A New Phase Space Density for Quantum Expectations

Abstract

Abstract. We introduce a new density for the representation of quantum states on phase space. It is constructed as a weighted difference of two smooth probability densities using the Husimi function and first-order Hermite spectrograms. In contrast to the Wigner function, it is accessible by sampling strategies for positive densities. In the semiclassical regime, the new density allows to approximate expectation values to second order with respect to the high frequency parameter and is thus more accurate than the uncorrected Husimi function. As an application, we combine the new phase space density with Egorov's theorem for the numerical simulation of time-evolved quantum expectations by an ensemble of classical trajectories. We present supporting numerical experiments in different settings and dimensions.

Symplectic semiclassical wave packet dynamics II: non-Gaussian states

“…The solutions of the semiclassical system (31) (solid and red) escape from the potential well inside, and show a very good agreement with the solutions (dashed and green) obtained by an algorithm based on Egorov's Theorem (or the IVR method) for a short time. the semiclassical solution of (38), and the expectation value Ĥ along the solutions of the Egorov-type algorithm [13]. One can see that the energies are conserved well by the numerical methods.…”

The spectrogram expansion of Wigner functions

“…for signal reassignment [9], filtering [10] and cross-entropy minimization [11]. However, to the best of our knowledge, apart from our preceding work [5], there are no results on the combination of spectrograms for approximating Wigner functions and expectation values.…”

“…a( 1 2 (y + q), p)e i(q−y)p/ε ψ(y)dy dp (5) with sufficiently regular symbol a : R 2d → C can be exactly expressed via the weighted phase space integral (2). Whenever W ψ is a probability density, (2) suggests to approximate expectation values by means of a Monte Carlo type quadrature, see §3.1.…”

“…where Φ t is the the flow of the underlying classical Hamiltonian systemq = p, p = −∇V (q). This approximation and its discretization has been studied in [5]. The spectrogram method (27) improves the Wigner function method (26) that has been widely used in chemical physics since decades under the name linearized semiclassical initial value representation (LSC-IVR) or Wigner phase space method; see, e.g., [2,3].…”

Computing quantum dynamics in the semiclassical regime

“…The classical propagation of the Husimi function, with suitable corrections that render an approximation that is second-order accurate with respect to ε, was carried out in Keller and Lasser (2013); see also Gaim and Lasser (2014) for a related approach to the numerical computation of fourth-order corrections to Egorov's theorem. The novel spectrogram method that combines initial sampling of positive phase space distributions with plain, uncorrected classical dynamics was proposed by Keller, Lasser and Ohsawa (2016). Higher-order spectrogram expansions have recently been analysed in Keller (2019).…”