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I was studying about perturbation theory when I read about the conditions to the state of a interacting system be orthogonal to the state of the non-interacting system.
That is, consider that our quantum system has a Hamiltonian: $$H = H_{0} + V$$ And let $\Psi_{0}$ be an eigenstate of $H$ and $\phi_{0}$ an eigenstate of $H_{0}$ (both being the respective ground states). So, I read that the product $\langle \phi_{0} | \Psi_{0} \rangle$ only will be different from zero if the states $\Psi_{0}$ and $\phi_{0}$ have the same symmetry. I don't know what it means. What would be a symmetry of a state? And how can I determine it?

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In perturbation theory, $|\psi_0\rangle$ is derived from the known $|\phi_0\rangle$. Let's say that $|\phi_0\rangle$ is symmetric(or antisymmetric), then the $|\psi_0\rangle$ must also be symmetric(or antisymmetric) or the $|\psi_0\rangle$ will not be a correct state representation of $H$.

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