Reconstructing The Wave Function Through The Momentum Weak Value

The wave function is the cornerstone of quantum mechanics, although it is still open to different interpretations. It has been shown that the amplitude and the phase of a wave function are manifested in the imaginary and real parts of the momentum weak value, respectively. Based on this, we develop a weak measurement scheme to reconstruct the wave function through measuring the momentum weak value. A proof-of-principle experiment is made to confirm the feasibility of our measurement scheme, in which the polarization states and the momentum of light are coupled. We believe the experimental protocol has the potential to be applied in diverse fields such as wave-front sensing, edge detection, and optical communication, and measurement of the momentum weak value may offer valuable insight to better understand quantum mechanics.