Clarification of the Pauli exclusion princple

[HomogenousCow]

TL;DR

Clarification on Pauli exclusion princple.

The exlcusion principle seems intuitive enough to me when the states being considered are eigenstates, however how does it work exactly with general states? It seems to me that if we're allowed to consider general quantum states then the principle breaks down, since we can always find states that are only infinitesimally distinct. For example if we had $N$ free fermions, couldn't we have them all in the same exact momentum state but simply with infinitesimally small but distinct contributions from other states? I ask this because I do not entirely understand why electrons in atomic orbits are always assumed to be in energy eigenstates and hence strongly affected by the exclusion principle, couldn't we have all the electrons in an atom in the ground state but each with a different superposition of spin up and down?

 

[PeroK]

Summary: Clarification on Pauli exclusion princple.

The exlcusion principle seems intuitive enough to me when the states being considered are eigenstates, however how does it work exactly with general states? It seems to me that if we're allowed to consider general quantum states then the principle breaks down, since we can always find states that are only infinitesimally distinct. For example if we had $N$ free fermions, couldn't we have them all in the same exact momentum state but simply with infinitesimally small but distinct contributions from other states? I ask this because I do not entirely understand why electrons in atomic orbits are always assumed to be in energy eigenstates and hence strongly affected by the exclusion principle, couldn't we have all the electrons in an atom in the ground state but each with a different superposition of spin up and down?

You have a conceptual misunderatnding about two-particle states. If you have two identical fermions, electrons say, in the same system, then you cannot talk about "the state of electron A" and "the state of electron B". There is only the state of the two-particle system.

It is not possible, therefore, to talk about electron A being in one state and electron B being in a slighty different state. There is only one state describing both electrons.

Now, the state of a two particle system is a combination of one particle states. And, for identical fermions it must be antisymmetric. And this implies that if you measure both particles you cannot find them in exactly the same state - regardless of how you construct the state of the two-particle system. Or, more precisely, you cannot get exactly the same quantum numbers for both electrons.

It's not a question of eigenstates, although because every state of a two-particle system is a linear combination of eigenstates, the eigenstates themselves must be antisysmmetric.

 

[HomogenousCow]

You have a conceptual misunderatnding about two-particle states. If you have two identical fermions, electrons say, in the same system, then you cannot talk about "the state of electron A" and "the state of electron B". There is only the state of the two-particle system.

It is not possible, therefore, to talk about electron A being in one state and electron B being in a slighty different state. There is only one state describing both electrons.

Now, the state of a two particle system is a combination of one particle states. And, for identical fermions it must be antisymmetric. And this implies that if you measure both particles you cannot find them in exactly the same state - regardless of how you construct the state of the two-particle system. Or, more precisely, you cannot get exactly the same quantum numbers for both electrons.

It's not a question of eigenstates, although because every state of a two-particle system is a linear combination of eigenstates, the eigenstates themselves must be antisysmmetric.

I may have misphrased my question but all of this is very familiar to me. What I'm asking about is why in textbooks it's often reasoned that due to the exclusion principle electrons must inhabit distinct atomic/spin orbitals, when the exclusion principle does not in principle demand this.

For example, at zero temperature why isn't it the case that all electrons simply inhabit the ground state but with different spin states? There's a continuum of spin states that one can come up with and as long as they're all different the totally anti-symmetric state is nonzero.

 

[DrClaude]

I may have misphrased my question but all of this is very familiar to me. What I'm asking about is why in textbooks it's often reasoned that due to the exclusion principle electrons must inhabit distinct atomic/spin orbitals, when the exclusion principle does not in principle demand this.

I don't get it. This is exactly what the Pauli exclusion principle demands. For electrons, it is of course an approximation to write the many-particle state as a product of single-electron eigenstates, as these states are going to be modified by the presence of the other electrons, but the result would be the same even accounting for electron-electron interaction.

There's a continuum of spin states that one can come up with

What is that continuum of spin states?

 

[HomogenousCow]

I don't get it. This is exactly what the Pauli exclusion principle demands. For electrons, it is of course an approximation to write the many-particle state as a product of single-electron eigenstates, as these states are going to be modified by the presence of the other electrons, but the result would be the same even accounting for electron-electron interaction.

But the exclusion principle simply demands that the single-particle states we use to build up our antisymmetric states are nonequal, there is absolutely no requirement that they be distinct eigenstates of the Hamiltonian. Since our states are defined on the field of conitnuous complex numbers there are an incountably infinite number of general quantum states.

What is that continuum of spin states?

The entire 2-dimensional space of spin states. You can write down as many distinct spin kets as you want.

 

[PeroK]

I may have misphrased my question but all of this is very familiar to me. What I'm asking about is why in textbooks it's often reasoned that due to the exclusion principle electrons must inhabit distinct atomic/spin orbitals, when the exclusion principle does not in principle demand this.

For example, at zero temperature why isn't it the case that all electrons simply inhabit the ground state but with different spin states? There's a continuum of spin states that one can come up with and as long as they're all different the totally anti-symmetric state is nonzero.

Fundamentally you are still talking about single electron states where these are not directly relevant to a multi-particle system such as an atom.

Your logic demands that the electrons can be considered separately. But they cannot. All the allowable states for a multi-electron system are antisymmetri combinations of single electron states. It's the rules governing the combination of states that imply the PEP. Not, as you insist, the single particle states for a single electron.

 

[DrClaude]

The entire 2-dimensional space of spin states. You can write down as many distinct spin kets as you want.

Could you write it down for two electrons?

 

[PeroK]

The entire 2-dimensional space of spin states. You can write down as many distinct spin kets as you want.

Which is almost entirely irrelevant when you have two or more identical fermions.

 

[HomogenousCow]

Your logic demands that the electrons can be considered separately. But they cannot. All the allowable states for a multi-electron system are antisymmetri combinations of single electron states. It's the rules governing the combination of states that imply the PEP. Not, as you insist, the single particle states for a single electron.

I understand this entirely, I am referring to the single-particle states that we use to form totally-antisymmetric tensor products.

 

[PeroK]

I understand this entirely, I am referring to the single-particle states that we use to form totally-antisymmetric tensor products.

If you understood that entirely you wouldn't be asking the question.

You need to do as @DrClaude suggests and actually construct the antisymmetric product for three electrons where only the spin component differs.

Or two electrons where a measurement could return the same spin for each.

 

[PeroK]

I understand this entirely, I am referring to the single-particle states that we use to form totally-antisymmetric tensor products.

PS note that the basis for your two electron spins states are the singlet and triplet states, not individual spin states as you keep insisting.

 

[HomogenousCow]

Oh I see now, thanks.