A New Wave Equation For a Continuous Nondemolition Measurement

A stochastic model for nondemolition continuous measurement in a quantum system is given. It is shown that the posterior dynamics, including a continuous collapse of the wave function, is described by a nonlinear stochastic wave equation. For a particle in an electromagnetic field it reduces the Schrödinger equation with extra imaginary stochastic potentials. According to the statistical interpretation of the wave function ψ(t,x), a quantum measurement of a simple observable X̂ of a particle reduces it to an eigenfunction of X̂ by a random jump (collapse, or reduction of the wave packet). Such a jump cannot be described by the Schrödinger equation because the latter corresponds to the unobserved particle. To ignore this fact gives rise to the various quantum paradoxes of the Zeno kind. The aim of this paper is to derive a dissipative wave equation with an extra stochastic nonlinear term describing the quantum particle under continuous nondemolition observation. This derivation can be obtained in the framework of quantum stochastic theory of continuous measurements developed in refs. [4, 2, 5], and the quantum nonlinear filtering method recently announced in [6]. The derivation can be done in general terms but for simplicity here we consider a quantum spinless particle in an electromagnetic field. The unperturbed dynamics of such a particle with mass m is given by the Schrödinger equation ih̄∂ψ/∂t = [(U + ih̄∇)/2m+ V ]ψ, (1) where U = eA/c and V = eΦ, A(t,x) and Φ(t,x) are the scalar and vector field potentials. ∗Permanent address: M.I.E.M., B. Vusovsky 3/12, Moscow 109028, USSR.