Time evolution of a detected particle

[PeterDonis]

Or just tunneling away

If the potential in the box walls has only a finite width, yes.

 

[anuttarasammyak]

If we had already the particle confined in a very narrow 1D box (to the extent of almost a point $x_0$), then the speed of particle seems meaningless let alone the light speed!

You must investigate by what materials you make the box so that it does not rupture by high pressure of the particle in it. Feeling pressure is measurement of particle momentum so velocity.

 

[Roberto Pavani]

what materials

I think there is a distinction between the mathematical model and the physical system. Once we start asking what materials the box is made of, we are talking about real-world physics. In that case, confinement requires a physical interaction and barriers are finite. Some of the arguments above, however, rely on idealized infinite walls that are mathematically useful but not physically realizable.