Resolution to the quantum phase problem

Defining the observable φ canonically conjugate to the number observable N has long been an open problem in quantum theory. The problem stems from the fact that N is bounded from below. Here we show how to define the absolute phase observable Φ ≡ |φ| by suitably restricting the Hilbert space of x and p like variables. This Φ is actually the absolute value of the phase and has the correct classical limit. A correction to the “cosine” C and “sine” S operators of Carruthers and Nieto is obtained. PACS numbers: 42.50.Dv, 03.65.Vf, 03.65.Ca, 03.65.Ta Typeset using REVTEX E-mail: [email protected] Tel: 972-2-6586547 , Fax: 972-2-5611519 1 In 1927, Dirac [1] proposed a phase observable φ, supposedly a canonical conjugate to the number operator N = aa. The idea was to decompose the annihilation operator in the following form: a = exp(−iφ)N1/2. Many years later, Susskind and Glogower [2] pointed out that such a procedure was not correct because E, defined by a = EN, is not a unitary operator, and thus cannot be expressed as exp(−iφ). Despite the nonunitary nature of E and E, Carruthers and Nieto [3] have defined Hermitian “sine” and “cosine” operators S = (1/2i)(E−E†) and C = (1/2)(E+E), respectively. However, these operators cannot represents the exact sine and cosine of the phase observable since, for example, S +C 6= 1. Since that time many new techniques have been developed to define a phase operator [4,5]. In a number of theories, such as Susskind and Glogower’s [2], the Barnett and Pegg formalism [6], etc, the Hermitian phase operator is not well defined. Some other theories do not pass the Barnett and Pegg “acid-test” [7]: the eigenstates of the number operator do not represent states of indeterminate phase. In an excellent critical review, Lynch [5] argues that there is as yet no satisfactory solution to the quantum phase problem. In the present work we devise a new approach which succeeds in giving the absolute value Φ ≡ |φ| of the phase observable. As Moshinsky and Seligman [8] have shown many years ago, the classical canonical transformations to action and angle variables for harmonic oscillator (and some other systems) turn out to be nonbijective (not one-to-one onto). Hence, the fact that it is possible to construct only the absolute phase observable, and not the phase observable itself, is not surprising even from the classical point of view. Two problems arise in the construction of a phase observable. The first one (also the easier to overcome), is that the phase operator φ is supposed to be an angle operator. Hence, it is restricted to a finite interval which is chosen, somewhat arbitrarily, to be (−π, π]. Thus, the matrix elements of [N, φ] in the number state basis |n〉, 〈n|[N, φ]|n〉 = (n− n)〈n|φ|n〉 (1) vanish for n = n because |〈n|φ|n〉| ≤ π. This implies that [N, φ] 6= i so that φ is not canonical to N. The same problem appears also in the commutation relation of an angular 2 momentum Jz component and its associated angleΘ [9,10]. The solution to this last problem is well known: the commutation relation is to be changed to [Jz,Θ] = ih̄ (1− 2πδ(Θ− π)) , −π < θ ≤ π, (2) where Θ can be expressed as a 2π-periodic function of an unrestricted angle operator [5]. The second problem arises due to the fact that the number operator is bounded from below. As we now show, the commutation relation like (2) does not hold. The matrix elements of [N, φ] taken now in the phase basis |φ〉, 〈φ|[N, φ]|φ〉 = (φ − φ)〈φ|N|φ〉 = iδ(φ− φ), −π < φ, φ < π (3) imply that 〈φ|N|φ〉 = −iδ(φ− φ ) φ− φ′ = i d dφ δ(φ− φ). (4) Defining a state |ψ〉 = ∫ π −π dφ ψ(φ)|φ〉 in the basis of the phase states (ψ(φ) is a complex function of φ), we find the expected result 〈ψ|N|ψ〉 = ∫ π −π dφ ∫ π −π dφ ψ(φ)ψ(φ)〈φ|N|φ〉 = ∫ π −π dφ ψ(φ) (