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arXiv:quant-ph/0202043 · Journal of Physics A Mathematical and General

Schwinger and Pegg-Barnett approaches and a relationship between angular and Cartesian quantum descriptions: II. Phase spaces

M. Ruzzi, D. Galetti

Published May 18, 2002• 17 citations
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Abstract

Following the discussion - in state-space language - presented in a preceding paper, we work on the passage from the phase-space description of a degree of freedom described by a finite number of states (without classical counterpart) to one described by an infinite (and continuously labelled) number of states. With this it is possible to relate an original Schwinger idea to the Pegg-Barnett approach to the phase problem. In phase-space language, this discussion shows that one can obtain the Weyl-Wigner formalism, for both Cartesian and angular coordinates, as limiting elements of the discrete phase-space formalism.

DOI
10.1088/0305-4470/35/21/311

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Following the discussion - in state-space language - presented in a preceding paper, we work on the passage from the phase-space description of a degree of freedom described by a finite number of states (without classical counterpart) to one described by an infinite (and continuously labelled) number of states. With this it is possible to relate an original Schwinger idea to the Pegg-Barnett approach to the phase problem. In phase-space language, this discussion shows that one can obtain the Weyl-Wigner formalism, for both Cartesian and angular coordinates, as limiting elements of the discrete phase-space formalism.
In this paper, we show that the Pegg–Barnett formalism accepts coherent states constructed as eigenstates of the annihilation operator, considering both the number and the phase. These operators are defined within a ( s + 1 ) -dimensional Hilbert space H s and with periodic conditions. The coherent states that we find are determined by the eigenvalue of the annihilation operator, which leads to a discrete spectrum. This approach allows calculation of the discrete-finite counterpart of the Wigner function in a phase space defined by the variables of number and phase.
The Weyl-Wigner description of quantum mechanical operators and states in classical phase-space language is well known for Cartesian systems. We describe a new approach based on ideas of Dirac which leads to the same results but with interesting additional insights. A way to set up Wigner distributions in an interesting non-Cartesian case, when the configuration space is a compact connected Lie group, is outlined. Both these methods are adapted to quantum systems with finite-dimensional Hilbert spaces, and the results are contrasted.
A comprehensive study is presented on the Weyl-Wigner formalism for rotation-angle and angular-momentum variables: the elements of kinematics are extended, the elements of dynamics are established, and the implications of rotational perodicity and angular-momentum quantization are investigated. Particular attention is paid to discreteness, and two of its consequences are emphasized: the importance of evenness and oddness, and the need to use two difference operators in a discrete domain, whereas one differential operator suffices in a continuous domain. These consequences are shown to strongly distinguish the Weyl-Wigner formalism for rotation-angle and angular-momentum variables from the well-known Weyl-Wigner formalism for Cartesian-position and linear-momentum variables. The point is made clear that the first of these formalisms cannot be regarded as a trivial and straightforward extension of the second. The rotational Wigner function is derived as the only bilinear form of the state vector that is real, has the natural invariances for rotational motion, and yields the correct distributions for the rotation-angle and angular-momentum variables as well as the appropriate expression for the transition probability between states. The conditions for its uniqueness are thus established. Its properties are described in detail and, in particular, its uniform boundedness is demonstrated. The rotational Wigner function and the associated correspondence between quantum operators and classical-like functions, as well as the relations they obey, are explored and are written so as to clearly exhibit the distinct contributions of evenness and oddness in the discrete domain of the angular-momentum eigenvalues, thus providing a most natural way to account for periodicity.Using the derivative, which acts on the continuous rotation-angle variable, and the forward and backward differences, which act on the discrete angular-momentum variable, the equation of motion for the rotational Wigner function is established. This equation is detailed for the following Hamiltonian forms: those that depend only on the angular-momentum variable, including, in particular, the free rotator, and those that are in the cosine of the rotation-angle variable. It is verified that the Weyl-Wigner formalism for rotation-angle and angular-momentum variables has the correct nonperiodic limit and that it properly reduces to the Weyl-Wigner formalism for Cartesian-position and linear-momentum variables. In order to illustrate the formalism, a careful analysis is carried out for the rotational Wigner function representing the energy eigenstates of a hindered rotator whose Hamiltonian is the sum of a term in the absolute value of the angular-momentum variable with a term in the cosine of the rotation-angle variable. For this hindered rotator, and within the approximation of a large absolute value of the angular-momentum variable, the equation of motion for the rotational Wigner function is solved for its stationary solutions, and the time-independent Schr\"odinger equation is also solved.
A simple formalism is introduced for describing the quantum phase of a single-mode field (or, equivalently, a harmonic oscillator) using a particular infinite-dimensional Hilbert space Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​. This space is spanned by vectors which represent states of the single field mode containing non-negative numbers of photons. It has a symmetry in the sense that it contains vectors in the neighborhood of both the vacuum state and the infinite photon number state. A striking property of Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​ is that it supports Hermitian and unitary phase operators. Indeed the Pegg-Barnett unitary phase operators are shown to converge strongly on the Hilbert space Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​ and so the correponding limit points are ordinary operators on Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​. This can be compared with the situation for the infinite-dimensional Hilbert space H conventionally used in quantum optics on which the Pegg-Barnett unitary phase operators converge weakly, in general.