$\renewcommand{ket}[1]{|#1\rangle}$
Item #4 in your list is best thought of as the *definition* of the word "particle".

Consider a classical vibrating string.
Suppose it has a set of normal modes denoted $\{A, B, C, \ldots\}$.
To specify the state of the string, you write it as a Fourier series

$$f(x) = \sum_{\text{mode } n=\in \{A,B,C,\ldots \}} c_n [\text{shape of mode }n](x) \, .$$

In the typical case, $[\text{shape of mode }n](x)$ is something like $\sin(n\pi x / L)$ where $L$ is the length of the string.
Anyway, the point is that you describe the string by enumerating its possible modes and specifying the amount by which each mode is excited by giving the $c_n$ values.

Suppose mode $A$ has one unit of energy, mode $C$ has two units of energy, and all the other modes have zero units of energy.
There are two ways you could describe this situation.

# Enumerate the modes (good)

The first option is like the Fourier series: you enumerate the modes and give each one's excitation level:
$$|1\rangle_A, |2\rangle_C \, .$$
This is like second quantization; we describe the system by saying how many units of excitation are in each mode.
In quantum mechanics, we use the work "particle" instead of the phrase "unit of excitation".
This is mostly because historically we first understood "units of excitations" as things we could detect with a cloud chamber or Geiger counter.
To be honest, I think "particle" is a pretty awful word given how we now understand things.

# Label the units of excitation (bad)

The second way is to give each unit of excitation a label, and then say which mode each excitation is in.
Let's call the excitations $x$, $y$, and $z$.
Then in this notation the state of the system would be
$$\ket{A}_x, \ket{C}_y, \ket{C}_z \, .$$
This is like first quantization.
We've now labelled the "particles" and described the system by saying which state each particle is in.
This is a *terrible* notation though, because the state we wrote is equivalent to this one
$$\ket{A}_y, \ket{C}_x, \ket{C}_z \, .$$
In fact, any permutation of $x,y,z$ gives the same state of the string.
This is why first quantization is terrible: particles are units of excitation so *it is completely meaningless to give them labels*.

Traditionally, this terribleness of notation was fixed by symmetrizing or anti-symmetrizing the first-quantized wave functions.
This has the effect of removing the information we injected by labeling the particles, but you're way better off just not labeling them at all and using second quantization.

# Meaning of 2$^{\text{nd}}$ quantization

Going back to the second quantization notation, our string was written
$$\ket{1}_A, \ket{2}_C$$
meaning one excitation (particle) in $A$ and two excitations (particles) in $C$.
Another way to write this could be to write a single ket and just list all the excitation numbers for each mode:
$$\ket{\underbrace{1}_A \underbrace{0}_B \underbrace{2}_C \ldots}$$
which is how second quantization is actually written (without the underbraces).
Then you can realize that
$$\ket{000\ldots \underbrace{N}_{\text{mode }n} \ldots000} = \frac{(a_n^\dagger)^N}{\sqrt{N!}} \ket{0}$$
and just write all states as strings of creation operators acting on the vacuum state.

Anyway, the interpretation of second quantization is just that it's telling you how many excitation units ("quanta" or "particles") are in each mode in *exactly* the same way you would do it in classical physics.

See this post.

This post imported from StackExchange Physics at 2015-06-08 13:22 (UTC), posted by SE-user DanielSank