From c-Numbers to q-Numbers: The Classical Analogy in the by Olivier Darrigol

By Olivier Darrigol

The heritage of quantum conception is a maze of conceptual difficulties, wherein Olivier Darrigol presents a lucid and realized consultant, monitoring the position of formal analogies among classical and quantum conception. From Planck's first advent of the quantum of motion to Dirac's formula of quantum mechanics, Darrigol illuminates not just the heritage of quantum thought but in addition the function of analogies in medical considering and thought change.Unlike prior works, that have tended to target qualitative, worldwide arguments, Darrigol's learn follows the strains of mathematical reasoning and symbolizing and so is ready to exhibit the motivations of early quantum theorists extra precisely--and provocatively--than ever earlier than. Erudite and unique, From c-Numbers to q-Numbers units a brand new typical as a philosophically perceptive and mathematically distinct heritage of quantum mechanics. For years yet to come it is going to impact ancient and philosophical discussions of twentieth-century physics.

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8 In the case of central forces the collision ''kind'' is characterized by the azimuth j of the plane of the trajectory of "2" in this reference frame, and by the angle q between the initial and final relative velocities, which is a definite function of the parameter b (and of the initial relative velocity v2 - v1). Let it be agreed that a collision starts when "2" crosses a conventional plane Z perpendicular to the relative velocity v2 - v1. Then, for "2" to collide with "1" within the time dt, and with a kind (q, j), defined with the uncertainty (dq, dj), it must be located within the "efficient volume" shaded in figure 2.

Even the enemies of this principle came to agree with this. However, Bohr soon attributed more fundamental functions to his "principle": bring more structure into the quantum theory, and show the overall harmony of its various assumptions. Characteristically, he regarded the formal connections obtained by analogy as part of the quantum theory, even before these connections could be expressed in a precise quantitative way. In his opinion a reasonable degree of conceptual clarity and consistency could be achieved even before the advent of a more complete and definitive theory.

Consequently, direct collisions (contributing to dn) and reverse collisions (contributing to dn') are simply related by permuting the roles of the initial and final velocities. Let us denote by and the elements in the space of velocities respectively corresponding to the elements d3v1 and d3v2 for a sharply defined kind of collision z. Then the number dn' of inverse collisions is given by a result of Maxwell's Ansatz (2), when and are taken as initial velocities, and the selected kind of collision is z'.

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