Let us look at a simple model of hydrogen atom: we neglect the spin of the proton and the electron, and relativistic effects. What remains is a point charge in an inverse-square force field-- the classical Kepler problem.

With these simplifications, the states of the hydrogen atom can be specified by giving three integer labels , with:

The labels are called

Sommerfeld's quantum conditions stated that these three classical quantities were restricted to integer values (and in fact the collection of values given above).

I should make one refinement to this description of the ``old quantum''
viewpoint. The quantum conditions apply only to so-called *stationary*
orbits. Bohr offered no details for what happened during a transition from
one stationary orbit to another (a ``quantum jump''), nor could he explain
why these orbits were stationary. Initially, physicists probably regarded
these questions as topics for future research.

In the post-1925 reformulation, we have to solve a particular instance of
Schrödinger's equation, subject to certain boundary conditions. The
space of all solutions is a Hilbert space, with a basis .
The vector is simultaneously an eigenvector of three operators:
`
`

`
, the energy
, the magnitude of the orbital angular momentum
, the -component of the orbital angular momentum
`

So we can say that if the hydrogen atom is in state , then it has a definite energy, and its angular momentum has a definite magnitude and -component. In fact, the eigenvalues for are:

where is a physical constant known as Rydberg's constant.

The modern equivalent of the old notion of ``stationary orbit'' is ``eigenstate of the energy operator''. Such eigenstates do not change with time, and they possess a definite value for the energy. (These facts are closely related.) Schrödinger's equation, in fact, amounts to , where is the energy operator.

Figure 1 gives a pictorial representation of the basis
in a form known as a *term scheme*. The horizontal lines
stand for basis vectors (or equivalently, stationary quantum states);
height gives energy, and transitions are indicated by slanted lines. (Only
three transitions are pictured, to avoid clutter.)

From this simple diagram, many treasures flow. The next few paragraphs give a taste.

© 2001 Michael Weiss