The wavefunction of an elementary particle has no physical meaning. It can't be measured. This is perhaps a single case where a solution to an equation has no meaning. It has to be further processed by taking a squared magnitude, which gives a probability. That is Born's idea. Why it has to be squared, nobody knows. The Schrödinger equation can give solutions to a situation that can't exist. For example, take any case from a quantum mechanics textbook, and solve it for energy