Minimum-uncertainty states for noncanonical operators.

Abstract

The usual uncertainty relation \ensuremath{\Delta}${\mathit{A}}^{2}$\ensuremath{\Delta}${\mathit{B}}^{2}$\ensuremath{\ge}〈C${\mathrm{〉}}^{2}$/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}${\mathit{A}}^{2}$ and \ensuremath{\Delta}${\mathit{B}}^{2}$ when 〈C〉=0. For those operators, therefore, the general uncertainty relation \ensuremath{\Delta}${\mathit{A}}^{2}$\ensuremath{\Delta}${\mathit{B}}^{2}$\ensuremath{\ge}[〈C${\mathrm{〉}}^{2}$+〈F${\mathrm{〉}}^{2}$]/4 where 〈F〉=〈AB+BA〉-2〈A〉〈B〉 is better suited to determine the lower bound on \ensuremath{\Delta}${\mathit{A}}^{2}$\ensuremath{\Delta}${\mathit{B}}^{2}$. 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.