Multiplicity of eigenvalues of angular momentum

Question

Reading Dirac's Principles of Quantum Mechanics, I encounter in § 36 (Properties of angular momentum) this fragment:

This is for a dynamical system with two angular momenta $\mathbf{m}_1$ and $\mathbf{m}_2$ that commute with one another, and $\mathbf{M}=\mathbf{m}_1+\mathbf{m}_2$. $k_1$ and $k_2$ are the magnitudes of $\mathbf{m}_1$ and $\mathbf{m}_2$, so the possible eigenvalues of $m_{1z}$ are $k_1$, $k_1-\hslash$, $k_1-2\hslash$, ..., $-k_1$, and similarly for $m_{2z}$ and $k_2$.

The question is about the two $2k_2+1$ terms shown in the second line of eq (46). Shouldn't the last one be $2k_2+2$?

Answer 1

I think $2k_2 + 1$ is correct. If you write down the pairs of $k_{1z}$ and $k_{2z}$ they are:

• ($k_1 - 1$, $-k_2$)
• ($k_1 - 2$, $-k_2 + 1$)
• etc
• ($k_1 - 2k_2$, $k_2 - 1$)
• ($k_1 - (2k_2+1)$, $k_2$)

So that's $2k_2+1$ of them.

Answer 2

                     m_2 ^     ^  M=m_1+m_2
                         |    /
                     \   |   /
 k_2 -----------------------------------------  
     |                 \ | /                 |
     |                  \|/                  |
-----------------------------------------------------> m_1
     |                  /|\                  |
     |                 / | \                 |
-k_2 ----------------------------------------- 
    -k_1             /   |   \               k_1 
                    /    |    v   M=m_1-m_2   

$\uparrow$ Fig. 1. A rectangle of eigenvalues $(m_1,m_2)$.

I) Let $k_1$ and $k_2$ be non-negative half-integers or integers. The book is basically counting an integral lattice

$$ (\mathbb{Z}+k_1) \times (\mathbb{Z}+k_2) $$

of $(2k_1+1) \times (2k_2+1)$ points sitting inside a $2k_1 \times 2k_2$ rectangle by using "light-cone" coordinates

$$M ~=~ m_1+m_2, $$ $$\Delta ~=~ m_1-m_2, $$

which are tilded $45^{\circ}$. (Sorry, the ASCII diagram (Fig. 1) doesn't reproduce the angle $45^{\circ}$ well. Assume $\hbar=1$.)

A multiplicity formula for the lattice points inside the rectangle is

$$\tag{1} {\rm mult}(M) ~=~ 1 + k_1 + k_2 - \max(|M|,|k_1 - k_2|) $$

as a function of the variable

$$M\in(\mathbb{Z}+k_1+k_2) \cap [ -k_1-k_2,k_1+k_2]. $$

The function (1) has a trapezoidal graph:

                      ^ mult(M)
                      |
            /---------|---------\   1 + k_1 + k_2 - |k_1 - k_2|
           /|         |         |\
          / |         |         | \
         /  |         |         |  \
      --|---|---------|---------|---|--------> M
 -k_1-k_2   |         |         |  k_1+k_2  
            |         |         |  
       -|k_1-k_2 |          |k_1-k_2 |

$\uparrow$ Fig. 2. Multiplicity ${\rm mult}(M)$ as a function of $M$.

II) Group-theoretically, it is important to know the Clebsch-Gordan fusion rule

$$\tag{2} \underline{2k_1+1} \otimes \underline{2k_2+1} ~=~ \oplus_{K=|k_1-k_2|}^{k_1+k_2} \underline{2K+1}$$

in terms of irreps. Here $\underline{1}$, $\underline{2}$, $\underline{3}$, $\ldots$, refer to the singlet irrep, doublet irrep, triplet irrep, $\ldots$, respectively. It is a nice exercise to check that the dimensions on the right- and left- hand sides of (2) match. See also this Phys.SE question.