The tunneling time for a wave packet as measured with a physical clock

We study the time required for a wave packet to tunnel beyond a square barrier, or to be reflected, by envisaging a physical clock which ticks only when the particle is within the barrier region. The clock consists in a magnetic moment initially aligned with the x-axis which in the barrier region precesses around a constant magnetic field aligned with the z-axis, the motion being in the y direction. The values of the x and y components of the magnetic moment beyond or in front of the barrier allow to assign a tunnelling or reflection time to every fraction of the packet which emerges from the barrier and to calculate tunnelling times tau(T,x) and tau(T,y) and reflection times tau(R,x) tau(R,y). The times tau(T,x) and tau(T,y) (tau(R,x) and tau(R,y)) are remarkably equal, and independent of the initial position (in front of the barrier) or the packet.