  We consider a single copy of a quantum particle moving in a potential and
show that it is possible to monitor its complete wave function by only
continuously measuring its position. While we assume that the potential is
known, no information is available about its state initially. In order to
monitor the wave function, an estimate of the wave function is propagated due
to the influence of the potential and continuously updated according to the
results of the position measurement. We demonstrate by numerical simulations
that the estimation reaches arbitrary values of accuracy below 100 percent
within a finite time period for the potentials we study. In this way our method
grants, a certain time after the beginning of the measurement, an accurate
real-time record of the state evolution including the influence of the
continuous measurement. Moreover, it is robust against sudden perturbations of
the system as for example random momentum kicks from environmental particles,
provided they occur not too frequently.
