Sean Carroll

So what is the difference?

So in one way of thinking about it, you think of a wave located in space.

That's the way we usually picture these things.

That's what we are used to growing up in physics with things like the electric field and the gravitational field.

We have the notion of a field.

A field is something that has a value at every location in space.

When you think of the wave function of an electron, boy, it sure looks like a field, doesn't it?

There's a little picture of how big it is in the pictures of the orbitals of atoms, and you can use that to calculate the probability of different measurement outcomes.

The description of the electron's wave function as either a field in space,

or a superposition of different possible measurement outcomes, the second one just sounds sort of unnecessarily fussy, like maybe it's true, right?

Okay, but what are we learning by doing that?

And the answer is, what happens when we have two electrons?

Because your guess would be there'd be a wave function, a field for electron number one, and a different field or a different wave function for electron number two, but that is not what quantum mechanics says.

It says that there is a single wave function for the combination of the two electrons.

And if you're worried about subtleties, about electrons being indistinguishable, just think of an electron and a proton, or an electron and a positron, whatever, two different particles, two distinguishable particles.

So what does it mean to say that there is only one wave function for two particles?

Well, if you think of the wave function not as a field living in space, but rather as the superposition of every possible measurement outcome, then when you have two electrons, these ideas, these concepts become distinguishable, right?

Because what are the possible measurement outcomes of a system of two particles?

It's every possible location of particle 1 and every possible location of particle 2 considered separately.

So for those of you who are a little bit mathematically inclined, the correct Greek letter to attach to the wave function is psi, P-S-I.