# Two distinguishable particles in a box.

• peripatein
In summary, there is a formula for calculating the energy states of two free, distinguishable particles in a box of length L. The number of states for these particles depends on whether they are fermions or bosons. If the two particles have spin 3/2, they are both fermions and due to Pauli's exclusion principle, they cannot occupy the same state. This means that there could be 4 particles per energy level. For indistinguishable particles, the number of energy states would be 4!/(2!2!), and for distinguishable particles it would be 4!2!, at the elementary level.
peripatein
Hi,

## Homework Statement

I would like to determine the number of energy states two free, distinguishable particles in a box of length L have. I would then like to determine the number of states two free, indistinguishable particles, with spin 3/2 each, have in that box at the elementary level. Finally, determine the number of states in case these two particles with spin 3/2 each are distinguishable.

## The Attempt at a Solution

I am familiar with the following formula for the energy states
En=(hbar)2π2n2/(2mL2)
but am not sure how to proceed. If the particles are distinguishable, does that entail that one is a fermion whereas the other is a boson? I am not sure.
Furthermore, if the two particles have spin 3/2 each, that means they are both fermions, right? If that is correct, then, due to Pauli's exclusion principle, the two could not be at the same state. I also know that for spin 3/2 there could be 4 particles per energy level. But again I am not sure how to coherently process the given data and would appreciate some guidance.

In case the spin of each particle is 3/2, would the number of energy states be 4!/(2!2!) for indistinguishable particles and 4!2! for distinguishable particles? (at the elementary level)

## 1. What is the concept of "two distinguishable particles in a box"?

The concept of "two distinguishable particles in a box" refers to a theoretical scenario in which there are two particles confined within a closed, finite space or box. These particles are distinguishable from each other, meaning that they have different properties such as mass, charge, or spin.

## 2. What is the significance of studying this scenario?

Studying the scenario of two distinguishable particles in a box allows scientists to explore the fundamental principles of quantum mechanics, specifically in terms of the behavior and interactions of particles at the atomic and subatomic level. It also has practical applications in fields such as quantum computing and materials science.

## 3. How do the particles behave in this scenario?

In this scenario, the particles are subject to the laws of quantum mechanics, which dictate that they can exist in multiple states or positions simultaneously. This is known as superposition. The particles also have the ability to become entangled, meaning that their properties become correlated and they can influence each other's behavior even when separated by a distance.

## 4. What are the possible states of the particles in this scenario?

The possible states of the particles in this scenario are determined by their quantum numbers, which describe their energy, spin, and spatial distribution. These quantum numbers can have discrete values, leading to a discrete set of possible states for the particles.

## 5. How does this scenario relate to the concept of quantum entanglement?

This scenario is closely related to the concept of quantum entanglement, as the particles in the box have the potential to become entangled with each other. This means that their properties become linked and they can no longer be described as individual, independent particles. The study of two distinguishable particles in a box can help us better understand the mysterious phenomenon of quantum entanglement.

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