The Old Quantum Theory

Bohr's 1913 model of the hydrogen atom really starts the story. Nineteenth and early twentieth century experimentalists heaped up data on the spectra of atoms. Besides the plain old spectra of hydrogen, helium, sodium, etc., these physicists studied ionized atoms, atoms in electric fields (the Stark effect), atoms in magnetic fields (the Zeeman effect), atoms in crossed electric and magnetic fields; they discovered the so-called ``fine-structure'' of spectra, where one spectral line under higher resolution splits into two or more spectral lines; and so on, and so on, for thousands of journal pages.

Bohr's model solidified regularities in this data already partly noted by earlier workers (notably Balmer and Ritz). Bohr's transition picture (as I will call it) states that an atom has a discrete set of energy levels. When the atom emits a photon, it loses energy, changing from (say) energy level $E_i$ to level $E_j$, so the photon has energy $E_i-E_j$ and hence frequency

$$\nu=(E_i-E_j)/h$$

(This is known as the Bohr frequency condition.) So to explain the spectrum of an atom, we must identify all its energy levels. Of course, this picture depends fundamentally on the Einstein relation. It is utterly incompatible with any classical picture in which an orbiting electron gradually changes from one orbit to another, emitting radiation as it decellerates. Indeed, classically there should be no stable orbits at all!-- as noted in all histories of quantum theory.

The transition picture survives in modern quantum mechanics, although naturally much refined. The electromagnetic field is a quantum system; so is the atom; the two are coupled to form one combined quantum system. The interaction between the atom's charged particles and the electromagnetic field shows up as a term in the Hamiltonian (the energy operator, which governs the time evolution of the system). If this so-called coupling term is neglected, then the mathematics predicts that the atom should have discrete stable energy levels (i.e., if the atom is in an eigenstate for a particular energy level, then it will stay in that eigenstate forever.)

Just as the energy of the atom is quantized, so too is the energy of the electromagnetic field at a particular frequency. This falls out from computations with the quantum version of Maxwell's equations. We interpret this result as saying that electromagnetic field consists of quanta (or photons). Einstein's relation $E=h\nu$ can be deduced, rather than postulated.

Finally, if the coupling term between the field and the charged particles is not neglected, transitions become possible between formerly stable eigenstates. Both the atom and the field change their state together. Bohr's frequency condition can be derived, placing a satisfying capstone on the theoretical development.

Returning to history's tangled tale...Bohr's 1913 paper did more than introduce the transition picture. He also derived the energy levels for the hydrogen atom, by a remarkably simple argument. He assumed that the orbital angular momentum of the atom's solitary electron was $n\hbar$, with $n$ a positive integer. He assumed circular orbits, and applied Newton's laws, and the formula for the energy-levels of the atom popped right out. And it was right! (Apart from second-order corrections⁵.)

Once Bohr opened the gates, other quantum soldiers flocked in to consolidate the victory. Sommerfeld (in 1916) extended Bohr's model to allow for elliptical orbits. He included relativistic corrections, and calculated the effects of magnetic fields. Sommerfeld assembled a cadre of students, in Munich. Bohr had his group at Copenhagen, and Born started one at Göttingen. Heisenberg and Pauli began their careers as students in Sommerfeld's seminar on spectra, and studied at all three centers.

Almost all work during the period 1913-1925 shared a common approach, now called the old quantum theory. One began with a classical treatment of a dynamical problem - for example, elliptical orbits. One slapped quantum conditions on top. Quantum conditions stated that certain classical quantities had to be integer multiples of $\hbar$ - like the orbital angular momentum of the electron, in Bohr's theory of the hydrogen atom. (Sommerfeld added other quantum conditions.) Quantum conditions plus classical physics yielded a discrete set of energy levels, which in turn yielded spectra via Bohr's frequency condition.

These forays met initially with great success, but ultimately with crushing defeat. Generally speaking, the Bohr-Sommerfeld approach handled two-body problems reasonably well, but collapsed when faced with three-or-more-body problems. The hydrogen atom is a two-body problem: an electron orbits a proton. Bohr himself extended the theory to ``hydrogenic'' atoms - atoms that consist of a core of tightly-bound electrons around the nucleus, and a single loosely-bound electron orbiting further out. Hydrogenic atoms are ``approximately'' two-body systems: core-plus-electron. As soon as the theoreticians turned to three-body problems, theory and experiment parted company. Helium (two electrons and a nucleus) and the singly-ionized hydrogen molecule (one electron and two nuclei) spelled doom for the old quantum theory.

© 2001 Michael Weiss

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