Bohr's formula for the energy levels
is quite simple:

This simplicity stems from the simplicity of the formula he used for the energy:

where is the square of the momentum, is the radial distance from the proton to the electron, and is the mass of the electron. The first term represents the kinetic energy, the second the potential energy due to the inverse square force (Coulomb potential). The quantum theorists came along later and said that , , and are all really operators, and the energy levels are really eigenvalues, but the source of the simplicity remains the same. ``Simplicity'' is too vague a word: ``symmetry'' serves better, as we will see.

For example, the formula for is spatially symmetric under rotations about the proton (because and are unchanged by such rotations). Thus the formula for cannot depend on , which specifies the orientation of the orbit. If we impose a magnetic field along the -axis, say, then acquires a new term (of the form , where is the magnetic field strength, and is the -component of orbital angular momentum). The new formula for now depends on , since is not symmetric under arbitrary rotations. The magnetic field is referred to as a perturbation.

We can see the effects of the magnetic field in changes to the term scheme:
some states move up in energy, some move down. Reduced symmetry has lead
to reduced degeneracy. Transitions that formerly had the same
now have slightly different 's (see figure 0). This
means that a single spectral line with no magnetic field will split into
multiple spectral lines when the field is turned on. This is the famous
*Zeeman effect*.

Another example: in 1916, Sommerfeld replaced Bohr's simple formula for
with its relativistic equivalent. This lead to a new formula for
, depending on both and (but not on , of course).
These relativistic corrections (or perturbations) account for the so-called
*fine structure* of the hydrogen spectrum, known already to
spectroscopists long before 1916.

A historical footnote: one modern author has written:

When Dirac developed relativistic quantum mechanics, the relativistic Coulomb problem proved to beexactly solvable...But the resulting formula for the energy levels was truly a surprise:The new answer was precisely the old Sommerfeld formula!

How could this possibly be? Clearly Sommerfeld's methods were heuristic (Bohr quantization rules), out-dated byThe author calls this the ``Sommerfeld puzzle'', and resolves it, but I will discuss it no further.tworevolutions (Heisenberg-Schrödinger nonrelativistic quantum mechanics and Dirac's relativistic quantum mechanics) and his methods obviously had no place at all for the electron spin ...So Sommerfeld's correct answer could only be a lucky accident, a sort of cosmic joke at the expense of serious minded physicists.

© 2001 Michael Weiss