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Summary:
 I am trying to convert an expression written in terms of field operators and derivatives to be written in terms of ladder operators, but one expression seems to be Hermitian while the other isn't, so something is wrong.
Main Question or Discussion Point
Suppose we have a Hamiltonian containing a term of the form
where ∂=d/dr and A(r) is a real function. I would like to study this with harmonic oscillator ladder operators. The naïve approach is to use
where I have set ħ=1 so that
This term is Hermitian because r and p both are.*
However, if we check for Hermicity in real space, we realize that
We integrated by parts in the third step. This is *not* Hermitian; it has an extra term.
What is the cause of this disagreement, and what is the proper way to study this Hamiltonian using the ladder operators?

(Note: The actual Hamiltonian term I have is
which *is* Hermitian, but it suffers from a similar problem when expressed in the conventional form with all derivatives on ψ(r). In that case, an extra nonHermitian term appears.)

*Edit:
Thank you to HomogeneousCow for pointing out the sloppiness of my notation which treated the position representation of momentum as if it were the general operator. While correcting this sloppy notation I realized that my error was in the section marked with an asterisk. Clearly
is not Hermitian. Lesson learned, sloppy notation leads to careless mistakes.
where ∂=d/dr and A(r) is a real function. I would like to study this with harmonic oscillator ladder operators. The naïve approach is to use
where I have set ħ=1 so that
This term is Hermitian because r and p both are.*
However, if we check for Hermicity in real space, we realize that
We integrated by parts in the third step. This is *not* Hermitian; it has an extra term.
What is the cause of this disagreement, and what is the proper way to study this Hamiltonian using the ladder operators?

(Note: The actual Hamiltonian term I have is
which *is* Hermitian, but it suffers from a similar problem when expressed in the conventional form with all derivatives on ψ(r). In that case, an extra nonHermitian term appears.)

*Edit:
Thank you to HomogeneousCow for pointing out the sloppiness of my notation which treated the position representation of momentum as if it were the general operator. While correcting this sloppy notation I realized that my error was in the section marked with an asterisk. Clearly
is not Hermitian. Lesson learned, sloppy notation leads to careless mistakes.
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