Explain Like I'm Five

I think my answer here does a disservice to your last two questions, but they’re pretty interesting, so I’d like to try answering them here. That being said, I believe a proper explanation would require a greater understanding of quantum field than I currently have.

Regardless, let’s start with

How many properties does it take to describe, for example, an electron?

Wavefunctions are typically described using just two parameters: position and time. That being said, the specifics of a theory will often add two more ‘parameters:’ spin and charge.

For every position in space and time, there are four possible electrons:

Negatively charged with up spin,

Negatively charged with down spin,

Positively charged with up spin, and

Positively charged with down spin

Now, there are a couple conservation laws that prevent the charge of an electron from changing, so we split these four into two categories based on charge, and call members of the latter (positively charged) category ‘positrons.’

There are no such conservation laws preventing an electron from changing its spin, however, so it’s not worth calling the up and down spin electrons separate particles;^1^ we instead keep track of an electron’s up-ness and down-ness by sticking a two-component vector into the wavefunction.

So, to clarify, you should only need to know the electron’s position and spin at a given time to know its state.

Or, rather, knowing these will give you the local value of the wavefunction of the electron, which is defined at all points in space, at all points in time, and for both possible spins.

This wavefunction describes the state that the particle is in, and it is the solution to our Schrödinger equation.

The magic of quantum mechanics is that there are much fewer possible wavefunctions than you might expect for a given system.^2^

In many cases, we can actually label each of the allowed wavefunctions with one (or a handful of) quantum number(s),^3^ which will be integers. The energy (n) and angular momentum (j) are common quantum numbers.

So we can describe states with quantum numbers while the actual wavefunctions associated with these states describe the electrons that occupy them via spin, position, and time.

This means that when our Pauli exclusion principle requires different wavefunctions for different electrons, it is also enforcing that they occupy different states i.e. have different quantum numbers.

But then how does spin let you have two electrons in the same state? It doesn’t

If two electrons have ‘different spins,’ it actually means they have different wavefunctions (remember! Wave-funcs are functions of spin) and are thus in different states. In the absence of a magnetic field, however, up-spin and down-spin are effectively identical in all other aspects, so you will often see that which is described as one state is actually two.

Now for your other question:

What kind of precision does it take to tell whether the two states are identical?

None!

States are quantized, meaning they are described by a (typically finite) set of integer-valued quantum numbers, and if any of the qn’s are different, so too are the states.

It is worth noting that two different quantum states will have zero expected positional overlap, so you can say that they are in different places even if both wavefunctions are defined everywhere.

That being said, the mathematically-enforced inability of electrons to occupy the same state will certainly look like an equal and opposite to any force that energetically favors an already-occupied state… which is why you can pick up your pencil.

Now, if you start to apply a whole lot of pressure on the system, you start to change the states that solve the Schrödinger equation in the first place. So your neutron star isn’t infinitely stable.

^1^ spin can also mix while charge can’t, so an electron can be partially up and partially down, but not partially negative and partially positive. This is really just a consequence of the (in)ability to switch between states, but I figured I would mention this for clarity.

^2^ if you let some infinities be ‘fewer than others

^3^ technically, any linear combination of states can constitute an allowed wavefunction (this is superposition), but, this will constrain the set of allowable wavefunctions for other electrons, meaning the total number of allowed wavefunctions remains the same.