Probability amplitudes, de Broglie and Schrödinger

[redtree]

What is the relationship between the "matter waves" described by de Broglie, the probability amplitude function and Schrödinger's wave equation?

I've read the following:

"The wavelengths postulated by de Broglie to be associated with the motions of particles are in reality the wavelengths of the probability amplitudes or wave functions."

I've also read:

"What is a wave function? The short answer is that it is a probability amplitude, that also happens to solve Schrödinger’s equation."

Are they all versions of the same thing?

 

[andrewm]

The wavefunction in one dimension is simply some function f(x) that solves Schroedinger's equation. It is called a "wave" function because the Schroedinger equation is mathematically similar to the so-called Wave Equation (Wikipedia explains).

The probability amplitude is the square of the wavefunction. This is a postulate, so you'll have to remember it, or remember an analogy.

In fact, it is analogous to the electric field (wave) E(x), since we think about $$|E(x)|^2$$ as the intensity of the wave.