Reply to "Comment on `Optimal probe wave function of weak-value amplification' "

TL;DR

This paper clarifies the optimal probe wave function in weak-value amplification, correcting previous variational calculations and emphasizing the practical relevance of normalizable solutions.

Contribution

It corrects the variational approach for finding optimal probe wave functions, identifying the physically relevant normalizable solutions over un-normalizable ones.

Findings

Normalizable solutions are optimal for weak-value amplification.

Un-normalizable solutions are mathematically possible but not practically relevant.

The phase translation freedom must be fixed for correct variational analysis.

Abstract

It is pointed out that the "counter example" presented in the Comment is a family of probe wave functions which are increasingly broad as the shift becomes large. Furthermore, the author's variational calculation is not correct in the sense that we have to gauge fix the freedom of the phase translation. It is shown that there are two kinds of solutions, normalizable and un-normalizable. The former is our optimal solution, and the latter is what he found. It seems only the former is relevant from a practical point of view.