# Find projection operators degenerate energy eigensubspaces [closed]

A given system has Hamiltonian $$H=\sum_{i=0}^{n}\sigma^{(i)}_{z}$$, where $$\sigma^{(i)}_{z}$$ are the usual Pauli matrices. Now I want to find the corresponding $$n+1$$ projection operators corresponding to each energy eigensubspace (numerically is fine). How do I do this?

Have you heard of the Dicke basis? Eg Collective angular momentum , Dicke states and indistinguishable particles. Or the Fock basis? You write down the symmetric superposition of having $$m$$ of the spins pointing up and the rest of the spins pointing down. There may be a mistake in terms of defining $$n$$: when you have a linear combination of Pauli matrices corresponding to particles labeled from 0 to $$n$$, there are $$n+2$$ possible energy levels. So we have states like $$|m,n+1-m\rangle\propto\sum_{\text{permutations}}|\uparrow\rangle^{(1)}\otimes|\uparrow\rangle^{(2)}\otimes\cdots \otimes|\uparrow\rangle^{(m)}\otimes|\downarrow\rangle^{(m+1)}\otimes\cdots \otimes|\downarrow\rangle^{(n+1)},$$ and we form the $$n+2$$ projectors as $$(m\in 0,1,\cdots,n+1)$$: $$P_m=|m,n+1-m\rangle\langle m,n+1-m|.$$
For reference, a sum over permutations considers all possible orderings of the spins that are up and down, and we need to make sure each state $$|m,n+1-m\rangle$$ is normalized to unity. So for example we would have $$|1,2\rangle=\frac{|\uparrow\rangle^{(1)}\otimes|\downarrow\rangle^{(2)}\otimes|\downarrow\rangle^{(3)}+ |\downarrow\rangle^{(1)}\otimes|\uparrow\rangle^{(2)}\otimes|\downarrow\rangle^{(3)} + |\downarrow\rangle^{(1)}\otimes|\downarrow\rangle^{(2)}\otimes|\uparrow\rangle^{(3)}}{\sqrt{3}}.$$