Planck's Theory.—It is just here where the theory of Planck steps in. According to Planck, a vibrating system, like the pen dulum with a small amplitude of oscillation, will normally behave as if it were not interfered with and will obey the classical laws, but the constant A is restricted to be an integral multiple of a small universal constant called the Quantum of Action. The energy is therefore expressed by where n is a positive integer or zero. While this normal behaviour continues, the system is said to be in one of its stationary states, and the corresponding value of the energy is termed an energy level. Energy can only be gained or lost by a transition from one stationary state to another. Such transitions do not conform to classical laws and their nature is still obscure. In the case of a simple harmonic vibration, of which the motion of the pendulum described above is an instance, the integer n can only change by unity in a single transition. The reason for this will be explained later. The universal constant It turns out to be very small, being equal to so that, when applied to a thing on as big a scale as an ordinary pendulum, Planck's theory makes no practical or observable difference. But it makes a profound difference when applied to things on the scale of atoms. The classical theory (if under this term we include the theory of relativity) is quite competent to deal with microscopic phenomena, and it has always been, in fact, a guiding principle in the development of the quantum theory to make it coalesce with the classical theory when applied to large scale phenomena.
In his later work Planck modified his earlier hypothesis. Pos sibly he may have felt that he had laid hands on the doctrines of Newton with more violence than was necessary, since the later modifications conceded that a vibrating system might absorb energy after the classical fashion.
Black Body Radiation.—The following description of the theory of black body radiation (see BLACK BODY), or full radia tion as it is now more usually termed, while embodying the essen tials of Planck's theory, differs from it in immaterial details. The radiant energy in a vacuous enclosure is associated with aether vibrations of all wave-lengths from zero upwards. The sort of picture we must form of a vibration of a definite fre quency (or wave-length) is very like that of the vibration of the column of air in a cylindrical resonance tube or, better still, the transverse vibration of a stretched cord (Melde's experiment). Let us consider such a stretched cord, fixed at both ends. Its possible states of vibration will be such that it is divided into a number of intervals, each of the length of half a wave, terminated by nodes, or points where the cord is not in motion. In such a case we have the relation /=n (3) 2 where 1 is the length of the cord, A is the length of the wave cor responding to the frequency of the vibration and 71 is the number of intervals. If v represents the velocity with which transverse waves of frequency v travel along the cord, we have the well known relation, v = Xv, (4) It is clear, therefore, that the total number of possible vibrations of all frequencies from the lowest possible one up to v will be represented by n in equation (5), and, therefore, the number corresponding to the narrow range of frequencies between v and v+dv will be represented by Strictly speaking this represents the number of vibrations when the motions of the cord are all parallel to one another, or all in one plane. The most general sort of motion of a point on the cord
can be regarded as made up of two independent motions in direc tions at right angles to one another, and, therefore, the expression (6) should be multiplied by 2 to give the number of independent vibrations. We are concerned, however, with a slightly more complicated problem than that of the vibrations of a stretched cord. We wish to find an expression for the number of aether vibrations in unit volume of the enclosure and in the range of frequencies between v and v+dv. The same sort of method as that just described leads to the result where C is the velocity of aether waves (light waves) in empty space. Or when we take account of the fact that aether waves are transverse like those along the cord, and that therefore, the displacements in such waves can be regarded as compounded of two independent motions at right angles to one another, we must multiply the expression (7) by 2, so that we get the final result We shall get a formula tor the energy per unit voiume associates with the vibrations of this range of frequencies if we multiply formula (8) by the average energy of a vibration. The classical kinetic theory of matter requires that the average energy of a vi bration shall be two-thirds of the average kinetic energy of trans lation of a molecule of a gas at the temperature in question. It must therefore be equal to kT, (9) where T is the absolute temperature and k is the "gas constant" reckoned for one molecule. If, however, we adopt Planck's hypothesis and make use, as he did, of Boltzmann's notions of the relation between entropy and probability, we get the expression The expressions (i i) and (12) represent, according to the classical theory and the quantum theory respectively, the energy of full radiation per unit volume and within the range of frequencies between v and v+dv. If we remember that the product of wave length and frequency is equal to the velocity of the waves, in this case it is easy to see that these expressions are equivalent to respectively, either of which represents the energy per unit volume associated with the range of wave-lengths between X and X-FdX, the former according to Rayleigh and the latter according to Planck. Planck's formula fits the observed facts of black body radiation extraordinarily well (see HEAT), and it is easily verified that it approaches Rayleigh's formula in the limiting case of large values of the product XT.