Adding Two Spin-1/2 Particles

When adding two spin-1/2 particles, the total spin can be found using the rules of quantum mechanics. Here’s a brief overview of the process:

Individual Spins

Each particle has a spin quantum number \( s = \frac{1}{2} \).

Product basis for \(S_z\) operator :

Possible Combinations for total spin \( S \)

When combining two spin-1/2 particles, the possible total spin quantum numbers \( S \) are given by the rules of angular momentum addition:

• \( S = s_1 + s_2 = 1 \) (triplet state, total spin 1)
• \( S = |s_1 - s_2| = 0 \) (singlet state, total spin 0)

Triplet State

When the total spin \( S = 1 \), there are three possible projections of the total spin along \(z\)-axis labelled by \( m \):

• \( m = 1 \) (both particles spin up, \( |\uparrow\uparrow\rangle \))
• \( m = 0 \) (one particle spin up, one particle spin down, \( \frac{1}{\sqrt{2}}(|\uparrow\downarrow\rangle + |\downarrow\uparrow\rangle) \))
• \( m = -1 \) (both particles spin down, \( |\downarrow\downarrow\rangle \))

Singlet State

When the total spin \( S = 0 \), there is only one possible projection:

• \( m = 0 \) (one particle spin up, one particle spin down, \( \frac{1}{\sqrt{2}}(|\uparrow\downarrow\rangle - |\downarrow\uparrow\rangle) \))

So, when combining two spin-1/2 particles, you get one singlet state (total spin 0) and three triplet states (total spin 1).

Coupled basis

Total spin operator \( S^2 \) becomes diagonal in the coupled basis.

The coupled basis states are linear combinations of product basis states. In the present case, they are :