# Is quantum uncertainty of particle location bound by the speed of light?

If I measure the location of a quantum particle and then measure its location 1 second later, is there a probability larger than zero that I find it in a location farther away from the first location than 1 light second?

Follow up question if the distance IS bound by the speed of light: in that case is the number of possible distinct locations in which I could find the particle finite rather than infinite?

If Schrödinger equation is used, then it depends on the boundary conditions. The limit of light speed would induce a moving boundary, this can't be solved easily analytically, or by using finite elements in time.

For the second question, I'm not at that level to know if string theories or others solve the question to know if space (time) is discrete or continuous. As far as I know a space basis is continuous in standard quantum mechanics, but there is Planck's length too.

First question. After a measurement, the probability of finding a particle at a distance $$d>ct$$ in a follow up measurement must be zero. Of course you would need relativistic quantum mechanics for that.

Second question. This does not put any restriction on the maximum delocalization of a particle. An electron described by a plane wave is "delocalized" in the whole space, i.e. it could be found in any point in the whole space if a measurement is performed. It however put a restriction on the amount of time it takes to "delocalize", i.e. to the electronic wave function to spread in space.

• But isn't any finite volume of space quantized in a way that there can be only a finite amount of different locations that are physically relevant? (I'd assume something like volume/planck_length^3 or so.. Commented Jul 4, 2022 at 20:00
• If you consider an infinite volume you have a continuum of energies. Of course, after the first measurement it would take infinite time to get a plane wave. In any case, even in a finite volume, a discrete energy spectrum does not mean a discrete number of locations. You still have a continuum of possible positions. Unless you want to discretize the space with some form of quantum gravity. But this is not something well known I believe. Commented Jul 4, 2022 at 20:39
• I think you never evaluated rhe Schrödinger propagator. S. 's equation is a non-relativistic equation. It would be surprising to derive the relativistically correct result from it. Commented Jul 4, 2022 at 20:57
• @matthias You may like to look at this page re the quantization of spacetime: physics.stackexchange.com/q/9720/123208 & some of the numerous links on that page. Commented Jul 5, 2022 at 0:20
• @GiorgioP no one spoke about the Schrodinger equation ... Commented Jul 5, 2022 at 10:13