As far as I know, particles vibrate with a frequency and wavelength determined by their energy level.
Is this vibration in 3D space?
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Your original question was :
One has to state whether you are talking classical particles or quantum mechanical elementary particles or quantum mechanical atoms and molecules.
Classical particles move in three dimensions, any motion. It might be constrained because of boundary conditions but it is three dimensions.
Elementary particles, i.e the ones of the standard model move through space with their kinetic energy, in three dimensions. When unbound, they do not vibrate. Quarks, which are always bound within nuclei, vibrate in three dimentional space.
Atoms and molecules have vibrational states which again are three dimensional but following boundary conditions.
All the above vibrations are with the normal definition of vibration, the center of mass being displaced from its average position with the energy of vibration, not connected with the de Broglie wavelength.
The "vibrations" you are asking, i.e . the de Broglie wavelength defined "displacement" is not similar to the above. The wave is a probability wave, in three dimensional space, but the wave nature appears in specially designed experiments in the distribution of the probabilities of finding elementary particles, atoms, molecules.
Here is what happens when electrons of the same energy are sent through the double slit:
The probability wave pattern, connected with the energy of the electrons, builds up slowly one electron at a time, proving that the electrons can appear as a probability wave, in three dimensions, but the electron itself is intact, hitting one spot at a time. There is no 3D shape to the electron itself. It is its interactions with the slits that show up the wave nature.
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