Composite States

$|s,m_s⟩=\sum _{m_1+m_2=m_s}{C}_{m_1{m}_{2}m}^{s_1{s}_{2}s}|s_1{s}_2{m}_1{m}_2⟩$

• $|s,m_s⟩$ is a composite state of the two states $|s_1{s}_2{m}_1{m}_2⟩$ which can also be represented as $|s_1,m_1⟩\otimes |s_2,m_2⟩$
• $C_{m_1{m}_{2}m}^{s_1{s}_{2}s}$ is the Clebsch-Gordan coefficients

$|s_1{s}_2{m}_1{m}_2⟩=\sum _s{C}_{m_1{m}_{2}m}^{s_1{s}_{2}s}|sm⟩,(m=m_1+m_2)$

Examples

$|1,1⟩=\uparrow \uparrow$

• $\uparrow$ represents a Half Spin

$|1,0⟩=\frac{1}{\sqrt{2}}(\uparrow \downarrow +\downarrow \uparrow )$

$|1,-1⟩=\downarrow \downarrow$