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 tunneling or reflection time to every fraction of the packet which emerges from the barrier and to calculate tunneling times $\tau_{\rm T,x}$ and $\tau_{\rm T,y}$ and reflection times $\tau_{\rm R,x}$ and $\tau_{\rm R,y}$. The times $\tau_{\rm T,x}$ and $\tau_{\rm T,y}$ ($\tau_{\rm R,x}$ and $\tau_{\rm R,y}$) are remarkably equal, and independent of the initial position (in front of the barrier) of the packet.