By Wolfgang Arendt, Wolfgang P. Schleich
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Extra info for Coherent Manipulation and Transport of Matter Waves
Still, abstract mathematics and applied sciences have attracted and repelled each other many times over the last centuries, following the alternation of rushing theoretical developments and real-world applications. May this book contribute to this everlasting interplay. 1 Introduction Weyl’s law is in its simplest version a statement on the asymptotic growth of the eigenvalues of the Laplacian on bounded domains with Dirichlet and Neumann boundary conditions. In the typical applications in physics one deals either with the Helmholtz wave equation describing the vibrations of a string, a membrane (drum), a mass of air in a concert hall, the heat radiation from a body in thermal equilibrium, the ﬂuctuations of the gravitational ﬁeld in cosmology, or the Schrödinger equation of a quantum system which may be a simple quantum billiard, an atom, a molecule or a compound nucleus.
The problem was of course a question of the correct boundary conditions in the electromagnetic case. In his book, Rayleigh writes: “Some of the natural notes of the air contained within a room may generally be detected on singing the scale. ”  And in a footnote he adds: “A remarkable instance is quoted in Young’s Natural Philosophy, II. p. 272, from Darwin’s Zoonomia, II. 487. ” And then Rayleigh continues: “In long and narrow passages the vibrations parallel to the length are too slow to affect the ear, but notes due to transverse vibrations may often be heard.
18). In fact, it differs from Wien’s law only by the –1 in the denominator. It has rightly been said , that even if Planck had stopped after October 19, he would forever be remembered as the discoverer of the radiation law. It is a true measure of his greatness that he went further. 21). That made him to discover the quantum theory. Already on December 14, 1900, Planck presented a theoretical derivation of his formula to the German Physical Society in Berlin  and shortly afterwards (7 January 1901) submitted his famous paper .