Composite States
#Physics
$\displaystyle \ket{s,m_{s}}=\sum_{m_{1}+m_{2}=m_{s}}C_{m_{1}m_{2}m}^{s_{1}s_{2}s}\ket{s_{1}s_{2}m_{1}m_{2}}$
- $\displaystyle \ket{s,m_{s}}$ is a composite state of the two states $\displaystyle \ket{s_{1}s_{2}m_{1}m_{2}}$ which can also be represented as $\displaystyle \ket{s_{1},m_{1}}\otimes \ket{s_{2},m_{2}}$
- $\displaystyle C_{m_{1}m_{2}m}^{s_{1}s_{2}s}$ is the Clebsch-Gordan coefficients
$\displaystyle \ket{s_{1}s_{2}m_{1}m_{2}}=\sum_{s}C_{m_{1}m_{2}m}^{s_{1}s_{2}s}\ket{sm},\quad (m=m_{1}+m_{2})$
Examples
$\displaystyle \ket{1,1}=,\uparrow\uparrow$
- $\displaystyle \uparrow$ represents a Half Spin