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Every wave is characterised by some periodically changing disturbance. For example, that entity is air pressure for sound waves and electric,magnetic fields for EM waves. What is that disturbance for matter waves?

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There are no matter waves. In quantum mechanics the wave equation that describes the measurable observables gives wave functions , i.e. complex sinusoidally varying mathematical functions; the complex conjugate square of these functions gives the probability of finding a particle of mass m in the location $(x,y,z)$ at time $t.$ When the experiment is done, the particle appears whole.

Every wave is characterised by some periodically changing disturbance

That is a false statement even for classical physics. Electromagnetic waves are not a disturbance of a medium, no luminiferous aether exists, as was proven by the Michelson Morley experiment.

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  • $\begingroup$ I think a question like this is a good time to point out that the matter wave is defined on configuration space, which is the true reason there isn't a thing waving in non relativistic quantum mechanics. $\endgroup$ – Timaeus Nov 8 '15 at 3:59
  • $\begingroup$ @Timaeus It confuses people with no graduate physics background. IMO it is better to answer at the level of the question. The term "matter waves" is bad for a general vocabulary. The mass itself is not "waving" even in configuration space. Waving implies extent (elementary particles are points). $\endgroup$ – anna v Nov 8 '15 at 5:16
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The oscillating quantity in matter wave is probability-amplitude, a complex number .

Suppose the electron is in state $|\psi\rangle.$ The wavefunction of finding the electron at any coordinate $x$ is given by $\langle x|\psi\rangle= Ae^{-iEt/\hbar}\cdot e^{ipx/\hbar}.$ Probability of finding the electron around $x$ is given by $$P(x,x+dx)= |\psi(x)|^2 dx= |\langle x|\psi\rangle|^2 dx.$$ Thus the probability amplitude whose square determines the probability-density changes with time & this is the oscillating element.

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Let's take this one step at a time. Bear with me.

Every wave is characterised by some periodically changing disturbance. For example, that entity is air pressure for sound waves

It's air. When an ocean wave moves through the sea, the sea waves. When a seismic wave moves through the ground, the ground waves. When a sound wave moves through the air, the air waves.

and electric,magnetic fields for EM waves.

Have ever looked at LIGO? See this web page:

"Gravitational waves are distortions or 'ripples' in the fabric of space-time caused by some of the most violent and energetic processes in the Universe. Albert Einstein predicted the existence of gravitational waves in 1916 in his general theory of relativity. Einstein's mathematics showed that massive accelerating objects (such as neutron stars or black holes orbiting each other) would disrupt space-time in such a way that 'waves' of distorted space would radiate from the source".

See that mention of waves of distorted space? When a gravitational wave moves through space, space waves. Note Einstein talking about space as an "aether" in 1920 here:

"Recapitulating, we may say that according to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an aether".

It's a popscience myth that Einstein dispensed with the aether. See this quantum vacuum section on the Wikipedia "aether theories" article. See the quote by Nobel prizewinner Robert Laughlin of Stanford:

"It is ironic that Einstein's most creative work, the general theory of relativity, should boil down to conceptualizing space as a medium when his original premise [in special relativity] was that no such medium existed".

Anyway, when a gravitational wave moves through space, space waves. And when an electromagnetic wave moves through space, space waves. There are no waves where something doesn't wave.

What is that disturbance for matter waves?

Let's limit ourselves to electrons. We can make electrons and positrons out of photons in gamma-gamma pair production. And we can annihilate the electron and positron to get the two photons back. And in atomic orbitals electrons "exist as standing waves". Plus we can diffract electrons. Electron diffraction refers to the wave nature of electrons. So, what's waving? The same thing as per electromagnetic waves. Space. Only the waving isn't so obvious because electrons exist as standing waves. Standing wave, standing field. And this is an electromagnetic field, or the electron field if you prefer. But not a probability field.

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