PROBABILITY BACKFLOW AND A NEW DIMENSIONLESS QUANTUM NUMBER

Pure states of a free particle in non-relativistic quantum mechanics a re described, in which the probability of finding the particle to have a negative x-coordinate increases over an arbitrarily long, but finit e, time interval, even though the x-component of the particle's veloci ty is certainly positive throughout that time interval. It is shown th at, for any state of this type, the greatest amount of probability whi ch can flow back from positive to negative x-values in this counter-in tuitive way, over any given time interval, is equal to the largest eig envalue of a certain Hermitian operator, and it is estimated numerical ly to have a value near 0.04. This value is not only independent of th e length of the time interval and the mass of the particle, but is als o independent of the value of Planck's constant. It reflects the struc ture of Schrodinger's equation, rather than the values of the paramete rs appearing there. Backflow of positive probability is related to the non-positivity of Wigner's density function, and can be regarded as a rising from a flow of negative probability in the same direction as th e velocity. Generalizations are indicated, to the relativistic free el ectron, and to non-relativistic cases in which probability backflow oc curs even in opposition to an arbitrarily strong constant force.