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An Introduction To Non-aristotelian Systems And General Semantics.

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The impulse becomes. The phase-orbit is repre-
sented by an ellipse in the p-g-plane and is given by the equation,
where the minor axis
In our family of orbits the phase area between two orbits is equal to the quantum of action h. Sommerfeld regards h as an elementary region or element of the phase area, and considers it as the definition of the Planck quantum of action h. If Wn represents the energy of the oscillator when it describes the fl-th orbit, then . In these orbits the energy appears as a whole
multiple of the elementary quantum of energy;
We call stationary states of the oscillator those states which the oscillator may pass through without cessation and without loss of energy, or, without radiation.
When an oscillator retains its stationary state, its energy is constant and its graph appears as an ellipse of the family in the phase plane. However, when the energy of the oscillator changes and jumps over to a smaller orbit, it emits energy. When it passes to a larger orbit it absorbs energy. The emission and absorption of energy occurs in multiples of the energy quantum, «.
The graphs of the system in the phase plane are restricted to certain 'quantised' orbits. Between each orbit and its successor there is an elementary region, of area h. The n-th orbit, if closed, has as area nh. Or, expressed symbolically, This integral is called the phase integral and is taken along the w-th orbit.
The quantum hypothesis can be structurally formulated so that the phase integral must be a whole multiple of the quantum of action h. This form of the classical quantum postulate is more general than the original formulation of Planck, although it includes the latter as a particular case.
In case of a rotating point-mass, a similar analysis gives us
and whenwhere v represents the rotation frequency
of the rotator, or the number of full revolutions per unit of 'time', and takes the place of the vibration number of the oscillator.
In the classical theory, the quantised states were distinguished from all other possibilities by the characteristic whole numbers, and so we had a network. In a quantum orbit the 'electron', if undisturbed, was supposed to move permanently without resistance and not to emit radiation. The phase-space, representing the manifold of the possible states, including non-stationary states, is crossed, mesh-like, by the graph curves of the stationary orbits. The size of the meshes is determined by Planck's constant
Section C. Matrices.
The older quantum mechanics forms an elaborate system, and we have a large accumulation of numerical data on record. Some of these data corroborated the older theories nicely, but some data were in contradiction to the classical