Solving My Confusion: States in Energy Band

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Discussion Overview

The discussion revolves around the concepts of electronic states in energy bands, specifically addressing the occupancy of states by electrons, the implications of the Pauli exclusion principle, and the nature of wavefunctions in quantum mechanics. The scope includes theoretical considerations and conceptual clarifications related to solid-state physics and quantum mechanics.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant expresses confusion about why a single quantum state can accommodate two electrons with opposite spins, likening it to a scenario where one chair is occupied by two individuals.
  • Another participant explains that the ability for two electrons to occupy the same state relates to the nature of fermions and their wavefunctions, suggesting that the wavefunctions can overlap without destructive interference due to differing spins.
  • A further contribution clarifies that two electrons in the same state differ only by their spin, referencing atomic levels where electrons share the same energy and quantum numbers except for spin.
  • One participant questions the nature of the wavefunction when two electrons occupy the same state, seeking clarification on whether it results in a single wavefunction or two distinct ones based on spin.
  • Another participant notes their limited knowledge of relativistic quantum mechanics but attempts to explain that in non-relativistic quantum mechanics, the wavefunction can be viewed as having components corresponding to different spins.
  • A participant reiterates the question about the implications of the Pauli exclusion principle, asking what the maximum number of electrons in a band would be if it did not apply, suggesting an infinite occupancy scenario for non-fermionic particles like photons.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation of electron occupancy in states and the implications of the Pauli exclusion principle. There is no consensus on the nature of wavefunctions when multiple electrons occupy the same state, and the discussion remains unresolved regarding the maximum number of electrons in a band without the exclusion principle.

Contextual Notes

Participants acknowledge limitations in their understanding of relativistic quantum mechanics and the complexities of spin interactions. There is also a distinction made between bands and states, which may affect the clarity of the discussion.

arierreF
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It's known that the number of states in a band is equal to the number of unit cells in crystal.

Here is my problem (confusion with trivial concepts)

Bloch function is a electronic state, or orbital.
The number of orbitals in a band inside the first zone is equal to the number os units cells in crystal.

each state can accommodate at most two electrons. So the max number of electrons that can occupy a single band is 2N

But here is my stupid question:

I know that a state can be occupied with two electrons with opposite spins, but why two?

In my mind (i know that I am thinking wrong) one state should be only occupied by one electron.

Is like having N chairs for 2N people (N male and N female). We are saying that one chair can be occupied by one male and one female simultaneously . But one chair is made for one individual and not two individuals.

Other question:

If the Pauli exclusion principle would not be valid for electrons, then what is the max number of electrons that can occupy a single band? infinite?
 
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arierreF said:
I know that a state can be occupied with two electrons with opposite spins, but why two?

It has to do with how the wavefunctions of fermions interact. I believe that with two possible spin states, two wavefunctions can overlap completely without destructively interfering with each other.
 
Actually, two electrons do not occupy the exact same state, the difference being the spin. What happens is that the "states" are spin-degenerated, that is, there are states with same quantum numbers, and differ only in spin. Think about the atomic levels for example: neglecting spin interaction, the 1s electrons occupy have same energy, angular momentum, and the only difference is spin.

Yes, particles that are bosons do not obbey the Pauli exclusion principle and therefore an inifinite number of particles can occupy the EXACT (including spin) state. This difference is the basis of the difference between Fermi-Dirac and Bose-Einstein statistics.
About many particles in the same state, take a look about the Bose-Einstein Condensates.
 
(Neglecting spin interaction) if we have a sate with two electrons, same band, same wave vector, but opposite spins and i want to "plot" the wave function of that state, what is the result?

A single wave function that resulted on the overlapping of the two electrons?

Or two wave functions, with different spins.
 
Well...about details related to spin I might commit errors becaue I have no deep knowledge of relativistic quantum mechanics, which is where spin is accounted more precisely as far as I know.
But in non-relativistic quantum mechanics (at least for spin 1/2, which is the case of the electron), when you treat different spin components it appears as a different componnent, that is, the wave function of the system has one spin +1/2 component and one spin -1/2 component. Think of it like a 2D vector: you can decompose it in x and y components.
That means in the case you ask about, the two wave functions would be the same if plotted.
 
arierreF said:
Other question:

If the Pauli exclusion principle would not be valid for electrons, then what is the max number of electrons that can occupy a single band? infinite?

There are only two bands : the valence band and the conduction band, each containing many states. Be careful not to confuse band and state.
A photon is a particle to which the Pauli exclusion principle does not apply. There is not limit to the number of photons in a system that can be in the same state.
 

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