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Decomposing the direct sum as direct product

  1. Oct 24, 2011 #1
    This is a basic question in angular momentum in quantum mechanics that I am studying.
    I know that [tex]\frac{1}{2}\otimes \frac{1}{2} = 1\oplus 0[/tex] What would be a strategy to proving the general statement for spin representations [tex]j\otimes s =\bigoplus_{l=|s-j|}^{|s+j|} l[/tex]
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  3. Oct 24, 2011 #2


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    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)).
    Last edited: Oct 24, 2011
  4. Oct 24, 2011 #3


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    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.
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