in part, of
atomic hydrogen. Is the hydrogen present as H(g) or H+(g)? What is the temperature
required so that the maximum wavelength of the emission of a blackbody ionizes
atomic hydrogen? In what region of the electromagnetic spectrum is this wavelength
found?
1-39. Calculate (a) the wavelength and kinetic energy of an electron in a beam of
electrons accelerated by a voltage increment of 100 V and (b) the kinetic energy of an
electron that has a de Broglie wavelength of 200 pm (1 picometer = 10-12 m).
1‒46. If we locate an electron to within 20 pm, then what is the uncertainty in its
speed?
MATHCHAPTER A
A-1. Find the real and imaginary parts of the following quantities:
(a) (2  i)3
(b) e i /2
(c) e2i /2
(d) ( 2  2i)ei /2
A-3. Express the following complex numbers in the form re i :
(a) 6i
(b) 4 - 2i
(c) -l - 2i
(d) π+ ei
A-4. Express the following complex numbers in the form x  iy :
(a) ee / 4i
(b) 6e 2 i /3
(c) e  ( /4)i  ln 2
(d) e2 i  e4 i
A-6. Show that
cos  
ei  ei
2
and that
sin  
ei  ei
2i