Planck’s principle in its original form applies only to certain kinds of systems, and if rashly generalized it gives wrong results. The right way to generalize it has been discovered by Sommerfeld, but unfortunately it is very difficult to express in non-mathematical language. It turns out that the principle, in its general form, cannot be stated as involving little parcels of energy; this only seemed possible because Planck was dealing with a special case. The general form requires a method of stating the principles of dynamics which is due to Hamilton. In this form, if the state of some material system is determined at any moment when we know, at that moment, how large certain quantities are (as for example the position of an aeroplane is known if we know its latitude and longitude and its height above the ground), then these quantities are called “coordinates” of the system. Corresponding to each coordinate, the system has at each moment a certain characteristic which may be called the corresponding “impulse-coordinate.” In simple cases this reduces to what is ordinarily called momentum; in a generalized sense, it may itself be called the “momentum” corresponding to the coordinate in question. It is possible to choose our coordinates in such a way that the momentum corresponding to a given coordinate at a given moment shall not involve any other coordinate. When the coordinates have been chosen in this way, the generalized quantum principle is applicable. We shall assume that such a choice has been made. When such a choice has been made, the coordinates are said to be “separated.”
The quantum-principle is only applicable to motions that are periodic, or what is called “conditionally periodic.” The motion of a system is periodic if, after a certain lapse of time, its previous condition recurs, and if this goes on and on happening after equal intervals of time. The motion of a pendulum is periodic in this sense, because, when it has had time to move from left to right and from right to left, it is in the same position as before, and it goes on indefinitely repeating the same motion. Wave-motions are periodic in the same sense; so are the motions of the planets. Any motion is periodic if it can be described by means of a quantity which increases up to a maximum, then diminishes to a minimum, then increases to a maximum again, and so on, always taking the same length of time from one maximum to the next. One “period” of a periodic process is the time taken to complete the cycle from one maximum to the next, or from one minimum to the next—for example, from midnight to midnight, from New Year to New Year, from the crest of one wave to the crest of the next, or from a moment when the pendulum is at the extreme left of its beat to the next moment when it is at the extreme left.
A system is called “conditionally periodic” when its motion is compounded of a number of motions, each of which separately is periodic, but which do not have the same period. For example, the earth has a motion compounded of rotation round its axis, which takes a day, and revolution round the sun, which takes a year. There are not an exact number of days in a year, if a year is taken in the astronomical and not in the legal sense; that is why we need a complicated system of leap-years to prevent errors from piling up. Thus when we take account of both rotation and revolution, the motion of the earth is “conditionally periodic.” We shall find later that the motions of electrons in their orbits, when we take account of niceties, are, strictly speaking, conditionally periodic and not simply periodic. The quantum-theory in its general form applies to motions that are conditionally periodic in terms of “separated” coordinates.
We can now state the generalized quantum-principle. Take some one coordinate of the system, and imagine the motion of the system throughout one period of this coordinate divided into a great number of little bits. In each little bit, take the generalized momentum corresponding to the coordinate in question, and multiply it by the amount of change in the coordinate during that little bit. Add up all these for all the little bits that make up one complete period. Then, in the limit, when the bits are made very small and very numerous, the result of the addition for one complete period will be exactly
or
or
or some other exact multiple of