Quantum Conditions for Simply and Multiply Periodic Motions

Quantum Conditions for Simply and Multiply Periodic Motions. 

We may stow by a simple example how this can be done. Consider a simple pendulum (fig. 3) whose length can be altered, say by drawing the thread over a pulley. If we shorten the thread 

The first term corresponds to the elevation of the position of equilibrium, which does not interest us. The second part of the expression, i.e. the product of AZ by the expression in brackets, represents the increase Alf in the energy of the pendulum notion. Now the energy of the undisturbed pendulum motion is 

The relation thus formulated is capable of immediate generalization. Consider in the first place, as an example with one degree of freedom, the case already treated above (p. 100), that of the rotator. Here the co-ordinate is the azimuth q== (f y to which belongs, as canonically conjugated quantity, the angular momentum (or, in other words, the moment of momentum) p. In the free rotation p is constant, i.e. independent of the angle turned through. Thus 

If we represent the motion in the pg-plane, this integral is to be taken over the straight line p = const., not over a closed curve. But we must observe that in this plane points with the same p, whose g-co-ordinates differ by 27t, represent the same state of the rotator. Properly speaking, therefore, we should consider, not a y g -plane but a g-cylinder (fig. 5) of circumference 27t, so that the integral has now to be taken over the circumference of the cylinder, and has the 

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