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I was taking class where I encountered that $$Y_l^mx(-)=\vert j_1=l,m_1=m\rangle\vert j_2=1/2,m_2=1/2\rangle.$$

I have read Spin, orbital angular momentum and total angular momentum and Why do we think Spin is angular momentum as opposed to some other quantity?; however, these did not give me sufficient answers.

My question was that:

  1. Why could spin and angular momentum add together? (Quantitatively, they were different as on was caused by spin $\mu$ and the other one was caused by "moving charge". )

  2. ACuriousMind mentioned $SO(3)$ group, I suppose what he meant was that $S$ on spin was the same as $L$ on $l,\,m$. However, could you give me some mathematical proof?(I'd like to see how group theory in physics looks like.)

  3. (In summary of question 1 and 3) What's the mathematical basis/proof that in the expression $J=L+S$, $L(l,m)$ and $S(j_2=1/2,s)$ could be numerically equivalently added together in a sense that $$Y_l^mx(-)=\vert j_1=l,m_1=m\rangle\vert j_2=1/2,m_2=1/2\rangle?$$

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This is a purely (Lie) algebraic result. The raising and lowering operators are $$ J_\pm = L_\pm + S_\pm $$ and likewise for $J_z$. These satisfy the $su(2)$ commutation relations. Thus, \begin{align} J_+\vert L,L\rangle \vert S S\rangle&=0\\ J_z \vert L,L\rangle \vert S S\rangle &= J\vert L,L\rangle \vert S S\rangle \end{align} implies $\vert L,L\rangle \vert S S\rangle=\vert L+S,L+S\rangle$. The remaining state $\vert L+S,M\rangle$ can be reached by lowering with $J_-$.

The state $$ \sqrt{\frac{L}{L+S}}\vert LL\rangle \vert S,S-1\rangle - \sqrt{\frac{S}{L+S}}\vert L,L-1\rangle \vert S,S\rangle $$ is killed by $J_+$ and has $J_z$-eigenvalue $L+S-1$ so must be $\vert L+S-1,L+S-1\rangle$, and all other states $\vert L+S-1,M\rangle$ are reached by lowering with $J_-$ etc.

This (algebraic) procedure does not depend on $L$ or $S$ being integer or half integer. Thus, angular momentum and spin can be added together in the usual way.

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When you say add spin and angular momentum the following equation pops into my head:

$$ \hat{\boldsymbol{J}} = \hat{\boldsymbol{L}}+\hat{\boldsymbol{S}} $$

which is shorthand for the three equations:

$$ \begin{bmatrix} \hat{J}_x\\ \hat{J}_y \\ \hat{J}_z \end{bmatrix} = \begin{bmatrix} \hat{L}_x\\ \hat{L}_y \\ \hat{L}_z \end{bmatrix} + \begin{bmatrix} \hat{S}_x\\ \hat{S}_y \\ \hat{S}_z \end{bmatrix} $$

So the question "why can spin and angular momentum add" boils down to the question of why can we write down

$$ \hat{J}_x = \hat{L}_x + \hat{S}_x $$

The short (kind of tongue in cheek) answer is that we can add $\hat{L}_x$ to $\hat{S}_x$ and define a new operator $\hat{J}_x$ as their sum because both $\hat{L}_x$ and $\hat{S}_x$ are operators which act on the same Hilbert space. Namely the Hilbert space which is the tensor product of the spin Hilbert space and the particle position Hilbert space.

$\mathcal{H}_{\text{tot}} = \mathcal{H}_{\text{mech}}\otimes\mathcal{H}_{\text{spin}}$

If we define $\hat{L}_x^{\text{mech}}$ to be the operator which explicitly operates only on $\mathcal{H}_\text{mech}$ and likewise for $\hat{S}_x^{\text{spin}}$ and $\mathcal{H}_{\text{spin}}$ we can write down

\begin{align} \hat{L}_x &= \hat{L}_x^{\text{mech}}\otimes \hat{I}^{\text{spin}}\\ \hat{S}_x &= \hat{I}^{\text{mech}}\otimes\hat{S}_x^{\text{spin}} \end{align}

These operators both act on $\mathcal{H}_{\text{tot}}$ so they can be added together.

There is a follow up questions is "why would we want to define a new operator which is the sum of mechanical angular momentum and spin?" This is a more physically (rather than mathematically) motivated question. The answer is that physically spin and angular momentum behave similarly. For example, they satisfy similar symmetry properties and charged particles with either spin or mechanical angular momentum both have magnetic moments. In the end the notion that spin is a type of angular momentum ends up being a physical intuition which is very useful. It then becomes useful to group all types of angular momentum into a single operator $\hat{\boldsymbol{J}}$ in certain cases.

One more note: I do not understand your notation $Y_l^m x(-)$ I don't know what the $x$ or the $(-)$ are refering to. Normally when I think of the spherical harmonics, $Y_l^m$ I think of a function of a polar and azimuthal angle, $Y_l^m(\theta, \phi)$ which tells you something about the real space spatial distribution of a wavefunction. In particular spherical harmonics come up when describing states with non-zero mechanical angular momentum. However, spherical harmonics are not necessary to describe intrinsic spin angular momentum*.

*Of course often spherical harmonics may show up in the discussion of spin angular momentum by way of noting that both spin operators/states transform in similar ways under rotations as the the spherical harmonics do.

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Mathematics is a self consistent discipline, where statements can be proven or declared false, dependent only on the mathematical axioms. Physics uses mathematics as a tool to model measurements and observations.

To do that, a new set of axioms is posited, called postulates,laws, principles, which chooses a subset of the valid mathematically solutions, solutions which will describe measurements and observations. Also posited are definitions of coordinates as reflecting space and time, of specific charges and quantum numbers as carried by specific particles , in particular the elementary particle table of the standard model, is posited to have the specific masses and quantum numbers, including spins, and it is posited that spins are angular momentum.

It is an axiomatic level input, not something that can be proven.

Thus there can be no proof other than the ones given for the particular choice of the symmetries that describe the standard model group structure.

The reason spin is intrinsic and axiomatic is that it was necessary to be able to see in the data conservation of angular momentum. The reason angular momentum has to be conserved it is because it comes out of Neother's theorem , it is a symmetry of the action that describes the physical system.

Thus the painstaking accumulation of data in the twentieth century came to the elementary particle table, and the symmetries describing it, and spin is an intrinsic identity for each particle in the basic table , on which everything rests.

Angular momentum and spin add up because of construction of spin, so that angular momentum is conserved in elementary particle interactions.

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