Fig. 3: Particle speed and dwell time.
From: Energy–speed relationship of quantum particles challenges Bohmian mechanics
a, Particle speed v at the potential step as a function of energy Δ. Blue markers indicate the speed values obtained from the parabolic fitting of the population increase in the auxiliary waveguide. The speed obtained in this way is found to be a mirror-symmetric function of the energy Δ. For sufficiently large |Δ|, the data align well with \(v=\sqrt{2| \varDelta | /m}\) (green line). For negative Δ, this means that the lower the speed of the incoming particles, the faster the particles move within the step potential. The lowest speed values are found in the vicinity of Δ = 0 and are close to the theoretical prediction35 \({v}_{\min }=\sqrt{\hbar {J}_{0}/m}\approx 777\,{\rm{km}}\,{{\rm{s}}}^{-1}\) for J0 = 2π × 6.34 GHz. Error bars indicate the standard error of the mean. b, Dwell time as a function of energy. The dwell time τdwell (squares) describes how long particles are stored in the step potential and can be determined using τdwell = N/jin, where N is the number of stored particles and jin is the incoming particle current. Based on the energy–speed relation in a, a semiclassical estimate of τdwell can be introduced using τλ = λ/v, where λ is the decay length (circles). This estimate is found to be in good agreement with τdwell for a wide range of energies. The two solid lines give the expected behaviour for the dwell time and its semiclassical estimate. The given formulas are derived for a single waveguide step potential. Error bars for τdwell indicate the standard error of the mean. Error bars for τλ are derived from error propagation.
