Propagation of Quantum Expectations with Husimi Functions

Abstract

We analyse the dynamics of expectation values of quantum observables for the timedependent semiclassical Schrödinger equation. To benefit from the positivity of Husimi functions, we switch between observables obtained from Weyl and Anti-Wick quantization. We develop and prove a second order Egorov type propagation theorem with Husimi functions by establishing transition and commutator rules for Weyl and Anti-Wick operators. We provide a discretized version of our theorem and present numerical experiments for Schrödinger equations in dimensions two and six that validate our results.

Paths of canonical transformations and their quantization

“…There is no analogue in the case of non-linear flows (see de Gosson [34]). However, we conjecture that it is possible to obtain asymptotic equalities in the limit → 0 (with arbitrary accuracy) using the method developed by Lasser and her coworkers [22,56], and which yield a refinement of Egorov's theorem [15] on transformation properties of pseudo-differential operators.…”

A New Phase Space Density for Quantum Expectations

“…Discretization. For the algorithmic discretization of Corollary 4.1 we proceed similarly as in [16,10,14]. We consider various smooth functions a : R 2d → R and evaluate the phase space integral on the right hand side of the semiclassical approximation…”

“…As the initial state we consider the Gaussian wave packet ψ 0 = g z with phase space center z = (1, 0, 0, 0). This setup has already been considered in [5,16,14,10]. We investigate the dynamics of the following symbols a : R 4 → R, (1) Position: a(q, p) = q 1 and a(q, p) = q 2 , (2) Momentum: a(q, p) = p 1 and a(q, p) = p 2 , (3) Kinetic and potential energy: a(q, p) = 1 2 |p| 2 and a(q, p) = V (q), (4) Total energy: a(q, p) = 1 2 |p| 2 + V (q), and compare the outcome of the new algorithm with the naive, first-order Husimi approximation…”

“…One-dimensional cubic well. For the spectrogram algorithm we employed 2 14 Halton points and the eighth-order integrator with time stepping 10 −2 , which results in a computational time of 2 seconds. The quantum references are generated by means of a Strang splitting with 2 15 Fourier modes on the interval [−40, 4].…”

“…Replacing the new density by the Husimi function in Corollary 4.1 deteriorates the approximation in the sense thate −iHt/ε ψ, op(a)e −iHt/ε ψ = R 2d (a • Φ t )(z)H ψ (z)dz + O(ε)as ε → 0. It requires an additional system of coupled ODEs involving higher order derivatives of the Hamilton function h to retain second order accuracy with respect to ε; see[14, Theorem 3].…”