Coherent States and Quantum Asymptotic Features by Weak KAM Theory
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Coherent States and Quantum Asymptotic Features by Weak KAM Theory Franco Cardin · Simone Vazzoler
Received: 14 December 2013 / Accepted: 6 March 2014 © Springer Science+Business Media Dordrecht 2014
Abstract We propose a sketch for a proof of an interesting theorem on the evolution of coherent states, whose statement has been first presented in Paul (Séminaire: Équations aux Dérivées Partielles. 2007–2008, pages Exp. No. IV, 21. École Polytech., Palaiseau, 2009), and give some further insights on the asymptotic behavior, involving weak KAM theory. Keywords Quantum Mechanics · Coherent states · Weak KAM theory
1 Coherent States We will use the following notations through the article: ε will be Planck’s constant, (q, p) ∈ R2n , x ∈ Rn (note that the point (q, p) belongs to a different space than x) and t ∈ R. Our wave functions will be in the Hilbert space L2 (Rn ) (for fixed t ). To study semiclassical behavior of Quantum Mechanics (QM from now on), we start looking at the most classical states of the theory: these are the wave functions ψ ∈ L2 (Rn ) that minimize Heisenberg inequality ε (Δψ Xi )(Δψ Pj ) ≥ δij 2 where Δ is the measurement error operator defined as Δψ Xi := ψ|Xi2 ψ − ψ|Xi ψ2
(1)
(2)
and Xi and Pj are the position and the momentum operators respectively defined as Xi ψ(x) := xi ψ(x) and Pi ψ(x) := −iε∂xi ψ(x).
B
F. Cardin ( ) · S. Vazzoler Dip. Matematica, Università degli Studi di Padova, via Trieste 63, 35121 Padova, Italy e-mail: [email protected] S. Vazzoler e-mail: [email protected]
F. Cardin, S. Vazzoler
Definition 1 The states that minimize inequality (1) are called coherent states and have the following form n/4 1 i i ψ(q,p) (x) = e ε p|x−q e 2ε x−q|Z(x−q) (3) πε where Z = iI and (q, p) ∈ R2n is the center of the coherent state. In the previous definition the point (q, p) is a parameter where q describes the center of the Gaussian, while p is the momentum of the wave function. We propose the following definition that we will use as an approximation of the evolution of coherent states (but this class of wave functions will not minimize Heisenberg inequality). Definition 2 Let (q, p) ∈ R2n . We define a generalized coherent state centered in (q, p) as the following wave function a,Z,Θ ψ(q,p) (x) =
1 πε
n/4 x −q i 1 a √ e ε (p|x−q+ 2 x−q|Z(x−q)+Θ) ε
(4)
a,Z,Θ L2 = 1, Θ ∈ R and Z is a complex n × n matrix. where a is a function such that ψ(q,p)
Note that all the constants that we find in the previous definition could be substituted with real function of time: in this way our generalized coherent state becomes a time dependent a(t),Z(t),Θ(t) wave function ψ(q(t),p(t)) (x). Now we focus our attention on finding an approximate solution to the following Cauchy problem ⎧ 2 ⎪ ⎨iε∂ φ(t, x) = H φ(t, x) = − ε Δφ(t, x) + V (x)φ(t, x) t 2 (5) ⎪ ⎩φ(0, x) = ψ (q,p) (x) 2
where H is the quantum mechanical operator defined as H = − ε2 Δ + V (x). The point is that if we choose as initial datum a coherent state, then its time evolution is no more a coherent state: i
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