Semiclassical theory for small displacements

TL;DR

This paper develops a semiclassical approximation for the quantum characteristic (chord) function of large Bohr-quantized states, accurately identifying blind spots and overlaps under displacements, with implications for quantum state analysis.

Contribution

It introduces a semiclassical method to approximate the chord function for large quantum states, accurately locating blind spots and overlaps, extending understanding of quantum displacements.

Findings

The approximation accurately predicts blind spots within a Planck area.

It verifies the method for Bohr-quantized states.

The approach depends on the Schrödinger covariance matrix.

Abstract

Characteristic functions contain complete information about all the moments of a classical distribution and the same holds for the Fourier transform of the Wigner function: a quantum characteristic function, or the chord function. However, knowledge of a finite number of moments does not allow for accurate determination of the chord function. For pure states this provides the overlap of the state with all its possible rigid translations (or displacements). We here present a semiclassical approximation of the chord function for large Bohr-quantized states, which is accurate right up to a caustic, beyond which the chord function becomes evanescent. It is verified to pick out blind spots, which are displacements for zero overlaps. These occur even for translations within a Planck area of the origin. We derive a simple approximation for the closest blind spots, depending on the Schroedinger covariance matrix, which is verified for Bohr-quantized states.