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?
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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 $$|1,2\rangle=\frac{|\uparrow\rangle^{(1)}\otimes|\downarrow\rangle^{(2)}\otimes|\downarrow\rangle^{(3)}+ |\downarrow\rangle^{(1)}\otimes|\uparrow\rangle^{(2)}\otimes|\downarrow\rangle^{(3)} + |\downarrow\rangle^{(1)}\otimes|\downarrow\rangle^{(2)}\otimes|\uparrow\rangle^{(3)}}{\sqrt{3}}.$$
Quantum Mechanic
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