By Monique Combescure

This e-book offers many of the sorts of coherent states brought and studied within the physics and arithmetic literature and describes their houses including program to quantum physics difficulties. it really is meant to function a compendium on coherent states and their functions for physicists and mathematicians, stretching from the fundamental mathematical buildings of generalized coherent states within the feel of Perelomov through the semiclassical evolution of coherent states to numerous particular examples of coherent states (hydrogen atom, quantum oscillator, ...).

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105) We have, using linearity of integration, ˆ z. dz ϕz , ψ Aϕ ˆ = (2π )−n Aψ From the triangle inequality, we have ˆ Aψ ≤ (2π )−n ≤ (2π )−n ˆ z Aϕ dz ψ , ϕz [|z−X0 |

Now if ψ, ϕ ∈ S(Rn ) then we easily get Wψ,ϕ ∈ S(R2n ). On the other side there exists Aj ∈ S(R2n ) such that Aj → A in S (R2n ). 21) to Aj and we go to the limit in j . What Wigner was looking for was an equivalent of the classical probability distribution in the phase space R2n . That is, associated to any quantum state a distribution function in phase space that imitates a classical distribution probability in phase space. Recall that a classical probability distribution is a non-negative Borel function ρ; Z → R+ , Z := R2n , normalized to unity: ρ(z) dz = 1, Z and such that the average of any observable A ∈ C ∞ is simply given by ρ(A) = A(z)ρ(z) dz.

In the following we denote by Wϕ the Wigner transform for ϕ, ϕ. What about the expected properties of (2π )−n Wϕ as a possible probability distribution in phase space? Namely: • positivity • normalization to 1 • correct marginal distributions Proposition 15 Let z = (x, ξ ) ∈ R2n and ϕ ∈ L2 (Rn ) with ϕ = 1. We have (i) (2π )−n 2 Rn Wϕ (x, ξ ) dξ = ϕ(x) , which is the probability amplitude to find the quantum particle at position x. (ii) (2π )−n 2 Rn ˜ ) , Wϕ (x, ξ ) dx = ϕ(ξ which is the probability amplitude to find the quantum particle at momentum ξ .

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