Properties of Phase Space Wavefunctions and Eigenvalue Equation of Momentum Dispersion Operator

Abstract

This paper is a continuation of our previous works about coordinate, momentum, dispersion operators and phase space representation of quantum mechanics. It concerns a study on the properties of wavefunctions in the phase space representation and the momentum dispersion operator, its representations and eigenvalue equation. After the recall of some results from our previous papers, we give most of the main properties of the phase space wavefunctions and consider some examples of them. Then we establish the eigenvalue equation for the differential operator corresponding to the momentum dispersion operator in the phase space representation. It is shown in particular that any phase space wavefunction is solution of this equation.

Linear Canonical Transformations in relativistic quantum physics

“…Referring to our previous works on phase space representation of quantum theory and timefrequency analysis [16][17][18][19][20][21][22][23]25] we introduce the angular frequency dispersion operator † and their eigenstate | z, :, Ω, ˆ‚ defined through the relations The concepts of reduced operators have been introduced in [16]. In our case, we may consider the reduced time operator |, reduced angular frequency } and reduced angular frequency dispersion operator † .…”

“…See for instance references [8][9][10][11][12][13][14][15]. In section 4 and 5, we will study the relation between LCTs and the phase space representation of quantum theory developed in our previous works [16][17][18][19][20][21][22][23]. The transformations laws of reduced operators and the concept of Isodispersion LCTs (ILCTs) will be considered in particular.…”