Operator Bonanza

Suppose \(\hat { H } { \hat { a } }_{ + }{ \psi }_{ n } = ({E}_{n}+h\omega) { \hat { a } }_{ + }{ \psi }_{ n }\), where \(\hat{H} {\psi}_{n} = {E}_{n} {\psi}_{n}\) and \(\hat{H}\) is a Hamiltonian operator for energy. Also, \({\psi}_{n}\) is a completely valid quantum state.

What can one say about \({\hat{a}}_{+}{\psi}_{n}\)?

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