Addition of Spin Angular Momentum | Quantum Science Philippines

## Addition of Spin Angular Momentum

Eric Alcantara, Carlo Paul Morente and Gibson T. Maglasang

If you have two particles of spin [eq]S_1[/eq] and [eq]S_2[/eq]. Let [eq]\bf{S}[/eq] be the combined spin of the particles. You can get values of [eq]\bf{S}[/eq] from [eq](S_1+S_2)[/eq] down to [eq](S_1-S_2)[/eq]:

[eq]\bf{S} =(S_1+S_2), (S_1+S_2-1), (S_1+S_2-2),\cdot\cdot\cdot(S_1-S_2)[/eq]                (1)

We can get then the combination of states from the total spin state [eq]\bf{S}[/eq]. In particular, a state [eq]|S\quad m\rangle[/eq] with total S and z-component m will be some linear combination of the composite states [eq]|S_1\quad m\rangle[/eq] [eq]|S_2\quad m\rangle[/eq].

Consider now two spin 3/2 particles at the ground state. To find the possible total spin states, we use (eqn. 1). Thus,

[eq]\bf{S} = (3/2+3/2)= 3 ;[/eq]

[eq]\bf{S} = (3/2+3/2-1)= 2;[/eq]

[eq]\bf{S}=(3/2+3/2-2)= 1;[/eq]

[eq]\bf{S} = (3/2+3/2-3) = 0[/eq].

(i) For [eq]\bf{S} = 3[/eq], the corresponding values for [eq]m_s[/eq] are 3, 2, 1, 0, -1, -2, -3.

We have the following states [eq]|\bf{S}[/eq]  [eq] m_s\rangle[/eq],

[eq]|3[/eq]  [eq] 3\rangle[/eq],

[eq]|3[/eq]  [eq] 2\rangle[/eq],

[eq]|3[/eq]  [eq]1\rangle[/eq],

[eq]|3[/eq]  [eq]0\rangle[/eq],

[eq]|3[/eq]  [eq]-3\rangle[/eq],

[eq]|3[/eq]  [eq]-2\rangle[/eq],

[eq]|3[/eq]  [eq]-1\rangle[/eq].

(ii) For [eq]\bf{S}[/eq] = 2,   [eq]m_s=2, 1, 0, -1, -2[/eq]. The following states are obtained,

[eq]|2[/eq]  [eq] 2\rangle[/eq],

[eq]|2[/eq]  [eq] 1\rangle[/eq],

[eq]|2[/eq]  [eq] 0\rangle[/eq],

[eq]|2[/eq]  [eq] -2\rangle[/eq],

[eq]|2[/eq]  [eq] -1\rangle[/eq].

(iii) For [eq]\bf{S} = 1[/eq],    [eq]m_s = 1, 0, -1[/eq]

[eq]|1[/eq]  [eq]1\rangle[/eq],

[eq]|1[/eq]  [eq]0\rangle[/eq],

[eq]|1[/eq]  [eq]-1\rangle[/eq].

(iv) For [eq]\bf{S} = 0[/eq], [eq]m_s = 0[/eq]. Only one state is available for [eq]\bf{S} = 0[/eq]. We have,

[eq]|0[/eq]  [eq]0\rangle[/eq]

Thus, we have successfully extracted the 16 possible spin states for two spin 3/2 particles at the ground state.  One can also get all the 16 spin states for this particular problem by looking up the Clebsch-Gordan table.