In Paper I of this investigation a new calculus for functions of noncommuting operators was developed, based on the notion of mapping of operators onto $c$ -number functions. With the help of this calculus, a general theory is formulated, in the present paper, of phase-space representation of quantum-mechanical systems. It is shown that there is a whole class of such representations, one associated with each type of mapping, the simplest one being the well-known representation due to Weyl. For each representation, the quantum-mechanical expectation value of an operator is found to be expressible in the form of a phase-space average of classical statistical mechanics. The phase-space distribution functions are, however, not true probabilities, in general. The phase-space forms of the main quantum-mechanical equations of motion are obtained and are found to have the form of a generalized Liouville equation. The phase-space form of the Bloch equation for the density operator of a quantum system in thermal equilibrium is also derived. The generalized characteristic functions of boson systems are defined and their main properties are studied. The equations of motion for the characteristic functions are also derived. As an illustration of the theory, a generalized stochastic description of a quantized electromagnetic field is obtained.In Paper I of this investigation a new calculus for functions of noncommuting operators was developed, based on the notion of mapping of operators onto $c$-number functions. With the help of this calculus, a general theory is formulated, in the present paper, of phase-space representation of quantum-mechanical systems. It is shown that there is a whole class of such representations, one associated with each type of mapping, the simplest one being the well-known representation due to Weyl. For each representation, the quantum-mechanical expectation value of an operator is found to be expressible in the form of a phase-space average of classical statistical mechanics. The phase-space distribution functions are, however, not true probabilities, in general. The phase-space forms of the main quantum-mechanical equations of motion are obtained and are found to have the form of a generalized Liouville equation. The phase-space form of the Bloch equation for the density operator of a quantum system in thermal equilibrium is also derived. The generalized characteristic functions of boson systems are defined and their main properties are studied. The equations of motion for the characteristic functions are also derived. As an illustration of the theory, a generalized stochastic description of a quantized electromagnetic field is obtained.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsNov 15, 1970285 citations

Quantum Optics in Phase Space

Wolfgang P. Schleich

Wiley eBooksFeb 9, 20011,626 citations

Quantum mechanics in phase space: the Schrodinger and the Moyal representations

Nuno Costa Dias, Maurice A. de Gosson, Franz Luef

We present a phase space formulation of quantum mechanics in the Schrödinger representation and derive the associated Weyl pseudo-differential calculus. We prove that the resulting theory is unitarily equivalent to the standard "configuration space" formulation and show that it allows for a uniform treatment of both pure and mixed quantum states. In the second part of the paper we determine the unitary transformation (and its infinitesimal generator) that maps the phase space Schrödinger representation into another (called Moyal) representation, where the wave function is the cross-Wigner function familiar from deformation quantization. Some features of this representation are studied, namely the associated pseudo-differential calculus and the main spectral and dynamical results. Finally, the relation with deformation quantization is discussed.We present a phase space formulation of quantum mechanics in the Schrödinger representation and derive the associated Weyl pseudo-differential calculus. We prove that the resulting theory is unitarily equivalent to the standard "configuration space" formulation and show that it allows for a uniform treatment of both pure and mixed quantum states. In the second part of the paper we determine the unitary transformation (and its infinitesimal generator) that maps the phase space Schrödinger representation into another (called Moyal) representation, where the wave function is the cross-Wigner function familiar from deformation quantization. Some features of this representation are studied, namely the associated pseudo-differential calculus and the main spectral and dynamical results. Finally, the relation with deformation quantization is discussed.

Jan 1, 201212 citations

P-representations in quantum optics

Apostol Vourdas

Journal of Physics A Mathematical and TheoreticalJan 6, 20171 citation

Quantization of Phases and Moduli in terms of the Group SO(1,2)

H A Kastrup

The problem of quantizing phase and modulus associated with the polar coordinates of a 2-dimensional phase space dates back to the first years of quantum mechanics: The symplectic space P = { φ ∈ ( -π,+π],p > 0} has the global structure S 1 × R + and cannot be quantized in the conventional naive way. The appropriate method is the group theoretical quantization which here leads to the group 50(1,2) and the positive discrete series of its irreducible unitary representations. The basic classical observables p cosφ.p sinφ and p correspond to the self-adjoint operators K 1 , K 2 and K < of the SO ( 1,2) Lie algebra. The approach provides appropriate quantum observables for the phase space P and a promising basis for the description of quantum optical structuresThe problem of quantizing phase and modulus associated with the polar coordinates of a 2-dimensional phase space dates back to the first years of quantum mechanics: The symplectic space P = { φ ∈ ( -π,+π],p > 0} has the global structure S 1 × R + and cannot be quantized in the conventional naive way. The appropriate method is the group theoretical quantization which here leads to the group 50(1,2) and the positive discrete series of its irreducible unitary representations. The basic classical observables p cosφ.p sinφ and p correspond to the self-adjoint operators K 1 , K 2 and K < of the SO ( 1,2) Lie algebra. The approach provides appropriate quantum observables for the phase space P and a promising basis for the description of quantum optical structures

CRC Press eBooksNov 30, 20030 citations

In Paper I of this investigation a new calculus for functions of noncommuting operators was developed, based on the notion of mapping of operators onto

c

-number functions. With the help of this calculus, a general theory is formulated, in the present paper, of phase-space representation of quantum-mechanical systems. It is shown that there is a whole class of such representations, one associated with each type of mapping, the simplest one being the well-known representation due to Weyl. For each representation, the quantum-mechanical expectation value of an operator is found to be expressible in the form of a phase-space average of classical statistical mechanics. The phase-space distribution functions are, however, not true probabilities, in general. The phase-space forms of the main quantum-mechanical equations of motion are obtained and are found to have the form of a generalized Liouville equation. The phase-space form of the Bloch equation for the density operator of a quantum system in thermal equilibrium is also derived. The generalized characteristic functions of boson systems are defined and their main properties are studied. The equations of motion for the characteristic functions are also derived. As an illustration of the theory, a generalized stochastic description of a quantized electromagnetic field is obtained.