Semiclassical Pseudodifferential Calculus

Abstract

As we have already explained, one of the main motivations of the pseudodifferential calculus is to get an algebraic correspondence between the classical observables and the quantum observables (one calls it a quantization of the classical observables). In particular, this would permit us to localize (within the limits allowed by the uncertainty principle) both in position and momentum variables any quantum state ψ: Take a smooth cutoff function χ = χ(x, ξ) (that is, χ ∈ C ^∞₀ (R ²ⁿ), the space of smooth compactly supported functions on R ²ⁿ, and χ is close to the characteristic function of some compact subset of R ²ⁿ). Then its associated quantum observable χ(x, ħD _x ) applied to ψ will have the effect of (essentially) cutting off the Cartesian product Supp ψ × Supp $\hat \psi$ outside Supp χ (here Supp stands for the support).