Next: Selection Rules Redux Up: Spin Previous: The Paschen-Back Limit

#### Quantization

Schrödinger titled his first great paper on quantum mechanics Quantization as an Eigenvalue Problem''. I have alluded to the meaning of this several times already. A state specified by has quantum number for operator if satisfies the eigenvalue equation . Borrowing the imagery of the old quantum theory, we say that has the value in the state specified by .

Other jargon says that is sharp'' or definite'' in the state . According to the quantum theory of measurement, if we prepare an ensemble of systems, all in the state , and measure in each one, we will always get the value . But if is not an eigenvector of , then we will various eigenvalues of with different probabilities. There is statistical spread, and the value of for is not sharp''.

What about stationary''? Here we must bring in another concept I've mentioned before: the energy operator governs the time evolution of a quantum system. A state is absolutely stationary (does not change at all) if and only if it is an eigenstate of the energy operator . To be a touch more precise: suppose . Then evolves like so:

Remember that and specify the same state for any non-zero complex . So the state doesn't change. (I am also assuming that the energy operator does not change with time. A changing calls for a different equation for .)

Bohr's states are almost'' stationary. Perturbation theory deals with this ambiguity. Express the energy operator as a sum of two parts:

where , the perturbation, is small'' relative to . Eigenstates of will not usually be eigenstates of , but they will change slowly'' because is small''13. Start the system in an eigenstate of , say. Wait awhile. The system is now in a superposition of eigenstates of - say, for simplicity, . Here is another eigenvector of , and and are complex numbers. If we measure , we have a certain probability of finding the system in the state .

In other words, the system has made a transition from one state to another, under the influence of a perturbation.

Bohr's transition picture is a special case of this. The perturbation is the electromagnetic term, representing the interaction between the electromagnetic field and the atom. In other words, is due to the ability of the atom to make transitions by absorbing or emitting a photon. Everything else is stuffed into : the attraction of the nucleus, spin-orbit coupling, the constant magnetic field of the Zeeman effect. (Classical electromagnetism gives an unambiguous way to separate this constant magnetic field from the travelling field of light.) Bohr's stationary states are the eigenstates of .

What has become of Bohr's notion of quantum number? For Bohr and Sommerfeld, stationary states had quantum numbers. Translated into Hilbert space language, this asserts that eigenvectors of are eigenvectors of all other operators of physical importance. Alas, life is not so easy. If operators and commute, one can generally pick a basis of eigenvectors of both and . (Trivial exercise: prove the converse.) Some phenomena do submit to such an approach. The Paschen-Back effect does not-- as we will soon see.

Next: Selection Rules Redux Up: Spin Previous: The Paschen-Back Limit