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

Wigner operator's new transformation in phase space quantum mechanics and its applications

Hong-Yi Fan

Using operators' Weyl ordering expansion formula (Hong-yi Fan,\emph{\}J. Phys. A 25 (1992) 3443) we find new two-fold integration transformation about the Wigner operator $Δ(q',p')$ ( $q$ -number transform) in phase space quantum mechanics, $\iint_{-\infty}^\infty dp' dq'/πΔ(q',p') e^{-2i(p-p') (q-q')} =δ(p-P) δ(q-Q),$ and its inverse $\iint_{-\infty}^\infty dq dp δ(p-P) δ(q-Q) e^{2i(p-p') (q-q')}=Δ(q',p'),$ where $Q,$ $P$ are the coordinate and momentum operators, respectively. We apply it to studying mutual converting formulas among $Q-P$ ordering, $P-Q$ ordering and Weyl ordering of operators. In this way, the contents of phase space quantum mechanics can be enriched.Using operators' Weyl ordering expansion formula (Hong-yi Fan,\emph{\}J. Phys. A 25 (1992) 3443) we find new two-fold integration transformation about the Wigner operator $Δ(q',p')$ ($q$-number transform) in phase space quantum mechanics, \[ \iint_{-\infty}^\infty dp' dq'/πΔ(q',p') e^{-2i(p-p') (q-q')} =δ(p-P) δ(q-Q), \] and its inverse \[\iint_{-\infty}^\infty dq dp δ(p-P) δ(q-Q) e^{2i(p-p') (q-q')}=Δ(q',p'), \] where $Q,$ $P$ are the coordinate and momentum operators, respectively. We apply it to studying mutual converting formulas among $Q-P$ ordering, $P-Q$ ordering and Weyl ordering of operators. In this way, the contents of phase space quantum mechanics can be enriched.

arXiv:0903.1769Mar 10, 20090 citations

Quantum Mechanical Operators

George H. Duffey

Springer eBooksJan 1, 19840 citations

Quantum Mechanics in Phase Space (Operator Algebras and Their Applications )

Frank Hansen

数理解析研究所講究録Oct 1, 19800 citations

Operators and quantum mechanics

Elsevier eBooksJan 1, 19750 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.

Using operators' Weyl ordering expansion formula (Hong-yi Fan,\emph{\}J. Phys. A 25 (1992) 3443) we find new two-fold integration transformation about the Wigner operator

Δ(q',p')

(

q

-number transform) in phase space quantum mechanics,

\iint_{-\infty}^\infty dp' dq'/πΔ(q',p') e^{-2i(p-p') (q-q')} =δ(p-P) δ(q-Q),

and its inverse

\iint_{-\infty}^\infty dq dp δ(p-P) δ(q-Q) e^{2i(p-p') (q-q')}=Δ(q',p'),

where

Q,

P

are the coordinate and momentum operators, respectively. We apply it to studying mutual converting formulas among

Q-P

ordering,

P-Q

ordering and Weyl ordering of operators. In this way, the contents of phase space quantum mechanics can be enriched.