Page 196 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

6.5 Spectra 187
Z 2
ÿ2
hr i nl ˆ (6:78)
1
3
n (l ‡ )a 2
2 0
Expression (6.71) for the expectation value of r ÿ1 may be used to calculate
the average potential energy of the electron in the state jnlmi. The potential
energy V(r) is given by equation (6.13). Its expectation value is
2
Z e9 2
2 ÿ1
hVi nl ˆÿZe9 hr i nl ˆÿ (6:79)
a 0 n 2
The result depends only on the principal quantum number n, so we may drop
the subscript l. A comparison with equation (6.57) shows that the total energy
is equal to one-half of the average potential energy
1
E n ˆ hVi n (6:80)
2
Since the total energy is the sum of the kinetic energy T and the potential
energy V, we also have the expression
2
Z e9 2
T n ˆÿE n ˆ (6:81)
2a 0 n 2
The relationship E n ˆÿT n ˆ (V n =2) is an example of the quantum-mechani-
cal virial theorem.

6.5 Spectra

The theoretical results for the hydrogen-like atom may be related to experimen-
tally measured spectra. Observed spectral lines arise from transitions of the
atom from one electronic energy level to another. The frequency í of any given
spectral line is given by the Planck relation
í ˆ (E 2 ÿ E 1 )=h
where E 1 is the lower energy level and E 2 the higher one. In an absorption
spectrum, the atom absorbs a photon of frequency í and undergoes a transition
from a lower to a higher energy level (E 1 ! E 2 ). In an emission spectrum, the
process is reversed; the transition is from a higher to a lower energy level
(E 2 ! E 1 ) and a photon is emitted. A spectral line is usually expressed as a
wave number ~ í, de®ned as the reciprocal of the wavelength ë
1 í jE 2 ÿ E 1 j
~ í ˆ ˆ (6:82)
ë c hc
The hydrogen-like atomic energy levels are given in equation (6.48). If n 1 and
n 2 are the principal quantum numbers of the energy levels E 1 and E 2 ,
respectively, then the wave number of the spectral line is