Pauli exclusion principle question

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SUMMARY

The discussion centers on the application of the Pauli exclusion principle in determining the electron configuration of five electrons in an infinite square well. The correct configuration is two electrons in n=1, two in n=2, and one in n=3, as opposed to placing three electrons in n=2. This is due to the exclusion principle, which allows only two electrons per energy level based on their spin quantum number (m_s). The infinite square well lacks spherical symmetry, making angular momentum (L) irrelevant in this context.

PREREQUISITES
  • Understanding of quantum mechanics principles, specifically the Pauli exclusion principle.
  • Familiarity with quantum numbers and their significance in electron configurations.
  • Knowledge of infinite square well potential and its implications on energy levels.
  • Basic concepts of angular momentum in quantum systems.
NEXT STEPS
  • Study the implications of the Pauli exclusion principle in multi-electron systems.
  • Learn about quantum numbers and their roles in various potential wells.
  • Explore the differences between spherical and non-spherical potentials in quantum mechanics.
  • Investigate the behavior of electrons in different quantum systems, such as atoms versus wells.
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Students and professionals in physics, particularly those studying quantum mechanics and atomic structure, will benefit from this discussion.

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I have a question in my book where five electrons are placed in a infinite square well and I am supposed to calculate the lowest energies. My problem is with the electron configuration. I think that the first two shall be placed in n=1 and then the rest shall be placed in n=2. This since l=0,1 and m=-1,0,1 (plus the spin). However my book gives me a different solution. Two in n=1, two in n=2 and one in n=3, due to Pauli exclusion. But three electrons can have different quantum numbers in n=2. So why shall one of the electroons be placed in n=3?
 
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Is the well spherical symmetric? No.

In the infinite square well, your quantum numbers are only n and m_s, you don't have L since it isn't spherical symmetric - hence angular momentum is not a good quantun number here.

So for each n, you can put 2 number of electrons due to 2 different m_s values.

So L only plays a role if your potential has spherical symmetry - in the atom for example - the nuclei is generating a spherical symmetric potential - and here L quantum number becomes important.
 
of course! thanks a lot
 

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