Undergrad C-G coefficients and Angular momentum

[QuarkDecay]

Can someone explain to me how we find it?
Examples

Y₁⁰X_-=
= |1,0>|1/2,-1/2> = √2/3 |3/2,-1/2> + √1/3|1/2,-1/2>

Y₁¹X_-=
= |1,1>|1/2,-1/2> = √1/3|3/2,1/2> + √2/3|1/2,1/2>

Y₂^-1X_- =
= √4/5 | 5/2, -3/2> + √1/5 | 3/2, -3/2>

I understand it goes like Y_l^mX_± = |l,m>|s,m_s> = a |j_max, ?> + b |j_min, ?> and a,b is from the (C-G) coefficients, but I can't find how the ? comes up. I thought it was m-m_s or m_s-m but it doesn't match all of the examples.

 

[kuruman]

When you add angular momenta, say $l$ and $s$ so that $\vec l+\vec s = \vec j$, you have terms like $| l, m_l\rangle |s, m_s\rangle$ that you want to write as a linear combination of the $|j, m_j\rangle$ states. First note that the terms that appear in the linear combination may have all the values of $j$ that satisfy the triangle inequality, i.e. $j_{min} \leq j \leq j_{max}$, not just the two extremal values. Also, the component of the total angular momentum along the quantization axis must be conserved, i.e. $m_l+m_s=m_j$. In all the examples you have posted the ? is that sum.

 

[stevendaryl]

Can someone explain to me how we find it?

Let $|j_1, m_1\rangle |j_2, m_2\rangle$ be the state in which the first system has angular momentum $j_1$ and z-component $m_1$, and the second system has angular momentum $j_2$ and z-component $m_2$. Let $|j, m\rangle$ in which the total angular momentum is $j$ and the total z-component is $m$.

Since the z-component of angular momentum, $m$, is additive, we know that

$|j_1, m_1\rangle |j_2, m_2\rangle =$ some combination of terms of the form $|j, m\rangle$ where
$|j_1 - j_2| \leq j \leq j_1 + j_2$ and
$-j \leq m \leq +j$

To start with, it's clear that if $m_1 = j_1$ and $m_2 = j_2$, then there is only one possibility:

1. $|j_1, j_1\rangle |j_2, j_2\rangle = |(j_1 + j_2), (j_1 + j_2)\rangle$ (The case with $j = j_1 + j_2$ and $m = m_1 + m_2$)

Now, we use the lowering operators:

2. $J^- = J_1^{-} + J_2^{-}$

You use the fact that
3. $J^{-} |j, m\rangle = \sqrt{j (j+1) - m (m-1)}|j, m-1\rangle$
4. $J_1^{-} |j_1, m_1\rangle |j_2, m_2\rangle = \sqrt{j_1 (j_1+1) - m_1 (m_1-1)}|j_1, m_1 - 1\rangle |j_2, m_2\rangle$
5. $J_2^{-} |j_1, m_1\rangle |j_2, m_2\rangle = \sqrt{j_2 (j_2+1) - m_2 (m_2-1)}|j_1, m_1\rangle |j_2, m_2 - 1\rangle$

($J_1^{-}$ doesn't affect $|j_2, m_2\rangle$ and $J_2^{-}$ doesn't affect $|j_1, m_1\rangle$)

So applying $J_1^{-} + J_2^{-}$ to the left side of equation 1, and applying $J^{-}$ to the right side gives us:

6. $\sqrt{j_1 (j_1+1) - m_1 (m_1-1)} |j_1, j_1 - 1\rangle |j_2, j_2\rangle$
$+ \sqrt{j_2 (j_2+1) - m_2 (m_2-1)} |j_1, j_1\rangle |j_2, j_2 - 1\rangle$
$= \sqrt{(j_1 + j _1)(j_1+j_2 +1) - (j_1 + j_2) (j_1 + j_2 -1)} |(j_1 + j_2), (j_1 + j_2 - 1)\rangle$

You can continue using $J^{-}$ to get all the coefficients for $|j, m\rangle$ with $j = j_1 + j_2$.

Now, let's look at the case where $j = j_1 + j_2 - 1$. How do you figure out that case? Well, let's look at the case $j = j_1 + j_2 - 1$ and $m = j_1 + j_2 - 1$. We know that that has to be some combination of $m_1 = j_1, m_2 = j_2 -1$ and $m_1 = j_1, m_2 = j_2$. Those are the only two ways to get $m = m_1 + m_2$. So we know that there must be coefficients $\alpha$ and $\beta$ such that:

$\alpha |j_1, j_1 - 1\rangle |j_2, j_1\rangle + \beta |j_1, j_1\rangle |j_2, m_2 - 1\rangle = |(j_1 + j_2 - 1), (j_1 + j_2 - 1)\rangle$

So we have to figure out what $\alpha$ and $\beta$ must be. That's two unknowns. But we also have two constraints: (1) This state must be orthogonal to the state $|(j_1 + j_2), (j_1 + j_2 - 1)\rangle$, and (2) $|\alpha|^2 + |\beta|^2 = 1$ (it has to be normalized). Those two constraints uniquely determine $\alpha$ and $\beta$ up to an unknown phase factor. (I'm not sure if there is some standard convention for picking the phase).

Once you know $|j, m\rangle$ for the case $j = j_1 + j_2 - 1$, $m = j_1 + j_2 - 1$, you can again use the lowering operators to find the states for all other values of $m$.

Then you can again use orthogonality to find the state $|j, m\rangle$ with $j = j_1 + j_2 - 2$ and $m = j_1 + j_2 - 2$.

Continuing in this way, you can find all the possibilities for $|j, m\rangle$.