Energy-speed relationship of quantum particles challenges Bohmian mechanics

Classical mechanics characterizes the kinetic energy of a particle, the energy it holds due to its motion, as consistently positive. By contrast, quantum mechanics describes the motion of particles using wave functions, in which regions of negative local kinetic energy can emerge¹. This phenomenon occurs when the amplitude of the wave function experiences notable decay, typically associated with quantum tunnelling. Here, we investigate the quantum mechanical motion of particles in a system of two coupled waveguides, in which the population transfer between the waveguides acts as a clock, allowing particle speeds along the waveguide axis to be determined. By applying this scheme to exponentially decaying quantum states at a reflective potential step, we determine an energy-speed relationship for particles with negative local kinetic energy. We find that the smaller the energy of the particles-in other words, the more negative the local kinetic energy-the higher the measured speed inside the potential step. Our findings contribute to the ongoing tunnelling time debate^2-6 and can be viewed as a test of Bohmian trajectories in quantum mechanics^7-9. Regarding the latter, we find that the measured energy-speed relationship does not align with the particle dynamics postulated by the guiding equation in Bohmian mechanics.