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

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In summary, the conversation discusses the concept of wave-packet spreading in quantum mechanics. The question is posed whether it is possible for a particle's wavefunction to become perfectly uniform in distribution over all of space in finite time. The response is that it is impossible, given the particle is isolated in an infinite space. This is similar to classical physics, where the probability of finding a particle in a specific volume is zero.
  • #1
asimov42
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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.
 
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  • #2
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
 
  • #3
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.
 

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

1. What is a wave-packet and how does it relate to particle behavior?

A wave-packet is a mathematical concept used to describe the behavior of a particle. It represents a localized disturbance or wave in space and time. It relates to particle behavior because it describes the probability of a particle's position and momentum at a specific point in time.

2. How does empty space affect wave-packet spreading?

In empty space, a wave-packet will spread out or disperse over time. This is due to the wave-like nature of particles and the uncertainty principle, which states that the more precisely we know a particle's position, the less we know about its momentum. As a result, the wave-packet will spread out to cover a larger area in space.

3. What factors influence wave-packet spreading?

Several factors can influence wave-packet spreading, including the initial size and shape of the wave-packet, the energy and momentum of the particle, and any external forces acting on the particle. Additionally, the quantum nature of particles and their interactions with empty space can also affect wave-packet spreading.

4. How does understanding wave-packet spreading contribute to our understanding of particle behavior in empty space?

Studying wave-packet spreading allows us to better understand the wave-like nature of particles and how they behave in empty space. It also helps us understand the impact of various factors on particle behavior and how they contribute to the overall behavior of a particle over time.

5. Are there any practical applications of studying wave-packet spreading?

Yes, there are practical applications of studying wave-packet spreading. Understanding how particles behave in empty space is crucial for various fields, including quantum mechanics, particle physics, and even technology development. It also has implications for understanding various phenomena in the natural world, such as the behavior of atoms and molecules.

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