Nonclassical Evolution of a Free Particle

A conditional kinetic energy is defined in terms of the Wigner distribution. It is shown that this kinetic energy may become negative for negative Wigner distributions. A free particle wave packet with negative kinetic energy will spread in a nonclassical manner. 1 The Free Particle In classical statistical mechanics, a particle is described in terms of a nonnegative phase space distribution P (x, p; t) ≥ 0. (1) This is the probability of the particle having position x and momentum p at the time t. The time evolution of the phase space distribution for a free particle is ∂P ∂t = − p m ∂P ∂x . (2) That quantum mechanics can be formulated in a similar manner was first discovered by Wigner [1]. However, the Wigner distribution W (x, p; t) has the nonclassical feature of being negative for certain quantum states. In fact, among pure states only the gaussian states have nonnegative Wigner distributions [2]. However, even if a free particle is in a state with negative Wigner distribution, it will not necessarily behave exhibit different dynamics than an ensemble of classical free particles [3]. For instance, the Wigner distribution obeys the same equation of motion (2) as a classical phase space distribution. Bracken and Melloy [4] recently found that the probability of observing a free particle in a certain region of space may increase even though there is zero probability for the particle to have momenta pointing towards this region. They explained this effect in terms of negative probability. Here we shall study another dynamical effect which defies explanation in terms of the classical model (1) and (2). To this end, we consider the modulus of x, 〈| x |〉 = ∫