MATHEMATICAL INTRODUCTION
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'■ Chapter X, p,:
general, and tie®
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not explained. For this reason Planck modified the exposition
of his theory, and introduced the conception of a quantum of
action. His statement of this new point of view, derived from
a study of statistical mechanics and probability may be
paraphrased as follows :
“ The probability of a continuous variable may be obtained
by considering independent elementary regions, of equal
probability. ... In finding these elementary regions in
classical dynamics, use is made of the theorem that two physical
states, of which one is the necessary effect of the other, are
equally probable. In a physical system, where q represents one
of the generalized co-ordinates, p the corresponding ‘ moment ’
or ‘ impulse/ according to the theorem of Liouville the region
dq considered at any instant, is an invariant with respect
to the time, if q and p vary according to Hamilton’s equations.
On the other hand, p and q can, at a given instant, take all
possible values, independently of one another. Hence it follows
that the elementary region of probability is the infinitely small
element of the phase space dp dq. The new hypothesis has the
effect of limiting the variability of p and q in such a way that
these variables change only by jumps. ... In this way the
number of elementary regions of probability is reduced, whilst
the extent of each is increased. The hypothesis of quanta
consists of supposing that these regions, which are all equal,
are no longer infinitely small but finite, and that for each
where h is constant.”
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4. The Generalized Form of the Quantum Theory
Recent progress in the development of the quantum theory
has been based on the generalized form of the theory put forward
independently in 1915 by W. Wilson,* and by Sommerfeld.f A
generalization of a somewhat similar character was proposed
about the same time by Ishiwara,t but this contained certain
features which are not altogether acceptable. The aim in all these
methods is to furnish a common basis on which the theories
of full (black body) radiation, spectral series and other pheno-
* W. Wilson, Phil. Mag., vol. 29, p. 795, June, 1915.
f Sommerfeld, Sitz. Ber. der Münchener Akad., Dec, 1915; Ann. d.
Physik, vol. 51, p. i, 1916.
J Ishiwara, Tokyo Sugaki-Buturigakkwai Kizi, 2nd ser., vol. 8, No. 4,
p. 106.