>>13657585Actually, it is perfectly valid to "operate on the left", given that you know what you're doing. The derivation by
>>13657711 essentially is using this fact.
For simplicity, suppose is a bounded linear operator on a Hilbert space . Its adjoint is defined by the relation for all . An operator is called self-adjoint (often physicists simply call this "Hermitian" instead of self-adjoint) if . Thus, it is clear that if you have a self-adjoint operator, then when evaluating any matrix element you can "act" to the right or left. This is why, when physicists use bra-ket notation, they have no qualms about the fact that doesn't precisely distinguish what the two vectors in the inner product are. They are very often working with self-adjoint operators (i.e., observables), so it doesn't matter how you interpret the entries on the sesquilinear form.
And even if you're not working with a self-adjoint operator, can nonetheless always be "moved" to the left, as long as you remember to take its adjoint as well. For instance, physicsts are also often interested in the transition amplitudes defined by a unitary operator , i.e., . If you're using the bra-ket notation and you want to compute this matrix element by acting first on rather than on , you simply have to remember to take the adjoint, i.e., .
