What Is the Ground State of Bound Fermions in a Box?

  • Context: Graduate 
  • Thread starter Thread starter Evgeny Naumov
  • Start date Start date
  • Tags Tags
    Principle
Click For Summary

Discussion Overview

The discussion revolves around the ground state of bound fermions in a box, exploring concepts related to quantum states, the Pauli exclusion principle, and the behavior of fermions and bosons in various contexts. Participants engage with theoretical aspects, applications, and implications of these principles in quantum mechanics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants clarify that two identical fermions cannot occupy the same quantum state, emphasizing the implications of the Pauli exclusion principle.
  • Others argue that while fermions cannot share quantum states, different fermions in separate atoms can occupy the same energy state, highlighting the importance of quantum numbers in defining states.
  • A participant points out that the energy eigenvalue does not uniquely specify a quantum state, as states can differ based on additional quantum numbers such as spin and angular momentum.
  • There is a discussion about the behavior of electrons in molecules, particularly in the H2 molecule, where overlapping electron states influence their spatial distribution and bonding characteristics.
  • Some participants question how to conceptualize the identity of electrons in unbound atoms described by a single wave equation, raising issues of electron belonging and state representation.
  • A later post introduces a practice problem regarding the ground state of a system of noninteracting bosons, which are bound states of fermions, and poses a question about the ground state in a box scenario.

Areas of Agreement / Disagreement

Participants generally agree on the fundamental principles of quantum states and the Pauli exclusion principle, but there are competing views regarding the implications of these principles in different contexts, particularly concerning the behavior of fermions and bosons. The discussion remains unresolved on several points, particularly regarding the identity of electrons in unbound systems and the specifics of the ground state in the proposed problem.

Contextual Notes

Limitations include the dependence on definitions of quantum states and the assumptions made regarding the interactions and configurations of particles in various scenarios. The discussion also touches on unresolved mathematical steps related to the proposed practice problem.

Evgeny Naumov
Messages
2
Reaction score
0
In terms of the Pauli exclusion principle, what does it mean, quantitatively, to possesses the same location in space?

Is there a definite distance which Fermions with the same states will simply not get closer than? Or is it some force of repulsion?
 
Physics news on Phys.org
No not "same location in space" but occupying the same Quantum State. Two identical fermions can not occupy the same quantum state.

Due to this, fermionic gas will constitute a pressure. Even when the fermions are free (not bound to an atom or similar) the states are still DISCRETE! In fact, they are DISCRETE in momentum space as well, so there will exists momentum greater than zero in the fermionic gas. And pressure is fore/area = (change in momentum / time)/are... c.f. classical gas molecules in a container, they will give rise to pressure at container walls.

The limit of what Pressure a collection of fermions can withstand are related first to the chandrasekhar limit - electrons gas withstanding gravity. Then you have the maximum mass of a neutron star, before it collapses to a black hole.

Those are some "applications" of pauli principle ;)
 
malawi_glenn said:
No not "same location in space" but occupying the same Quantum State. Two identical fermions can not occupy the same quantum state.

Two different fermions in two different atoms can occupy the same energy state.
 
Phrak said:
Two different fermions in two different atoms can occupy the same energy state.

A state is NOT specified by its energy!

The state in an atom is specified by n,L, m_L and S.

So there can be several eletrons having same energy, but they then differ in S (spin) or L (m_L) (angular momentum). Hence only one fermion/state.

In hydrogentic atoms, L can take vaules n-1,n-2,..,0
n can take 0,1,...
etc.

Basic exercise in intro QM
 
Phrak said:
Two different fermions in two different atoms can occupy the same energy state.

As malawi_glenn wrote the energy eigenvalue does not specify the quantum state uniquely, not even in an atom. In this case, the relevant thing is that the wavefunctions of the "same state" would still be different because the atoms are located at different places. So, if one wavefunction is psi(x), the other wavefunction is psi(x-a), if the other atom is at position a.
 
OK, 'quantum states'. But these atoms are two identical bosons...
 
Phrak said:
OK, 'quantum states'. But these atoms are two identical bosons...

and??
 
malawi_glenn said:
and??

I thought you could tell me. If you have two identically prepared, neutral atoms described by a single wave equation, does it make sense to talk about whether a particular electron is a constituent of one atom or the other?
 
Look at the H2 molecule for instance, there this effect occurs. The overlapping of each atoms electron state forces the electrons to "be away from each other" and thus makes each atom to work as a weak dipole -> which will make the atoms bind together. So to answer your question: This was an example where the electrons "can't decide" on which atom they belong to.
 
  • #10
malawi_glenn said:
Look at the H2 molecule for instance, there this effect occurs. The overlapping of each atoms electron state forces the electrons to "be away from each other" and thus makes each atom to work as a weak dipole -> which will make the atoms bind together. So to answer your question: This was an example where the electrons "can't decide" on which atom they belong to.

Yes. I was thinking more along the lines of unbound atoms. So if the atoms themselves can only be described by a single wave equation, what of their constituent electrons?
 
  • #11
Phrak said:
Yes. I was thinking more along the lines of unbound atoms. So if the atoms themselves can only be described by a single wave equation, what of their constituent electrons?

Then the electrons "know" what atom they belong to. You have the "same" pheneomenon in solids, when putting many atoms close toghter, this force the energies of atoms to form a band.
 
  • #12
Another practice problem for our QM students: Suppose we have N noninteracting Bosons in a box. Then it is easy to find the ground state of this system: all the Bosons occupy the single particle ground state. Now, suppose that the Bosons are really bound states of two Fermions. Let's say that the Fermons are bound by a harmonic potential. We assume that the average distance of the bound Fermions in a boson, which is of order
sqrt[hbar/(m omega)], is much less than the dimension (L) of the box.

What is the ground state of this system?
 

Similar threads

Undergrad Two hydrogen atoms and Pauli's exclusion principle
  • · Replies 17 ·
Replies
17
Views
4K
Graduate How can a 'Principle' produce a 'Force'?
  • · Replies 12 ·
Replies
12
Views
2K
Undergrad How does the Pauli exclusion principle work?
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Virtual Fermions and Pauli Principle
  • · Replies 2 ·
Replies
2
Views
2K
High School Can Electrical Repulsion Distort Electron Orbits in Atoms?
  • · Replies 3 ·
Replies
3
Views
2K
High School Bypassing Pauli Exclusion principle.
  • · Replies 3 ·
Replies
3
Views
1K
Graduate Pauli exclusion principle question (again)
  • · Replies 8 ·
Replies
8
Views
3K
Graduate What the heck is meant by Pauli force/effect ?
  • · Replies 4 ·
Replies
4
Views
3K
Graduate Violation of Pauli's Exclusion Principle possible?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Pauli exclusion principle explained
  • · Replies 2 ·
Replies
2
Views
2K