Can a particle's wavefunction become perfectly flat in empty space over time?

[asimov42]

Hi everyone,

I have a rather naive question regarding wave-packet spreading. I understand that a particle's wavefunction will spread out in space over time (assuming no measurements are made) due to momentum uncertainty. Now my question:

If we have an isolated particle in otherwise empty space, and we measure, say, it's position, immediately after the measurement, the particle's wavefunction with start to spread out in space. Is it every possible for the particle's wavefunction to become perfectly `flat' (i.e. to have an exactly uniform distribution over all of space) in finite time? (considering the particle alone - if the rest of the universe were empty)

Assuming space is infinite, this would seem impossible to me - if we integrated the square of the wavefunction over any finite region of space, the probability of finding the particle there would be zero?

Thanks.

J.

 

[per.sundqvist]

No. not in finite time, size ~ sqrt(h*t/m), where t=time.

Why is that impossible? Its the same as in classical physics. You have one particle (normalization in QM) in an infinite box, What is the probability you find it in a volume of 1dm^3? =0