Until what point can the de Broglie wave be thought as a real wave?
I mean, is it made of something? What amplitude does it have? Is it a sine wave?
How can it be related to the wavefunction of the particle?
Answer
All matter is made of waves—at least it can be represented that way, and it behaves that way. Of course, matter also behaves like particles. This is one of the odd-but-true conclusions of quantum mechanics.
The de Broglie wavelength gives the wavelength of any "matter wave." These waves aren't waves in the classical sense with amplitude and the like; they are wave functions, which express the probable location of a particle as something that looks like a wave. You can think of it in a sense that, looking very closely, the location and motion of a particle becomes blurry and starts to look like a wave instead of just one point.
We usually think of matter as a wave only when making observations on a scale comparable to the de Broglie wavelength, which is very small for most things. Using the de Broglie relation $\lambda=\frac{h}{p}$, you can calculate the wavelength of a particle with momentum $p$. As an example, a electron with a kinetic energy of 10 eV has a wavelength of 0.39 nm. That is the wavelength of the electron's wavefunction. If you perform an experiment that would highlight that wave behavior, such as pass a beam of electrons of that energy through a diffraction grating with a spacing comparable to that wavelength, you would see an interference pattern, just as you do with light, because electrons behave like waves (when we want them to).
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