Coulomb force vs. Pauli principle in atoms

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Homework Help Overview

The discussion revolves around the relationship between the Coulomb force and the Pauli exclusion principle in atomic systems. Participants are tasked with demonstrating that a system of 2N atoms has a lower energy than two separate N-atom systems, while considering the implications of the energy scaling as N^(5/3) in the absence of the Pauli principle.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants attempt to calculate the energy for a system of 2N atoms and compare it to the energy of two N-atom systems, questioning the validity of their reasoning when they find a contradiction. They also raise questions about the nature of the constant c, specifically whether it can be positive or negative.

Discussion Status

The discussion is ongoing, with participants exploring the implications of their calculations and questioning the assumptions about the constant c. There is no explicit consensus yet, but participants are engaging with the problem and each other's reasoning.

Contextual Notes

Participants are considering the effects of Coulomb repulsion and the characteristics of atomic systems, including the role of positive nuclei in determining the sign of c. There is an acknowledgment that energy might depend on N in more complex ways than initially stated.

Heirot
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Homework Statement



In the absence of the Pauli principle, the Coulomb force would not be saturated, and the energy of a system containing N atoms would vary as N^(5/3).

Show that a system of 2N atoms would have an energy lower than that of two N - atom systems.

Homework Equations



E = c * N^(5/3)

The Attempt at a Solution



E(2N) = c * (2N)^(5/3) = c * 2^(5/3) * N^(5/3)
2 * E(N) = 2 * c N^(5/3)

Since, 2^(5/3) > 2, then obviously E(2N) > 2 E(N), contrary to the what had to be shown. Where's the error in my reasoning?
 
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Heirot said:

Homework Statement



In the absence of the Pauli principle, the Coulomb force would not be saturated, and the energy of a system containing N atoms would vary as N^(5/3).

Show that a system of 2N atoms would have an energy lower than that of two N - atom systems.

Homework Equations



E = c * N^(5/3)

The Attempt at a Solution



E(2N) = c * (2N)^(5/3) = c * 2^(5/3) * N^(5/3)
2 * E(N) = 2 * c N^(5/3)

Since, 2^(5/3) > 2, then obviously E(2N) > 2 E(N), contrary to the what had to be shown. Where's the error in my reasoning?

Is [itex]c[/itex] positive or negative, and why?
 
I can't see how could c be anything but positive due to Coulomb repulsion of like charges.
 
Heirot said:
I can't see how could c be anything but positive due to Coulomb repulsion of like charges.
True, but this is an atom: it has a large positive nuclei
 
But then then energy wouldn't depend on N as N^(5/3) but also as N^2
 

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