# Two Particles in a Box: Similar Velocities, Momentum, or Kinetic Energy?

• nonequilibrium
In summary, when considering two interacting particles, their momenta would likely approach equality over a large number of interactions, while their kinetic energies would tend to be equal due to a theorem in Thermodynamics.
nonequilibrium
Hello,

Say you have a box with two (interacting) particles in them. If you had to venture a guess, what would be most reasonable: that they both have similar velocities, similar momenta or similar kinetic energies? (Or perhaps none of the above?)

NB: with "similar" I mean "of comparable magnitude"

I'd say that this question can't probably be answered in a general way. If the interaction involved is physical (like the two particles bouncing into each other), my guess would be that over an arbitrarily large amount of interactions the two particles momenta would approach equality. I would say neither of the other two values would approach each other, and that this situation would change if the interaction changed.

And why would you think that the momenta would approach equality?

More a guess than anything, it seems to be than in realistic collisions between two bodies the difference in momentum would be less after the collision.

Actually, there is a theorem in Thermodynamics that states that on average, available energy is distributed equally between available degrees of freedom. That means that if particles are identical, they will, on average, have equal kinetic energies.

## 1. What is the Two Particles in a Box experiment?

The Two Particles in a Box experiment is a thought experiment in quantum mechanics that involves two identical particles confined in a one-dimensional box. The particles are assumed to have similar velocities, and the experiment explores the relationship between their momenta and kinetic energy.

## 2. What is the significance of similar velocities in this experiment?

The assumption of similar velocities in this experiment allows us to simplify the calculations and focus on the relationship between momentum and kinetic energy. It also allows us to explore the quantum mechanical principles of indistinguishability and symmetry in the behavior of identical particles.

## 3. How does momentum change in this experiment?

In this experiment, the momentum of each particle remains constant. However, when the particles are confined to the same box, their momenta combine to produce a collective momentum that is different from the individual momenta. This phenomenon is known as the "momentum paradox" and is a key concept in quantum mechanics.

## 4. What is the relationship between momentum and kinetic energy in this experiment?

The relationship between momentum and kinetic energy in this experiment is described by the Heisenberg Uncertainty Principle. This principle states that the more precisely we know the momentum of a particle, the less we know about its position, and vice versa. Therefore, the two particles in the box will have a range of possible kinetic energies, depending on the accuracy of their momenta measurements.

## 5. How does this experiment relate to real-world scenarios?

Although this experiment is a thought experiment, it has real-world applications in fields such as quantum computing and particle physics. It allows us to understand the behavior of identical particles in confined spaces and provides insights into the fundamental principles of quantum mechanics. Additionally, the concepts explored in this experiment have practical applications in areas such as cryptography and data encryption.

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