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American Journal of Physics

Wigner distribution for angle coordinates in quantum mechanics

Narasimhaiengar Mukunda

Published Feb 1, 1979• 74 citations
Publisher

Abstract

The method of Wigner distribution functions, and the Weyl correspondence between quantum and classical variables, are extended from the usual kind of canonically conjugate position and momentum operators to the case of an angle and angular momentum operator pair. The sense in which one has a description of quantum mechanics using classical phase-space language is much clarified by this extension.

DOI
10.1119/1.11869

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The method of Wigner distribution functions, and the Weyl correspondence between quantum and classical variables, are extended from the usual kind of canonically conjugate position and momentum operators to the case of an angle and angular momentum operator pair. The sense in which one has a description of quantum mechanics using classical phase-space language is much clarified by this extension.
The problem of constructing physically and mathematically well-defined Wigner functions for the canonical pair angle $\ensuremath{\theta}$ and angular momentum ppp is solved. While a key element for the construction of Wigner functions for the planar phase space ${(q,p)\ensuremath{\in}{\mathbb{R}}^{2}}$ is the Heisenberg-Weyl group, the corresponding group for the cylindrical phase space ${(\ensuremath{\theta},p)\ensuremath{\in}{S}^{1}\ifmmode\times\else\texttimes\fi{}\mathbb{R}}$ is the Euclidean group E(2)E(2)E(2) of the plane and its unitary representations. Here the angle $\ensuremath{\theta}$ is replaced by the pair $(cos\ensuremath{\theta},sin\ensuremath{\theta})$, which corresponds uniquely to the points on the unit circle. The main structural properties of the Wigner functions for the planar and the cylindrical phase spaces are strikingly similar. A crucial role is played by the sincsincsinc function, which provides the interpolation for the discontinuous quantized angular momenta in terms of the continuous classical ones, in accordance with the famous Whittaker cardinal function well known from interpolation and sampling theories. The quantum mechanical marginal distributions for the angle (continuous) and angular momentum (discontinuous) are, as usual, uniquely obtained by appropriate integrations of the $(\ensuremath{\theta},p)$ Wigner function. Among the examples discussed is an elementary system of simple cat states.
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.
Abstract Coherent states, in Wigner representation of quantum mechanics which is classical in structure with Wigner function playing the role of classical phase space probability distribution, and standard quantum mechanical expression for average values of corresponding Weyl quantum operators, general expression for matrix elements of the operator corresponding to phase of the oscillator by Weyl procedure, are obtained in the |n〉 basis. The general expression for matrix elements in the same basis of any other operator is also obtained. As the only mathematical technique necessary in the proposed procedure is a simple change of variables to polar coordinates in the corresponding integrals, this way to introduce Weyl phase operator, and some other operators in Weyl quantization, greatly simplifies necessary derivations and calculations.
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.
The Weyl correspondence for obtaining quantum operators from functions of classical coordinates and momenta is known to be incorrect. To calculate quantum-mechanical expectation values as phase-space averages with the Wigner density function, one cannot use classical functions but must use Weyl transforms. These transforms are defined and their properties derived from quantum mechanics. Their properties are expressed in terms of a Hermitian operator Δ(Q, K) whose Weyl transform is a δ function. The Wigner function is the transform of the density operator. Every Weyl transform is exhibited as a difference of two functions which are nonnegative on the phase spece. Weyl transforms do not obey the algebra of classical functions. In the classical limit ℏ → 0, Weyl transforms become classical functions, the Wigner function becomes nonnegative throughout the phase space, and the Hilbert space is spanned by an orthonormal set of vectors which are simultaneous eigenkets of the commuting coordinate and momentum operators.