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Proceedings of the Samahang Pisika ng Pilipinas

Independence of minimum uncertainty states on generalized Wigner functions

Herbert B. Domingo

Published May 10, 2017• 0 citations
OpenAlex

Abstract

With the correspondence between operator ordering rules and probability distribution functions, moments can be calculated in many ways. We show here the independence of minimum uncertainty relations on the choice of Hermitian ordering rule. Generalized Wigner functions were expressed in terms of ordering real-valued function ϴ(x) for x ϵ R satisfying ϴ(0)=1. Uncertainties of position and momentum are obtained for a Gaussian state. Results confirm that the minimum uncertainty relation holds even when the Hermitian ordering rule is modified.

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With the correspondence between operator ordering rules and probability distribution functions, moments can be calculated in many ways. We show here the independence of minimum uncertainty relations on the choice of Hermitian ordering rule. Generalized Wigner functions were expressed in terms of ordering real-valued function ϴ(x) for x ϵ R satisfying ϴ(0)=1. Uncertainties of position and momentum are obtained for a Gaussian state. Results confirm that the minimum uncertainty relation holds even when the Hermitian ordering rule is modified.
The eigenvalues and eigenvectors of a normalized gaussian operator do not seem to have been previously considered. I solve this problem for 1-dimensional translational systems. I also address the question as to whether a gaussian operator is a density operator. To answer that question, it is first necessary to be sure what conditions must be satisfied, so a short review of density operators is given. Since position and momentum do not commute in quantum mechanics, it is useful to start with the consequences of the noncommutation, which is generally the Schrödinger-Robertson uncertainty relation, which is also briefly reviewed. It is found that the question of whether a gaussian operator is a density operator is directly tied to this uncertainty relation. Since the Wigner function is the phase space representation of a translational density operator, it is natural to consider the gaussian phase space function associated with a gaussian operator and to compare the phase space and operator properties. Throughout such discussions, the independent parameters in these functions are the first and second moments of position and momentum. The application of this formalism to the free translation and spreading of a gaussian packet is given and shows the formal similarity between classical and quantum behavior, whereas the literature standardly only considers the pure state case (equivalent to a single wavefunction).
The mathematical content of the uncertainty principle is contained in special states akin to, and including, states of minimum uncertainty. Specific uncertainty relations are produced by averaging observables in these states. The approach is shown to give more flexible and complete expressions of indeterminacy than the familiar generalized Heisenberg inequality: statistical measures other than variances can be employed, the method is not limited to describing uncertainties in terms of products of statistical dispersions, and the resulting inequalities have the most stringent bounds attainable independent of the system state. Applications are then made to find uncertainty relations for position-momentum average displacements from the means, time-energy indeterminacy, and uncertainties in angular momentum components.
We discuss two sets of conditions that are necessary and sufficient for a function defined on phase space to be a Wigner distribution function (WDF). The first set is well known and involves the function itself; the second set is less familiar and involves the function's ``symplectic'' Fourier transform. After explaining why these two sets are equivalent, we explore some properties and applications of the second one. Among other things, we show that that set includes the position-momentum uncertainty relations as a special case, and in doing so we give a new derivation of them. This derivation itself serves as the starting point for the discussion of a quantum-mechanical moment problem. It also enables us to exhibit a real-valued phase-space function that obeys the uncertainty relations but that is not a WDF.
The usual uncertainty relation \ensuremath{\Delta}A2{\mathit{A}}^{2}A2\ensuremath{\Delta}B2{\mathit{B}}^{2}B2\ensuremath{\ge}〈C〉2{\mathrm{〉}}^{2}〉2/4 between two Hermitian operators A and B satisfying the noncanonical commutation relation [A,B]=iC, where C is not a constant multiple of the unit operator, fails to give a nontrivial lower bound on the product of the variances \ensuremath{\Delta}A2{\mathit{A}}^{2}A2 and \ensuremath{\Delta}B2{\mathit{B}}^{2}B2 when 〈C〉=0. For those operators, therefore, the general uncertainty relation \ensuremath{\Delta}A2{\mathit{A}}^{2}A2\ensuremath{\Delta}B2{\mathit{B}}^{2}B2\ensuremath{\ge}[〈C〉2{\mathrm{〉}}^{2}〉2+〈F〉2{\mathrm{〉}}^{2}〉2]/4 where 〈F〉=〈AB+BA〉-2〈A〉〈B〉 is better suited to determine the lower bound on \ensuremath{\Delta}A2{\mathit{A}}^{2}A2\ensuremath{\Delta}B2{\mathit{B}}^{2}B2. The implications of the general uncertainty relation and the properties of the minimum-uncertainty states, i.e., the states for which the general uncertainty relation is satisfied with equality, are discussed. The minimum-uncertainty states are found to fall in two different classes. One is the usually studied class for which 〈F〉=0, that is, the case when the usual uncertainty relation holds. The other is the hitherto unnoticed class for 〈C〉=0, that is, when the usual uncertainty relation is redundant. The squeezing properties of the present class of minimum-uncertainty states is discussed by defining squeezing in the light of the general uncertainty relation.