Distinguishability of electrons in an atom

[Ananthan9470]

Hi. I am struggling to understand the concept of distinguishability in quantum mechanics. If the wave functions of two particles overlap, those become indistinguishable from what I can understand. So if, in an atom, two electrons occupying an orbital are also indistinguishable. right? But can't we use the fact that both have opposite spin to distinguish them?

 

[Nugatory]

Hi. I am struggling to understand the concept of distinguishability in quantum mechanics. If the wave functions of two particles overlap, those become indistinguishable from what I can understand. So if, in an atom, two electrons occupying an orbital are also indistinguishable. right? But can't we use the fact that both have opposite spin to distinguish them?

No. We know that there are two electrons there, but we have no way of telling them apart. If the spin-up electron were to flip its spin to spin-down and the spin-down one were to flip to spin-up, we'd still have two electrons, one up and one down, and we'd never know that the switch happened.

This goes beyond the classical notion that if I put two identical coins in a box and shake it, I may lose track of which one is which. In that case, the state "coin A is here and coin B is there" is a different state than "coin B is here and coin A is there", and at least in principle I could put a dab of paint on one the coins and distinguish the states. In the case of the two electrons, there is only one state: "A spin-up electron and a spin-down electron".

 

[Jano L.]

Hi. I am struggling to understand the concept of distinguishability in quantum mechanics. If the wave functions of two particles overlap, those become indistinguishable from what I can understand. So if, in an atom, two electrons occupying an orbital are also indistinguishable. right? But can't we use the fact that both have opposite spin to distinguish them?

It depends on the $\psi$ function assumed. If we assume two electrons are described by the function

$$
\psi(\mathbf r_1,\mathbf r_2) = f(\mathbf r_1)g(\mathbf r_2)
$$
where $f,g$ are two different functions of 3 coordinates, then the two electrons are distinguishable in description and theory predicts in general different probability density for a pair of permuted configurations.

If we assume two electrons are described by the function

$$
\psi(\mathbf r_1,\mathbf r_2) = f(\mathbf r_1)g(\mathbf r_2)+g(\mathbf r_1)f(\mathbf r_2)
$$

or
$$
\psi(\mathbf r_1,\mathbf r_2) = f(\mathbf r_1)g(\mathbf r_2)-g(\mathbf r_1)f(\mathbf r_2)
$$
then the two electrons are indistinguishable in description and theory predicts the same probability density for any pair of permuted configurations.

In calculations it is advantageous and customary to focus on Hamiltonian eigenstates. These can all be chosen to be symmetric or antisymmetric. In case of calculation of electronic $\psi$ functions, antisymmetric functions of coordinates and spin variable were found to be most useful as a basis.

 

[Xu Shuang]

This can best be understood with measurement.

In quantum mechanics, there is no this particle or that particle. (of a particular kind)
There is only:
if you measure somewhere, what's the probability of finding N particles.
In the case of electron, its either 0 or 1 at the same exact place.
In the case of photon, it can be any natural number.

 

[TreeDweller]

So each particle is identical basically? There's no way to tell them apart?

 

[DrClaude]

It depends on the $\psi$ function assumed. If we assume two electrons are described by the function

$$
\psi(\mathbf r_1,\mathbf r_2) = f(\mathbf r_1)g(\mathbf r_2)
$$
where $f,g$ are two different functions of 3 coordinates, then the two electrons are distinguishable in description and theory predicts in general different probability density for a pair of permuted configurations.

While this is a useful representation, and allows for instance to write down the electronic configuration of an atom, it is not a proper wave function for a many-electron system. As per the Pauli principle, the actual wave function has to be anti-symmetric upon exchange of two identical fermions, so this is not an actual wave function for electrons.

These can all be chosen to be symmetric or antisymmetric. In case of calculation of electronic $\psi$ functions, antisymmetric functions of coordinates and spin variable were found to be most useful as a basis.

It is not a question of usefulness, it is a question of obeying the Rules of Quantum Mechanics.

The conclusion is that the electrons in an atom have to be indistinguishable. Otherwise, we end up with a wave function that doesn't obey the Pauli principle.

 

[Jano L.]

While this is a useful representation, and allows for instance to write down the electronic configuration of an atom, it is not a proper wave function for a many-electron system. As per the Pauli principle, the actual wave function has to be anti-symmetric upon exchange of two identical fermions, so this is not an actual wave function for electrons.

I was explaining how to find out whether electrons are distinguishable or not based on the function $\psi$. I never wrote that $\psi$ was an "actual" wave function. Actual wave function is a problematic concept that was not needed for that explanation.

It is not a question of usefulness, it is a question of obeying the Rules of Quantum Mechanics.

The conclusion is that the electrons in an atom have to be indistinguishable. Otherwise, we end up with a wave function that doesn't obey the Pauli principle.

I was explaining why many-electron Hamiltonian eigenstates are sought in the form of combination of anti-symmetric functions and not symmetric ones. There is no reason for this rule in non-relativistic quantum theory other than it leads to better results(more compatible with observations), hence the word "useful".

 

[Jano L.]

So each particle is identical basically? There's no way to tell them apart?

Identical does not mean "no way to tell them apart"; that would be the meaning of "indistinguishable". Identical means the objects have the same intrinsic properties like mass, charge etc. but not location. Two xerox copies of a master blueprint are identical, yet they can be put in different places and you can tell them apart just by holding them apart each with different hand or placing them in two different rooms (and assigning different labels to the rooms). Different label means you can tell them apart and distinguish them, but they remain identical (same size, mass, color ...).