Question

please answer with short verbal comment:
can Schrodinger's equation have real wave functions psi(x,t) as solutions?

Answer 1

Suppose   is real. Then the energy is given by

or the energy is:

Since is real, its time derivative will also be real. Therefore the energy will be imaginary because of the factor of i in the above equation. But the energy cannot be imaginary. Hence cannot be real.

Note that however, the time-independent part of the wavefunction can be real. The full wave function which includes time dependance cannot be real.