Old Quantum Theory - Basic Principles

Basic Principles

The basic idea of the old quantum theory is that the motion in an atomic system is quantized, or discrete. The system obeys classical mechanics except that not every motion is allowed, only those motions which obey the old quantum condition:

where the are the momenta of the system and the are the corresponding coordinates. The quantum numbers are integers and the integral is taken over one period of the motion at constant energy (as described by the Hamiltonian). The integral is an area in phase space, which is a quantity called the action and is quantized in units of Planck's constant. For this reason, Planck's constant was often called the quantum of action.

In order for the old quantum condition to make sense, the classical motion must be separable, meaning that there are separate coordinates in terms of which the motion is periodic. The periods of the different motions do not have to be the same, they can even be incommensurate, but there must be a set of coordinates where the motion decomposes in a multi-periodic way.

The motivation for the old quantum condition was the correspondence principle, complemented by the physical observation that the quantities which are quantized must be adiabatic invariants. Given Planck's quantization rule for the harmonic oscillator, either condition determines the correct classical quantity to quantize in a general system up to an additive constant.