By Olivier Darrigol
The historical past of quantum concept is a maze of conceptual difficulties, in which Olivier Darrigol offers a lucid and realized advisor, monitoring the function of formal analogies among classical and quantum concept. From Planck's first advent of the quantum of motion to Dirac's formula of quantum mechanics, Darrigol illuminates not just the background of quantum conception but in addition the function of analogies in clinical pondering and concept switch. in contrast to prior works, that have tended to target qualitative, worldwide arguments, Darrigol's research follows the traces of mathematical reasoning and symbolizing and so is ready to express the motivations of early quantum theorists extra preciselyand provocativelythan ever sooner than. Erudite and unique, From c- Numbers to q-Numbers units a brand new commonplace as a philosophically perceptive and mathematically targeted historical past of quantum mechanics. For future years it's going to impression ancient and philosophical discussions of twentieth-century physics.
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Extra resources for From c-Numbers to q-Numbers: The Classical Analogy in the History of Quantum Theory (California Studies in the History of Science)
Sample text
Indeed, the physical problem has no definite solution as long as nothing is known about the resonator but its eigenfrequency w 0 and its damping constant r ; that our theory is able to determine the approximate course of phenomena while only these two constants [71] ― 53 ― are given, must be regarded as a great advantage. For the same reason this theory cannot tell us more about the resonator than the determination of w0 and r . [72] Thus Planck believed that the indetermination of the internal structure of his resonators made room for an everlasting disorder, which would make the evolution of the accessible properties of the system strictly irreversible.
Therefore, where q denotes the angle between z and the electric moment of the resonator. The corresponding energy absorbed by the resonator in a time unit, ― 46 ― , is given by the right-hand side of the "fundamental equation" (76): The total energy radiated by the resonator in a time unit is given by the damping term r w0U in the fundamental equation. As results from the expression (41) of dipolar radiation, the radiation emitted in the direction z is completely polarized in the plane defined by this direction and the electric moment, and its intensity varies with the azimuthal angle q like sin2q Consequently, the energy radiated in the solid angle dW around z amounts to where the factor 3/8p gives the proper normalization, as results from For the intensity of this emitted radiation to be defined, the finite breadth (Breite ) of the source, that is, its efficient cross section times its spectral width, must be taken into account.
This calculation gave two necessary conditions for entropy increase: the form (113) for the function L (I ) corresponding to the function S (U ), and the convexity condition for the latter function, d2/dU 2 < 0, which provides dS T < 0 (since dUd U < 0). As Planck now noticed, these conditions were also sufficient for entropy increase. Indeed, the proof of the electro-magnetic H -theorem earlier given may be adapted in the following way. Starting from the equations (86) and (87) for energy and entropy balance, one can use the first condition (113) to derive Then, the two conditions together imply that L(I) - IdS/dU (at a constant U ) is a convex function of I , with an absolute maximum for I = v2U/c 2 These properties are the only ones necessary to the rest of the ― 60 ― proof of entropy increase.