How can we experimentally confirm that atoms/molecules in a solid actually "move"? The atoms in a solid are so attracted to each other that they "vibrate" and don't move past each other.
How do scientists "measure" that atomic vibration in a solid (let's say at room temperature)?
As a raw, uneducated person it is easy for me to conclude that the solid is completely at rest and no part of it is "moving". So, what is the experimental evidence which shows that my conclusion is totally wrong and that the tiny invisible atoms are actually "jiggling"?
In the case of the Brownian motion, it is somehow easier (more intuitive and common sense) to assume that the invisible atoms are "moving" and thus "hitting" the colloidal particles. However, regarding a solid... I can't even imagine how I can detect that atomic "vibrations" because I can't see them or feel them.

 A: For me, the most salient fact arising from molecular “jiggling” is simply thermal radiation. It has the advantage of being relatively easy to observe (using thermal imaging at room temperature, and just your eyes at red-hot temperatures and higher), but I suppose whether you consider it convincing evidence of molecular oscillation depends on how comfortable you are with the fact that accelerating charges produce radiation.
A: Another way is to look at the quantum efficiency of photoelectric sensors using indirect band gap materials like silicon. For such materials, a long wavelength photon needs the assistance of a phonon (lattice vibration) to produce an electron-hole pair. A consequence is that the sensor's quantum efficiency varies with temperature.
A: The motion of atoms can be studied using various techniques based on neutron scattering. Unlike X-ray scattering, where X-rays are reflected by the electronic clouds surrounding atoms, neutrons are scattered primarily by the nuclei. Time-resolved versions of neutron scattering (like spin echo) allow observing how displacement of atoms happens in time.
Collective motion of atoms, such as, e.g., vibrations of crystals (phonons), can be also studied using infrared spectroscopy or Brillouin scattering (which is similar to Raman scattering, but involving absorption/emission of phonons).
Finally, nowadays atoms can be viewed under electronic or atomic force microscope (although they "jiggle" too fast to actually see them moving in real time).
A: We talk about “jiggling atoms” because the classical harmonic oscillator explains how solids store heat at high temperatures.  Bolstering the argument, the quantum-mechanical oscillator explains why the heat capacity is reduced at low temperatures, as the energy between the ground and first excited states of the oscillator becomes large relative to the thermal energy available.
The pop-science statement that the atoms in a solid are “always jiggling” as related to the result that a quantum harmonic oscillator has nonzero energy $\frac12\hbar\omega$ in its ground state.
This may not be very convincing evidence of atomic harmonic motion — after all, many things are usefully approximated as harmonic oscillators.  However, a collective excitation of the oscillators in a material is known as a “phonon.” The name intentionally suggests that a phonon is a “quantum of sound” in a way analogous to the photon, the quantum of light.  There are a number of connections between the “incoherent” phonons which describe heat transfer within a material and the “coherent” phonons which describe the propagation of sound. We have excellent reasons to think of sound as macroscopic vibrations in a material.
