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Is there a way to lower the total spin of the state and fixing the $S_z$ rather than lowering the $S_z$ by spin ladder operator? Or in other words, how to connect the $S=1$ state with $S=2$ or $S=0$ state? Is there such an operation though it might be unphysical?

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For a spin-1 particle, the S=1 state has dimension 3, and the S=0 state has dimension 1. You can't connect these with a nice operator. –  Peter Shor Jun 13 at 1:44
    
I mean multiple spins. Actually, from the tensor product, one sees that s=0 and s=1 states are in block form for S^2 and Sz. So I would guess this questions does not make any sense. However, I do not know whether there is a mathematical way to connect the two like ladder operators. –  user45765 Jun 13 at 2:48

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