Prompted by several personal inquiries, let me expand on an earlier post
in which I pointed out that the Shroedinger equation can be turned into
two coupled equations, free of imaginary quantities. These coupled
equations illustrate yet another member of the family of real,
differential "wave equations":
IE., if you substitute into the free particle Sch eq : Phi(x,t) = R(x,t)
+ i*M(x,t) you get two coupled equations for (real) R and M :
(d/dx)^2(R) = dM/dt and (d/dx)^2 (M) = - dR/dt (all real constants
have been replaced by unity)
These equations are at first glance a set of coupled "diffusion"
equations, free of any imaginary quantities. However, unlike the
"stand-alone" single (real) diffusion equation, the coupled pair supports
traveling wave solutions. One family of such solutions is of the form:
R(u) = f(u) and M(u) = df(u)/du , having defined u(x,t) = x - t .
An example using circular functions is: R(x,t) = COS(x-t) and M(x,t)
= SIN(x-t)
*******
PS.
The above coupled pair of DEs is curiously reminiscent of the (totally
unrelated) pair of DEs:
(d/dt)^2(x) = dy/dt and (d/dt)^2 (y) = - dx/dt
These (with all constants set to unity) describe the circular orbit of a
charged particle in a uniform magnetic field. The orbit is in the (x,y)
plane; the magnetic field is in the z direction. x(t) and y(t) are the
particle's coordinates. (This is apropos to "cyclotron" motion.)
The quantities { R(x,t) and M(x,t) } in the "reified" SE are of course
of a completely different kind than these { x(t) and y(t) } , but to a
sufficiently "wierd" mind the similarity of the juxtaposition of
derivatives may speak to some unspeakable physical/conceptual/mathematical
connection between Shroedinger QM and Newtonian cyclotron motion :)