Consider two operators \hat{a} and \hat{b}. the operation \hat{a}\hat{b} |ψ> means first operate on state |ψ> with operator \hat{b} to produce a new state |φ> ≡ \hat{b} |ψ> , then operate on the new state |φ> with operator \hat{a} to produce yet another state |φ'> ≡ \hat{a} |φ> , i. e. |φ'> = \hat{a} |φ> = \hat{a}\hat{b} |ψ> . note that, in general, the order of operation matters, i. e., \hat{a}\hat{b}\neq\hat{b}\hat{a} (aˆ and bˆ do not commute).
(a) using the definition that the hermitian adjoint of aˆ |ψ> is < ψ| \hat{a}^\dagger , prove that (aˆbˆ)^† = bˆ†aˆ† .
(b) suppose that aˆ and bˆ are both hermitian and that they do not commute aˆbˆ \neq bˆaˆ. is the product aˆbˆ hermitian? is the sum of products aˆbˆ + bˆaˆ hermitian?
(c) suppose that aˆ and bˆ are both hermitian and that they commute aˆbˆ = bˆaˆ. is the product aˆbˆ hermitian?
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