A probabilistic approach to some results by Nieto and Truax

TL;DR

This paper introduces a probabilistic method to analyze generating functions for coherent and squeezed states, replacing traditional operational techniques with expectation operators, offering new insights into their mathematical structure.

Contribution

It presents a novel probabilistic approach to results by Nieto and Truax, replacing exponential derivative operators with expectation operators for generating functions.

Findings

Operational approach can be replaced by probabilistic expectation operators.

Provides new insights into the connection between operational and probabilistic calculus.

Simplifies the analysis of generating functions for quantum states.

Abstract

In this paper, we reconsider some results by Nieto and Truax about generating functions for arbitrary order coherent and squeezed states. These results were obtained using the exponential of the Laplacian operator; more elaborated operational identities were used by Dattoli et al. \cite{Dattoli} to extend these results. In this note, we show that the operational approach can be replaced by a purely probabilistic approach, in the sense that the exponential of derivatives operators can be identified with equivalent expectation operators. This approach brings new insight about the kinks between operational and probabilistic calculus.