The real and imaginary parts of the time-dependent Schrodinger equation do
have simple physical significance. It is discussed briefly in Messiah,
Vol.1, "Quantum Mechanics", pages 222 to 224. Briefly put, what one does is
write the wave function in the form psi = Aexp(iS/h) and substitute into
the equation. Equating real and imaginary parts, one gets two real coupled
equations that the functions S and A obey.
The result from equating the imaginary part is just the equation of
continuity, if one notes that the current density is
2
J = A (grad S)/m
where m is the particle mass.
The equation obtained by equating the real parts has a nice physical
interpretation if the velocity field is defined by v = J/P, where
P is the square of A and is the probability density. Then, the result is
2
m dv/dt = - grad V plus terms proportional to h ,
where V is the potential. Thus, Newton's Second Law appears from this
equation, in the classical limit when h goes to zero. I am using h for
Planck's constant divided by 2pi.
There is also a small literature on the "hydrodynamic formulation of
quantum mechanics" in which the term proportianal to the square of h is
kept and so you have an exact formulation of quantum mechanics involving
only the real quantities J and P. The price you pay for this "non-complex"
formulation is that the two coupled equations for J and P are non-linear.
Nonetheless, it is possible, for example, to solve for the hydrogen atom
energy levels using the hydrodynamic formulation.