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Q: Quantenoptik

Q 513: Quanteneffekte IV

Q 513.5: Vortrag

Freitag, 8. März 2002, 15:00–15:15, HS 11/215

Quantizing Phases and Moduli — •Hans Kastrup — DESY Theorie, Notkestrasse 85, D-22603 Hamburg

> A typical classical interference pattern of 2 waves with intensities I₁, > I₂ and relative phase ϕ = ϕ₂− ϕ₁ may be > characterized by the 3 observables p=√I₁ I₂,   pcosϕ and > −psinϕ. A group theoretical quantization of the symplectic space > {(ϕ ∈ R mod2π,  p>0)} in terms of irreducible unitary > representations of the group SO^↑(1,2) provides a solution to > the old and controversial problem of quantizing the conjugate pair phase > ϕ and modulus p:

The Poisson brackets of the classical > observables pcosϕ,−sinϕ and p >0 form the Lie algebra of > the group SO^↑(1,2). The corresponding self-adjoint generators > pcosϕ = K₁, −psinϕ = K₂ and p > = K₃ of that group may be obtained from its irreducible unitary > representations. For the positive discrete series the modulus operator > K₃ has the spectrum {n+k, n=0,1,2, …;k>0 }. Self-adjoint > operators > cosϕ and sinϕ can be defined as (K₃⁻¹K₁ + > K₁K₃⁻¹)/2 and −(K₃⁻¹K₂ + K₂ K₃⁻¹)/2 which have the > desired properties for k ≥ 0.5. The approach described here solves, > e.g. the modulus-phase quantization problem for the harmonic oscillator > and provides a full quantum theoretical basis for the semi-classical > operational approach to the phase quantization problem by Noh, Fougère > and Mandel. >