# What is the correct name of this 'basis label operator'?

What is the correct name for this operator I am calling the 'basis label operator' which returns the constant function of the eigenvalue for all vectors in a (momentum) eigenspace?

$$\hat{O} : \hat{O}(\psi) \rightarrow \frac{-i\hbar}{\psi}\frac{\partial\psi}{\partial x}$$

for those who would like me to be a little more formal / explicit in my formulation, I could write exactly the same definition as:

$$\hat{O} : \hat{O}(\psi) \rightarrow \left( \phi: \phi(a) \rightarrow \left. \frac{-i\hbar}{\psi(a)}\frac{\partial\psi}{\partial x} \right \rvert_{x=a}\right)$$

• Assuming it’s extended to all vectors by linearity, this is the expectation value of momentum. Jul 9, 2018 at 9:50
• Hi @knzhou Is there a theorem that this will give the expectation value of momentum for non-eigenfunctions? I wouldn't have expected that because I don't think it is a linear operator. I'm not sure what you meant by "extended to all vectors by linearity" but I think this operator is defined for all states in the Hilbert space without being linear. Jul 9, 2018 at 14:26
• This is a very unusual object, in what context would this come up? I don't think that it has a name. Jul 9, 2018 at 14:40
• @user183966 Sorry, I shouldn't have said "extended by linearity" because it's not linear. Jul 9, 2018 at 14:42
• What happens when $\psi(x)=0$? Jul 9, 2018 at 16:56

Maybe it helps that $$\hat O = \hat p \circ \log_x$$ where $\log_x$ is defined by its action in position space: $$\log_x( |\psi\rangle ) = \int |x'\rangle\, \log(\langle x' | \psi \rangle)\, \mathrm dx' .$$
However. Be warned that, if something like this comes up in your calculations, and you are not 100% sure what you are doing, I think it is likely that what you are doing does not make sense at all. Non-linear operators come up only very rarely. Also note that I was careful in specifying that the action of the operator depends on the position space representation of $|\psi\rangle$. It is very unnatural to consider an operator whose action depends on a specific representation of the Hilbert space.