many deficiencies for other atoms, which is to say, almost all atoms of interest other
than hydrogen. The problems with the Bohr atom for these cases were:
- There were lines in the spectra corresponding to transitions other than simply
between twonvalues (cf. Eq.4.14). This was rationalized by Sommerfeld in
1915, by the hypothesis of elliptical rather than circular orbits, which essentially
introduced a new quantum number k, a measure of the eccentricity of the
elliptical orbit. Electrons could have the samenbut differentk’s, increasing
the variety of possible electronic transitions;kis related to what we now call the
azimuthal quantum number,l;l¼k"1). - There were lines in the spectra of the alkali metals that were not accounted for by
the quantum numbersnandk. In 1925 Goudsmit and Uhlenbeck showed that
these could be explained by assuming that the electron spins on an axis; the
magnetic field generated by this spin around an axis could reinforce or oppose
the field generated by the orbital motion of the electron around the nucleus. Thus
for eachnandkthere are two closely-spaced “magnetic levels”, making possible
new, closely-spaced spectral lines. The spin quantum number,ms¼þ1/2 or"1/2,
was introduced to account for spin. - There were new lines in atomic spectra in the presence of anexternalmagnetic
field (not to be confused with the fields generated by the electron itself). This
Zeeman effect (1896) was accounted for by the hypothesis that the electron
orbital plane can take up only a limited number of orientations, each with a
different energy, with respect to the external field. Each orientation was asso-
ciated with a magnetic quantum numbermm(often designatedm)¼"l,"(l"1),...,
(l"1),l). Thus in an external magnetic field the numbersn,k(laterl) andmsare
insufficient to describe the energy of an electron and new transitions, invoking
mm,are possible.
The only quantum number that flows naturally from the Bohr approach is the
principal quantum number,n; the azimuthal quantum numberl(a modifiedk), the
spin quantum numbermsand the magnetic quantum numbermmare all ad hoc,
improvised to meet an experimental reality. Why should electrons move in ellipti-
cal orbits that depend on the principal quantum numbern? Why should electrons
spin, with only two values for this spin? Why should the orbital plane of the electron
take up with respect to an external magnetic field only certain orientations, which
depend on the azimuthal quantum number? All four quantum numbers should
follow naturally from a satisfying theory of the behaviour of electrons in atoms.
The limitations of the Bohr theory arise because it does not reflect a fundamental
facet of nature, namely the fact that particles possess wave properties. These limi-
tations were transcended by the wave mechanics of Schr€odinger,^16 when he devised
his famous equation in 1926 [ 12 , 13 ]. Actually, the year before the Schr€odinger
(^16) Erwin Schr€odinger, born Vienna, 1887. Ph.D. University of Vienna. Professor Stuttgart, Berlin,
Graz (Austria), School for Advanced Studies Dublin, Vienna. Nobel Prize in physics 1933 (shared
with Dirac). Died Vienna, 1961.
4.2 The Development of Quantum Mechanics. The Schr€odinger Equation 97