The Correspondence Principle for One Degree of Freedom

The fundamental postulate of the stability of atoms referred to in the introduction is satisfied by the two principles of atomic mechanics given in 10. We now inquire to what extent they are in agreement with the other fundamental postulate, that the classical theory shall appear as a limiting case of the quantum theory. 

Planck s constant h occurs as a characteristic magnitude in both quantum principles, and is a measure of the separation of the quantum states. Our requirement signifies that the quantum laws shall tend into the classical ones as limits as h-+0 the discrete energy steps then converge to the continuum of the classical theory. The frequency condition requires special examination we have to see if the frequencies calculated by it agree in the limit with those to be expected from the classical theory. 

The radiation from a system of electrically charged particles with charges ek at the points tk is determined, according to the classical theory, by the electric moment 

If the energy radiated in the course of one period is small, the damping may for the present be neglected. For a system of one degree of freedom, such as we consider here, the rectangular co-ordinates of the charged particles will be periodic functions of 

The CT s are complex numbers since, however, the electric moment is 

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