Probability backflow and a new dimensionless quantum number

Abstract

Ahshct. Pure states of a free particle in non-relativistic quantum mechanics are demibed, in which the probability of finding the particle to have a negative x-coordinate increases over an arbivarily long, but finite. time interval, even though the x-component of the particle's velocity is certainly positive throughout that time internal It is shown that, for any s w of this type, the greatest amount of probability which can Bow back f" positive to negative x-values in this counter-intuitive way, over any given time interval, is equal b the largest eigenvalue of a certain Hermitian operator, and it is estimated numerically to have a value near 0.04. This value is not only independent of the length of the time interval and the mass of the particle, but is also independent of the value of Planck's constant. It r e f l a the smcture of Schriidinger's equation, rather than the values of the paramete~s appearing there. Backtlow of positive probability is related to the non-positivity of Wigner's density function, and can be rewded as arising from a ROW of negative probability in the same direction as the velocity. Generalizations are indicated, to the relativistic free electron, and to non-relativistic cases in which probability backflow occurs even in opposition to an arbitrarily strong constant force.

Quantum Inequalities in Quantum Mechanics

“…We remark that-again in contrast to [7]-our result is kinematical rather than dynamical: no specific Hamiltonian is invoked. Here, 'kinematic' refers to the kinematics of the Schrödinger representation, i.e., the (unique) regular representation of the Heisenberg commutation relations.…”

“…From this point of view, it is natural to seek bounds on its magnitude and extent. Bracken and Melloy [7] approached this question by showing that the probability P (t) of finding a right-moving particle in the left-hand half-line obeys Figure 1: Evolution of a wavepacket, illustrating the backflow phenomenon. From left to right, the plots show the position probability density at times t = −0.1, t = 0 and t = 0.1. for all t ≥ 0, where the dimensionless constant λ is the largest positive eigenvalue of the equation…”

“…In fact, µ x (∆) is simply the diagonal P ∆ (x, x) of the integral kernel of the spectral projection P ∆ of H on ∆, 7 given by…”

“…We will return to this phenomenon, which has come to be known as backflow [3,7,30], in Section 2.1. Another, related, phenomenon occurs in quantum field theory.…”

Bounds and enhancements for negative scattering time delays

“…A physical interpretation of t out a as an average detection time is not straightforward, since the flux J T is not a positive definite quantity, even for wave packets composed entirely by positive momenta [36]. One can show however that these "averages" do coincide with the ones calculated with the positive definite "ideal" time-of-arrival distribution of Kijowski [37,38].…”