  Central to quantum theory, the wavefunction is the complex distribution used
to completely describe a quantum system. Despite its fundamental role, it is
typically introduced as an abstract element of the theory with no explicit
definition. Rather, physicists come to a working understanding of the
wavefunction through its use to calculate measurement outcome probabilities via
the Born Rule. Presently, scientists determine the wavefunction through
tomographic methods, which estimate the wavefunction that is most consistent
with a diverse collection of measurements. The indirectness of these methods
compounds the problem of defining the wavefunction. Here we show that the
wavefunction can be measured directly by the sequential measurement of two
complementary variables of the system. The crux of our method is that the first
measurement is performed in a gentle way (i.e. weak measurement) so as not to
invalidate the second. The result is that the real and imaginary components of
the wavefunction appear directly on our measurement apparatus. We give an
experimental example by directly measuring the transverse spatial wavefunction
of a single photon, a task not previously realized by any method. We show that
the concept is universal, being applicable both to other degrees of freedom of
the photon (e.g. polarization, frequency, etc.) and to other quantum systems
(e.g. electron spin-z quantum state, SQUIDs, trapped ions, etc.). Consequently,
this method gives the wavefunction a straightforward and general definition in
terms of a specific set of experimental operations. We expect it to expand the
range of quantum systems scientists are able to characterize and initiate new
avenues to understand fundamental quantum theory.
