# Where did the bases states go?

I wrote out the bases states for 4 1/2-spin particles in $$|S,m\rangle$$ representation. I know I should have a 16-dimensional Hilbert space, but I only have 9: $$|2,2\rangle,\dots,|2,-2\rangle; |1,1\rangle,|1,0\rangle,|1,-1\rangle; |0,0\rangle$$ Where did the other 7 bases go?

• Dec 22, 2021 at 15:03
• @CosmasZachos, that post, with tensor powers, etc. is NOT helpful. The simplest explanations are the best. Dec 22, 2021 at 22:57
• But unless you appreciate addition of angular momentum, a simple explanation will fail you…. Dec 23, 2021 at 3:48

Some values of $$S$$ will be repeated. For instance $$S=1$$ occurs three times (that makes 6 more states you did not account for) and $$S=0$$ occurs twice (this is the last state unaccounted for).
Since $$1/2\otimes 1/2=1\oplus 0$$ you can decompose $$(1\oplus 0)\otimes (1\oplus 0)$$ to clearly see that $$S=0$$ occurs twice: once from $$0\otimes 0$$ and once from $$1\otimes 1$$. Likewise the three copies of $$S=1$$ occur in $$1\otimes 1$$, $$1\otimes 0$$ and $$0\otimes 1$$ respectively.
Finally be aware that, even if you have multiple copies of the same value of $$S$$, the states in each copy will be different and in fact can be made orthonormal.
• nonetheless different sets of basis states. at here are three copies of the $S=1$ states you one can make an arbitrary but common transformation in each $m_s$ subspace to generate different sets. Dec 23, 2021 at 3:05