Experimental verification for the de-Broglie - Einstein relation $E= h \nu$ for a particle Is there any experiment that confirms the relation $E=h \nu$ for material(I mean the matter waves) objects? I mean, from the photo electric effect experiment, we came to know that photon carries energy $E= h\nu$ for a given frequency, but how did they verify it for a particle ?
 A: The most striking experiment [1] I know of consist in diffracting fullerenes molecules, formula $\text{C}_{60}$, the famous buckyballs. The de Broglie wavelength is about 400 times smaller than the actual diameter of the molecule. Here is how the calculation goes: the average speed of the molecules was $v=220\  \mathrm{m/s}$, determined by a time-of-flight method; the mass of one molecule is $m=1.2\times 10^{-21}\ \mathrm{g}$, resulting in a momentum $p=\gamma m v\approx mv=2.6\times 10^{-19}\ \mathrm{gm/s}$, and therefore in a de Broglie wavelength $\lambda=h/p=2.5\times 10^{-12}\ \mathrm{m}$. From that value, one can predict the diffraction pattern and check it compares well with the experimental measurement, which it does according to the article.
What about the other de Broglie relation you were interested in then? $E=\gamma m c^2$ but $\gamma\approx 1$ here (I already used that in the computation of the momentum $p$), i.e. only the rest energy matters, and we get $E=6.7\times 10^{11}\ \mathrm{eV}$, which is way higher than the electromagnetic binding energies, so we are ok with just using the bare mass of the 60 Carbon atoms. That gives $\nu=1.6\times 10^{26}\ \mathrm{Hz}$. But none of that was measured as the diffraction patterns depends on the wavelength.
Moreover this molecule is a very complex object with many internal degrees of vibrations and a lot of very close energy levels. Thus it can be argued that this is as close to a classical object as we can get for this type of experiment.
[1] Markus Arndt, Olaf Nairz, Julian Vos-Andreae, Claudia Keller, Gerbrand van der Zouw, and Anton Zeilinger, Wave-particle duality of c60 molecules, Nature 401 (1999), 680--682
A: The clearest examples of this are various forms of the photoelectric effect, either in ionization from solids, ionization of atoms and molecules in the gas phase, or in excitation and de-excitation of bound-bound transitions in the gas phase.
If you buy into the matter-wave hypothesis from the beginning, then it is clear that these phenomena couple the frequency of the matter wave to that of the incoming electromagnetic radiation. There's a number of ways you can do this, depending on exactly what you're looking for, but they all play with the same set of fundamental ideas. 
As an example, you can take a sample of hydrogen atoms at rest and bombard them with electrons with a narrowband tunable kinetic energy, and then observe for fluorescence from the gas; you will find that there are certain specific regions of the bombardment energy that produce a much higher fluorescence, and that each of them produces a different colour in the fluorescence.
It is always a bit tricky to distinguish between experiments that couple to the wavelength (like the double-slit interference experiments in Luc and anna's answers) and experiments that couple directly to the frequency, but for experiments in the gas phase it is very hard to see how the wavelength of the light can be relevant for the coupling (since it's much longer than the size of each atom) so the only way the atomic dynamics can couple to the light is through its frequency.
Here, if you buy into wave mechanics, the connection is clear: you have formed a superposition of atomic states which oscillate at different frequencies, and their interference couples to the EM field, to a first approximation as I described here.
If you don't really buy the wave mechanics and you're looking for evidence to convince you, then it gets a bit tricky - ultimately, the reason we buy the wave mechanics isn't because of a single experiment but because of the full edifices of experiment and theory and how they interact with each other - but you nevertheless have a phenomenon that (i) clearly depends on the supplied mechanical energy, (ii) is clearly resonant, as a confined wave would, together with a Lorentzian lineshape if your experiment is precise enough, and (iii) its effects wind up imprinted in the temporal frequency of an external system.
A: That the electron trajectories obey a  wave probability is shown clearly in this accumulation of single electrons scattering on two slits.


Results of a double-slit-experiment performed by Dr. Tonomura showing the build-up of an interference pattern of single electrons. Numbers of electrons are 11 (a), 200 (b), 6000 (c), 40000 (d), 140000 (e).

From top to bottom, where the footprint of the single electrons are detected , the accumulation of the distribution of the scattered electrons ( a probability distribution) shows the interference pattern of wave nature. 
The wavelength is consistent with the quantum mechanical predictions. (original paper copies in pdf exist when searching  title and author , where the experiment is described and the conclusions drawn. 
