Classical hallmarks of macroscopic quantum wave function propagation

Abstract

The precise connection between quantum wave functions and the underlying classical trajectories often is presented rather vaguely by practitioners of quantum mechanics. Here we demonstrate, with simple examples, that the imaging theorem (IT) based on the semiclassical propagator provides a precise connection. Wave functions are preserved out to macroscopic distances but the variables, position and momentum of these functions describe classical trajectories. We show that the IT, based on an overtly time-dependent picture, provides a strategy alternative to standard scattering theory with which to compare experimental results to theory.

Trajectories and the perception of classical motion in the free propagation of wave packets

“…In Ref. 5–7, it is shown that the IT remains valid for typical weak particle‐extraction fields and where more than one trajectory contributes to the final wave function at fixed $$\bm{r}$$ and $$t$$. For the simple case of completely free propagation of interest here, the IT is valid in the far zone of wave propagation in complete analogy to the transition to beam optics in light wave propagation.…”

“…This equation is known as the IT approximation for the final wavefunction $$\Psi (\bm{r}, t)$$ 5–7 . Actually, suitably generalised, it is valid for motion in arbitrarily applied fields and where more than one trajectory contributes to the final wave function 5–7 .…”

“…This is also true for motion which is not free but steered by weak extraction fields. [5][6][7] The Bohmian velocity and trajectories for a Gaussian wave packet…”

“…This equation is known as the IT approximation for the final wavefunction Ψ(r, t). [5][6][7] Actually, suitably generalised, it is valid for motion in arbitrarily applied fields and where more than one trajectory contributes to the final wave function. [5][6][7] For the simple case of free motion, each final position r, t along a single classical trajectory corresponds to a unique initial momentum p ′ emanating from r ′ ≈ 0, t ′ = 0 (the initial time is taken to be zero).…”

“…This equation is known as the IT approximation for the final wavefunction $$\Psi (\bm{r}, t)$$ 5–7 . Actually, suitably generalised, it is valid for motion in arbitrarily applied fields and where more than one trajectory contributes to the final wave function 5–7 . For the simple case of free motion, each final position $$\bm{r},t$$ along a single classical trajectory corresponds to a unique initial momentum $$\bm{p}^{\prime }$$ emanating from $$\bm{r}^{\prime } \approx 0, t^{\prime }=0$$ (the initial time is taken to be zero).…”