How do we determine the conjugate of radial component of linear momentum?

Question

How do we describe the radial part of linear momentum? Here I found the following description:

Classically

$$p_r=\hat{D}_r = \frac{\hat{r}}{r} \cdot\hat{p} = \frac{\hbar}{i}\frac{\partial}{\partial r}$$

However $\hat{D}_r$ is not hermitian. Consider the adjoint

$$\hat{D}_r^\dagger= \hat{p}\cdot\frac{\hat{r}}{r} =\frac{\hbar}{i} \left ( \frac{\partial}{\partial r}+\frac{2}{r} \right )$$

Now we know from linear algebra how to construct a hermitian operator from an operator and its adjoint:

$$\hat{p}_r = \frac{\hat{D}_r^\dagger+\hat{D}_r}{2}=\frac{\hbar}{i} \left ( \frac{\partial}{\partial r}+\frac{1}{r} \right )$$

Can anyone clarify how conjugate is obtained?

Answer 1

It is well known that there is no way to make the radial momentum operator self-adjoint. As to how to construct the formal conjugate and and (if it exists) the true conjuage see our book (a draft is avaiable online here) chapter 4.