Decomposing the direct sum as direct product

[elduderino]

This is a basic question in angular momentum in quantum mechanics that I am studying.
I know that $$\frac{1}{2}\otimes \frac{1}{2} = 1\oplus 0$$ What would be a strategy to proving the general statement for spin representations $$j\otimes s =\bigoplus_{l=|s-j|}^{|s+j|} l$$

 

[dextercioby]

This is typically treated in every book of quantum mechanics. Use for example the second volume of Cohen - Tannoudji's text. The famous grid-proof is there. Or any books on group theory with applications to physics (this is typically the Clebsch-Gordan theorem for SU(2)).

 

[Bill_K]

elduderino, The decomposition can be obtained just by counting the number of states. A system with spin s has 2s + 1 states, one for each value of m running from m = s, s-1,... down to m = -s. Now look at the states in the product ℓ ⊗ s. There will be one state with m = ℓ+s, two states with m = ℓ+s-1, ... down to s states with m = ℓ-s (assuming s <= ℓ). If you try grouping these states into sets with definite j values, you will find it necessary to use j = ℓ+s, ... down to j = ℓ-s, and each of these j values will be needed exactly once.