Exploring Wave-packet Spreading: Understanding Particle Behavior in Empty Space

[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

 

[Entropia]

Smith,

I can provide some insight into this question about wave-packet spreading. First, it is important to understand that the behavior of particles in empty space is governed by quantum mechanics, which is a probabilistic theory. This means that even though we can make predictions about the behavior of particles, we cannot determine their exact position or momentum at any given time. Instead, we can only calculate the probability of finding a particle in a certain location or with a certain momentum.

In the scenario described, where we have an isolated particle in empty space, the particle's wavefunction will indeed spread out over time due to momentum uncertainty. However, it is highly unlikely that the wavefunction will ever become perfectly flat or have a uniform distribution over all of space. This is because the probability of finding the particle in any finite region of space will always be non-zero, even if it is very small.

Furthermore, the concept of infinite space also plays a role in this scenario. If we were to integrate the square of the wavefunction over any finite region of space, the probability of finding the particle there would indeed be zero. However, in an infinite space, there are an infinite number of regions to integrate over, making it impossible for the probability to truly be zero in all of them.

In conclusion, while it is theoretically possible for the particle's wavefunction to become perfectly flat in infinite time, it is highly improbable and goes against the principles of quantum mechanics. The particle's wavefunction will continue to spread out in space, but it will never become completely flat or uniform in distribution. I hope this helps to clarify the concept of wave-packet spreading and particle behavior in empty space.