Kramers–Heisenberg formula Kramers–Heisenberg formula The Kramers–Heisenberg dispersion formula is an expression for the cross section for scattering of a photon by an atomic electron. It was derived before the advent of quantum mechanics by Hendrik Kramers and Werner Heisenberg in 1925, based on the correspondence principle applied to the classical dispersion formula for light. The quantum mechanical derivation was given by Paul Dirac in 1927. The Kramers–Heisenberg formula was an important achievement when it was published, explaining the notion of "negative absorption" (stimulated emission), the Thomas–Reiche–Kuhn sum rule, and inelastic scattering — where the energy of the scattered photon may be larger
Heisenberg group connection with the theta functions. If , are integers (in the ring Z) then one has the discrete Heisenberg group H(Z). It is a non-abelian nilpotent group. It has two generators, and relations where is the generator of the center of H. (Note that the inverses of "x", "y", and "z" replace the 1 above the diagonal with −1.) By Bass's theorem, it has a polynomial growth rate of order 4. One can generate any element through If one takes "a, b, c" in Z/"p" Z for an odd prime "p", then one has the Heisenberg group modulo p. It