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Unitary vector commuting with Hamiltonian and effect on system
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[QUOTE="happyparticle, post: 6853274, member: 728762"] [B]Homework Statement:[/B] U unitary operator that commute with H. ##| \psi_n \rangle## an eigenstate of H with eigenvalue ##E_n## ##| \phi_n \rangle = U | \psi_n \rangle## Thus, ##| \phi_n \rangle = \sum_i \alpha_i |\psi_n^i \rangle## [B]Relevant Equations:[/B] ##| \phi_n \rangle = \sum_i \alpha_i |\psi_n^i \rangle## Hi, I'm not sure to understand what ##| \phi_n \rangle = \sum_i \alpha_i |\psi_n^i## means exactly or how we get it. From the statement, I understand that ##[U,H] = 0## and ##H|\psi_n \rangle = E_n|\psi_n \rangle## Also, a linear combination of all states is also an solution. If U commutes with H then they have the same eigenstates (and same eigenvalues ?) Thus, ##U|\psi_n \rangle = E_n | \psi_n \rangle## I have hard time to put all those things together or seeing what that really means. Thank you [/QUOTE]
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Unitary vector commuting with Hamiltonian and effect on system
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