Two distinguishable particles in a box.

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SUMMARY

The discussion focuses on calculating the number of energy states for two distinguishable and indistinguishable particles in a box of length L, specifically with spin 3/2. The formula for energy states is given as En=(ħ²π²n²)/(2mL²). It is established that both particles with spin 3/2 are fermions, adhering to the Pauli exclusion principle, which prevents them from occupying the same state. The correct calculations for indistinguishable particles yield 4!/(2!2!), while for distinguishable particles, it is 4!2!.

PREREQUISITES
  • Understanding of quantum mechanics principles, particularly particle statistics.
  • Familiarity with the Pauli exclusion principle and its implications for fermions.
  • Knowledge of the formula for energy states of particles in a box.
  • Basic combinatorial mathematics for calculating permutations and combinations.
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  • Study the implications of the Pauli exclusion principle on fermionic systems.
  • Learn about the statistical mechanics of indistinguishable particles.
  • Explore advanced topics in quantum mechanics, such as spin and its role in particle behavior.
  • Investigate the derivation and applications of the energy state formula En=(ħ²π²n²)/(2mL²).
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Students and researchers in quantum mechanics, particularly those studying particle statistics and energy state calculations in confined systems.

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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.


Homework Equations





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.
 
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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)
 

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