Uncertainty , Trajectories ,
نویسنده
چکیده
First we show explicitly how uncertainty can arise in a trajectory representation. Then we show that the formal utilization of the WKB like hierarchy structure of dKdV in the description of (X,ψ) duality does not encounter norm constraints. 1. BACKGROUND In a previous paper [4] (working with stationary states and ψ satisfying the Schrödinger equation (SE) (A0) − (~2/2m)ψ′′+V ψ = Eψ) we suggested that the notion of uncertainty in quantum mechanics (QM) can be phrased as incomplete information. The background theory here is taken to be the trajectory theory of Bertoldi-Faraggi-Matone-Floyd (cf. [1, 2, 3, 5, 11, 12, 13, 14]). The idea in [4] was simply that Floydian microstates satisfy a third order quantum stationary Hamilton-Jacobi equation (QSHJE) (1.1) 1 2m (S 0) 2 + W(q) +Q(q) = 0; Q(q) = ~ 2 4m {S0; q}; W(q) = − ~ 2 4m {exp(2iS0/~); q} ∼ V (q)−E where (A1) {f ; q} = (f /f ) − (3/2)(f /f ′)2 is the Schwarzian and S0 is the Hamilton principle function. If one recalls that the EP of Faraggi-Matone can only be implemented when S0 6= const one may think of (A2) ψ = Rexp(iS0/~) with Q = −~2R′′/2mR and (R2S′ 0) ′ = 0 where (A3) S 0 = p and mQq̇ = p with mQ = m(1− ∂EQ) and t ∼ ∂ES0. Thus microstates require three initial or boundary conditions in general to determine S0 whereas the SE involves only two such conditions. Hence in dealing with the SE in the standard QM Hilbert space formulation one is not using complete information about the “particles” described by microstate trajectories. The price of underdetermination is then uncertainty in q, p, t for example. In the present note we will make this more precise and add further discussion. 2. SOME CALCULATIONS It is shown in [12] that one has generally a formula (2.1) e0 = e w + il̄ w − il Date: August, 2003. email: [email protected].
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تاریخ انتشار 2003