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arXiv:0712.2704 · Journal of Physics A Mathematical and Theoretical

On Bargmann representations of the Wigner function

Fernando Parisio

Published Jan 23, 2008• 1 citation
arXivHTMLPDF

Abstract

By using the localized character of canonical coherent states, we give a straightforward derivation of the Bargmann integral representation of the Wigner function ( W ). A non-integral representation is presented in terms of a quadratic form W ∝ V † F V , where F is a self-adjoint matrix whose entries are tabulated functions and V is a vector depending, in a simple recursive way, on the derivatives of the Bargmann function. Such a representation may be of use in numerical computations. We discuss a relation involving the geometry of the Wigner function and the spatial uncertainty of the coherent state basis we use to represent it.

DOI
10.1088/1751-8113/41/5/055305

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By using the localized character of canonical coherent states, we give a straightforward derivation of the Bargmann integral representation of the Wigner function ( W ). A non-integral representation is presented in terms of a quadratic form W ∝ V † F V , where F is a self-adjoint matrix whose entries are tabulated functions and V is a vector depending, in a simple recursive way, on the derivatives of the Bargmann function. Such a representation may be of use in numerical computations. We discuss a relation involving the geometry of the Wigner function and the spatial uncertainty of the coherent state basis we use to represent it.
It is shown that the representation matrices of the quantum mechanical group in the coherent state basis can be used as bases for the expansion of arbitrary square integrable functions on the group. Although this basis is nonorthogonal the expansion can be inverted in a manner similar to an orthogonal basis. Next a coherent state ‘‘wave function’’ is defined in analogy with the Schrödinger wave function and the above matrix elements are used to determine the action of the group in the space of the coherent state wave functions. The function space in this realization is Bargmann’s Hilbert space of analytic functions. It is shown that the mixed basis matrix element between the coordinate and coherent state bases is essentially the integral kernel of Bargmann. The transpose of this matrix element yields the kernel for the inversion of the transform. In this construction the Bargmann transform appears as the unitary transformation connecting the Schrödinger and coherent state wave functions. This considerably simplifies Bargmann’s original arguments and yields the integral transform as well as its inversion formula in a simple way.
The first part is a systematical representation of the Wigner quasiprobability in representations by canonical variables and pairs of complex conjugate variables with particular attention to the reconstruction of the density operator from the Wigner quasiprobability to show their equivalence. All formulae are derived from basic postulates of quantum mechanics. Problems of operator orderings are touched. The second part considers the integration over the amplitude in the Wigner quasiprobability and its specialization for coherent states and for even and odd coherent states and provides phase distributions which are presented in graphics that are new in their kind. It is demonstrated that the Heitler number phase uncertainty relation is incorrect. The problem of the transition from quantum to classical distribution functions by ℏ→0 is discussed with the conjecture that the inertial point of the distributions plays a basic role.
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.
Representations of quantum state vectors by complex phase space amplitudes, complementing the description of the density operator by the Wigner function, have been defined by applying the Weyl–Wigner transform to dyadic operators, linear in the state vector and anti-linear in a fixed 'window state vector'. Here aspects of this construction are explored, and a connection is established with Gabor's 'windowed Fourier transform'. The amplitudes that arise for simple quantum states from various choices of windows are presented as illustrations. Generalized Bargmann representations of the state vector appear as special cases, associated with Gaussian windows. For every choice of window, amplitudes lie in a corresponding linear subspace of square-integrable functions on phase space. A generalized Born interpretation of amplitudes is described, with both the Wigner function and a generalized Husimi function appearing as quantities linear in an amplitude and anti-linear in its complex conjugate. Schrödinger's time-dependent and time-independent equations are represented on phase space amplitudes, and their solutions described in simple cases.
Starting from Wigner's definition of the function named now after him we systematically develop different representation of this quasiprobability with emphasis on symmetric representations concerning the canonical variables ( ) , q p of phase space and using the known relation to the parity operator.One of the representations is by means of the Laguerre 2D polynomials which is particularly effective in quantum optics.For the coherent states we show that their Fourier transforms are again coherent states.We calculate the Wigner quasiprobability to the eigenstates of a particle in a square well with infinitely high impenetrable walls which is not smooth in the spatial coordinate and vanishes outside the wall boundaries.It is not well suited for the calculation of expectation values.A great place takes on the calculation of the Wigner quasiprobability for coherent phase states in quantum optics which is essentially new.We show that an unorthodox entire function plays there a role in most formulae which makes all calculations difficult.The Wigner quasiprobability for coherent phase states is calculated and graphically represented but due to the involved unorthodox function it may be considered only as illustration and is not suited for the calculation of expectation values.By another approach via the number representation of the states and using the recently developed summation formula by means of Generalized Eulerian numbers it becomes possible to calculate in approximations with good convergence the basic expectation values, in particular, the basic uncertainties which are additionally represented in graphics.Both considered examples, the square well and the coherent phase states, belong to systems with ( ) 1,1 SU symmetry with the same index 1 2 k = of unitary irreducible representations.