On the 'orthogonalization' of the minimum uncertainty states between the position and the rational function of the momentum

Abstract

この論文では,位置演算子Qと運動量演算子Pの有理関数g(P)を用いた演算子Q+ig(P)の固有ベクトル系を,形式的に直交化する方法,すなわち,Qとg(P)の最小不確定状態を直交化する方法を提案する。この種の直交化は,空間をテンソル積により拡張することによりQとg(P)を「可換化」し,付加された空間の真空状態に射影することをもとにしている。特に,連続ウェーブレット変換に対応するg(P)=-kP^<-1>の場合について詳細に調べる。

In this paper, a method of the extension of a Hilbert space is proposed for a formal orthogonalization of the eigenvector system of the operator Q+ig(P) where Q denotes the position-coordinate operator and the and g(P) is a kind of rational function of the momentum operator, in other words, for the orthogonalization of the minimum uncertainty states between Q and g(P). This kind of orthogonalization is based on the 'commutabilization' between Q and g(P) by the space extension by tensor product and the projection into the analogue of the vacuum vector in the additional space. Especially, the special case with g(P)=-kP^<-1>, which is corresponding to a kind of Naimark extension of the continuous wavelet transfonntion, is investiceted in detail.