In the first graduate course of quantum mechanics I remember asking the question: “Can we consider the wave function as a description of a single quantum system?” I got no answer. Twelve years later, in South Carolina, after I completed my Ph.D. studies at Tel Aviv University under the supervision of Yakir Aharonov in which we developed the theory of weak measurements , I asked Aharonov: Can we use weak measurement to observe the wave function of a single particle?
At that time I had already become a strong believer in the many-worlds interpretation
(MWI) of quantum mechanics  and had no doubt that a single system is
described by the wave function. Yakir Aharonov never shared with me the belief in
the MWI. When we realised that using what is called now protective measurement,
we can, under certain conditions, observe the wave function of a single quantum
system, he was really excited by the result. At 1992 I was invited to a conference
on the Foundations of Quantum Mechanics in Japan where I presented this
result: “The Schr¨odinger wave is observable after all!”. Then I went home to Tel Aviv where I finished writing a letter which received mixed reviews in Phys. Rev. Lett., while Jeeva Anandan, working on the topic with Aharonov in South Carolina, wrote a paper accepted in Phys. Rev. A . After acceptance of the PRA paper it was hard to fight the referees in PRL, but PLA accepted it immediately .
Note: This is an excerpt of the second chapter of the anthology Protective Measurement And Quantum Reality (CUP, 2014). Here is the abstract: My view on the meaning of the quantum wave function and its connection to protective measurements is described. The wave function and only the wave function is the ontology of the quantum theory. Protective measurements support this view although they do not provide a decisive proof. A brief review of the discovery and the criticism of protective measurement is presented. Protective measurements with postselection are discussed. Full text in PDF can be found here.