I was flicking through Zettili's book on quantum mechanics and came across a 'derivation' of the momentum operator in the position representation on page 126. The author derived that ⟨→r|ˆ→P|ψ⟩=−iℏ→∇⟨→r|ψ⟩ (I've omitted the full derivation). However, from this relationship he concluded that ˆ→P=−iℏ→∇. I'm sure this is very basic but why can you immediately conclude this? Surely this assumes that ⟨→r|(−iℏ→∇)|ψ⟩=−iℏ→∇⟨→r|ψ⟩. I'm not sure why this is necessarily true.
Answer
−iℏ→∇|ψ⟩ is not a valid notation. The nabla operator is defined in three-dimensional Euclidean space, not the in the Hilbert space of quantum states.
When the author says ˆP=−iℏ→∇ he does not mean the momentum operator defined in the state space, but the space of wavefunctions. Then ˆPψ(r)=−iℏ→∇ψ(r).
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