so I'm currently learning quantum mechanics in the context of a physical chemistry course. So it's not very mathematical which leads to a lot of little traps that just confuse me.
Let me first introduce the Bra-Ket Notation I'm going to use. The inner product of two complex valued functions $\varphi_m$ and $\varphi_n$ can be written as
$$\int\varphi_m^*\varphi_nd\tau := \langle \varphi_m | \varphi_n\rangle := \langle m | n \rangle \tag{1}$$
whereas $m$ and $n$ are indices for the eigenfunctions and eigenvalues and $\varphi_m^*$ is the complex conjugate.
For an operator $\hat{A}$ we get:
$$\int\varphi_m^*\hat{A}\varphi_nd\tau := \langle \varphi_m | \hat{A}\varphi_n\rangle := \langle m | \hat{A} | n \rangle \tag{2}$$
So that's everything we ever learned about the Dirac-Notation.
Now if we have a complete orthonormal basis consisting of the eigenfunctions $\{ \varphi_n \}$ of a operator $\hat{A}$, we can write $\hat{A}$ as an matrix:
$$ A_{nm} = \int \varphi^*\hat{A}\varphi_md\tau = \langle n|\hat{A}|m\rangle \tag{3} $$
Now, so far so good. Now consider the following problem:
Let $\hat{J}$ be a general angular momentum. Furthermore we denote the spin "electron spin angular momentum" (I guess you could just think of spin) with $\vec{s}$, the "orbital angular momentum" with $\vec{l}$ and the total angular momentum with $\vec{j}$.
We want to calculate the matrices for $\hat{l_x},\hat{l_y},\hat{l_z},\hat{l}^2,\hat{s_x},\hat{s_y},\hat{s_z},\hat{s}^2$.
Since the operators $\hat{l}^2$ and $\hat{l_z}$ commute, we know they share the same basis. (Similar for $\hat{s}^2$ and $\hat{s_z}$). So we can use $|l, m_l\rangle$ respectively $s|m_s\rangle$ as a basis.
We further know that for a general angular Momentum $\hat{J}$ (i.e. especially for $\hat{s_z}, \hat{l_z}$) we have:
$\hat{J}_\pm = \hat{J}_x \pm i\hat{J_y} \tag{3}$
$\hat{J_\pm}|J,M\rangle = \hbar\sqrt{J(J+q) - M(M\pm1)}|J,M\pm 1\rangle \tag{4}$
$\hat{J_z}|J,M\rangle = \hat M|J,M\rangle \tag{5}$
Now, it'd be easy to calculate the matrices for $\hat{l_x},\hat{l_y},\hat{l_z},\hat{l}^2,\hat{s_x},\hat{s_y},\hat{s_z},\hat{s}^2$.
For the spin, we would get a 2 dimensional vector space $V_s$ and for the angular momentum we would get a 3 dimensional vector space $V_l$. We could now ask about the total angular momentum. What space does that one live in? The answer would be: $ V_j = V_s \otimes V_l$ (I hope the notation is correct). It's basis would be
$|l, m_l\rangle s, m_s\rangle \tag{6}$
So for me it's clear, that if we look at the total angular momentum, we basically look at the "union" of both other vector spaces. Now comes the actual question: What they did was the following. First they gave us the following basis:
As you can see, they used
$|l,m_l,s,m_s\rangle = |l, m_l\rangle s, m_s\rangle \tag{6}$
Which already confuses me. Sure it makes sense that a eigenfunction of the total angular momentum basis on the numbers $l, m_l, s, m_s$ but what does the notation $|l,m_l,s,m_s\rangle$ actually notate?
For me $|l, m_l\rangle s, m_s\rangle$ is a "multiplication" of two functions. Which would be an inner product (since we are in a Hilber space), but such an inner product would be donated e.g. as $\langle l, m_l | s, m_s\rangle$ but that obviously is in general not the same since we can't just "flip" a Bra to a Ket like that.
Further more, if we look at the defintion of the Bra-Ket Notation, we suddendly have 4 indices. (Sure, two are fixed and we could shorten it to $|m_l, m_s\rangle$, but there is no mention of such a thing being done.) How should I now apply the Bra-Ket Notation?
Anyway, to get the matrix of my $\hat{l_z}$ operator in the basis $|l, m_l\rangle s, m_s\rangle$ I'd do the following:
We know: $\hat{\ell}_z| l,m_l\rangle = c|l, m_l+1\rangle$ (with $c$ according to (5))
We multiply it with $|s, m_s\rangle$ and get
$\hat{\ell}_z| l,m_l\rangle|s, m_s\rangle = c|l, m_l+1\rangle|s, m_s\rangle$
Then
$\langle s,m_s|\langle l,m_l|\hat{\ell}_z| l',m_l'\rangle|s', m_s'\rangle$
$ = c\langle s,m_s|\langle l,m_l|l', m_l'+1\rangle|s', m_s'\rangle$
$= c\langle s,m_s|\delta_{l,l'}\delta_{m_l,(m_l'+1)}|s', m_s'\rangle$
$= c\delta_{l,l'}\delta_{m_l,(m_l'+1)}\langle s,m_s|s', m_s'\rangle$
$= c\delta_{l,l'}\delta_{m_l,(m_l'+1)}\delta_{s,s'}\delta_{m_s,m_s'}$
$= c\delta_{m_l,(m_l'+1)}\delta_{m_s,m_s'}$
So that works fine. But I don't see how I could do the same thing for the spin operator. I don't know how to handle the Dirac-Notation correctly here.
I hope you see my problem. It was a bit of a big text, so let's summarize my questions:
Q1: What does $|l,m_l,s,m_s\rangle$ denote exactly?
Q2: What does $|l, m_l\rangle s, m_s\rangle$ exactly?
Q3: To get the matrix representation of an operator we'd use (2). I'm confused on how to get to such an expression if we have more than two quantum numbers. How would I get the expression for the matrix?