Does the Schroedinger's equation have the advanced wave solution that travels backward in time?
My answer is not.
When a Schroedinger equation is written like,
then the solution may be written like,
The solution (2) for equation (1) is, in a sense, a retarded wave solution that travels forward in time. Physicist like Erwin Schroedinger in the era of pioneerong works of quantum mechanics already noticed that the complex conjugate function of the normal solution (2), that is, the follwing function,
represents an advanced wave solution that travels backward in time. But please be careful. The function (3) is never a solution for the original Schroedinger equation (1), but is a solution for the following different equation.
The equation (1) is usual Schroedinger equation, but I do not know how to call the equation (4). The equation (1) and equation (4) are definitely different. So, a theory that proposes that a quantum mechanical wave function could travel backward in time is not valid.