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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=12.

Product basis for Sz operator :

Sz operator is diagonalisable in the product basis. |↑↑,  |↑↓,  |↓↑,  |↓↓
R. Shankar's book denotes these as : |++,  |+,  |+,  |

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:

Triplet State

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

Singlet State

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

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 S2 becomes diagonal in the coupled basis.

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

|↑↑,  12(|↑↓+|↓↑),  12(|↑↓|↓↑),  |↓↑