Page 202 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

Problems 193

6.12 Show explicitly for a hydrogen atom in the 1s state that the total energy E 1 is
equal to one-half the expectation value of the potential energy of interaction
between the electron and the nucleus. This result is an example of the quantum-
mechanical virial theorem.
6.13 Calculate the frequency, wavelength, and wave number for the series limit of the
Balmer series of the hydrogen-atom spectral lines.
‡
6.14 The atomic spectrum of singly ionized helium He with n 1 ˆ 4, n 2 ˆ 5, 6, .. . is
known as the Pickering series. Calculate the energy differences, wave numbers,
and wavelengths for the ®rst three lines in this spectrum and for the series limit.
6.15 Calculate the frequency, wavelength, and wave number of the radiation emitted
2‡
from an electronic transition from the third to the ®rst electronic level of Li .
Calculate the ionization potential of Li 2‡ in electron volts.
6.16 Derive an expression in terms of R 1 for the difference in wavelength,
Äë ˆ ë H ÿ ë D , between the ®rst line of the Balmer series (n 1 ˆ 2) for a
hydrogen atom and the corresponding line for a deuterium atom? Assume that
the masses of the proton and the neutron are the same.