A conjugate for the Bargmann representation

TL;DR

This paper introduces a new conjugate representation to the Bargmann representation in quantum mechanics, reversing the roles of creation and annihilation operators, with applications to simple systems and semiclassical propagators.

Contribution

It proposes an alternative conjugate representation of quantum states that reverses the roles of creation and annihilation operators in the Bargmann framework.

Findings

Derived inner product expressions preserving Hilbert space distances

Applied the conjugate representation to simple quantum systems

Utilized the approach for semiclassical propagator calculations

Abstract

In the Bargmann representation of quantum mechanics, physical states are mapped into entire functions of a complex variable z*, whereas the creation and annihilation operators $\hat{a}^\dagger$ and $\hat{a}$ play the role of multiplication and differentiation with respect to z*, respectively. In this paper we propose an alternative representation of quantum states, conjugate to the Bargmann representation, where the roles of $\hat{a}^\dagger$ and $\hat{a}$ are reversed, much like the roles of the position and momentum operators in their respective representations. We derive expressions for the inner product that maintain the usual notion of distance between states in the Hilbert space. Applications to simple systems and to the calculation of semiclassical propagators are presented.