Quantum mechanics and everyday nature Is there a phenomenon visible to the naked eye that requires quantum mechanics to be satisfactorily explained? I am looking for a sort of quantic Newtonian apple. 
 A: The definitive, visual proof that quantum mechanics is required to describe our world is the observation of superfluidity in liquid helium that has been cooled to below the lambda point. Below this temperature (2.17K at STP) a macroscopic fraction of the atoms have condensed into the ground state. This leads to macroscopic correlations that cause the fluid to flow in highly non-classical, unusual ways. For example, the fluid can flow up (against gravity) the sides of it vessel to a nearby reservoir:

In a more elaborate set-up, we can see the fountain effect:

I find this to be the most convincing, visible phenomenon that requires QM.
A: This video and this article both show you how to make a quantum eraser experiment at home using only a laser pointer and some polarising filters, which in the video are obtained from 3D glasses*.
One could argue that this doesn't really count, since if you really think about it, you should expect the same result if light were just a wave. However, if you accept that light is photons then it very nicely demonstrates that the interference pattern disappears if there's a way to know which path the photon took, which is a very distinctly quantum phenomenon.
(*) In my experience, most 3D glasses tend to have circular polarising filters rather than linear ones. This doesn't seem to be addressed in the video, but it probably changes what you'd need to do in order to see the result. However, I have used at least one pair with linear filters, which were from an IMAX cinema.
A: Among the naked-eye visible effects that require a quantum explanation are fluorescence, phosphorescence, and electroluminescence. Concepts like band gap energy and the connection between energy and wavelength are needed to form plausible, detailed hypotheses which address the readily observed aspects of these phenomena.
A: The transparancy of glass is a quantum phenomenon. It is owed to the fact that the electrons in the silicon crystals require an inordinate amount of energy in order to get exited into a higher orbital. This means that low energy photons like visible light can pass through unhindered. Meanwhile, UV light has enough energy to be absorbed.
Glass is transparent, but you don't get a sunburn.
A: As an illustration of Dan's answer I get myself a diffraction grating and start looking at florescent lamps. I was frankly really disappointed because you really don't get to see the lines clearly with a grating alone. What you would see is more or less like this

But I moved on and told myself that I can make myself a spectroscope and I did just that! I had a big poster and rolled it to a tube. From a brochure I cut myself a thin slit and taped them together.
 

It didn't take 10 minutes but the result was really satisfactory:

I was also able to see some of the Fraunhofer lines while I was looking at the sun at 1 or 2 hours before sunset but I couldn't take a photo of it because of my poor cellphone camera. 
I hope this helps illustrating Quantum Mechanics from daily life with the help of daily objects!
A: If you are looking for something visible to the naked eye, but that not necessarily happens naturally around you, then the quintessential experiment displaying the quantum nature of light in a way that is visible to the naked eye is Young's double slit experiement.
The great thing about this experiment is that it can easily be performed in the comfort of your own home.  See this Physics.SE post: Is it possible to reproduce Double-slit experiment by myself at home?
A: One very "simple" example is that of the reason why we are not going through the ground while atoms (and hence matter) are mostly empty.
Although it is still debated which dominates in this effect between electrostatic repulsion and the so called Pauli exclusion Principle (a quantum effect), it is pretty much admitted that electrostatics only is not sufficient.
Quantitative estimates of this quantum repulsion are done on a daily basis by people calculating ab-initio (solving equation of quantum mechanics for the electrons) intermolecular potential parameters to be put then in simulations at the molecular scale (basically these calculations explain why it is almost fair to represent atoms as hard beads and therefore they already explain how come two empty atoms cannot overlap).
Another simple case is that of a piece of metal whose rigidity (or at least some measure of it) owes a lot to the exclusion principle (see Eq. 494 of the link and the following sentence) satisfied by the conducting electrons in the system.
A: Superconductivity of course. Classically, you cannot explain perfect diamagnetism and perfect conduction from a disordered system at the same time.
The most striking experiment is the levitation of a superconductor on a magnetic field, aka Meissner effect. You just need a high-Tc superconductor and a few N-liquid. The stricking fact is the disapearance of the effect when the nitrogen entirelly vaporized.
A lot of videos on internet about that. See e.g. this one: http://www.ted.com/talks/boaz_almog_levitates_a_superconductor.html
By the way, superconductivity is THE demonstration of quantun weirness at the macroscopic level.
Quantum Hall effect is an other interesting effect, but it requires more materials (fridge, ...).
In general, one can safely say that almost (if not) all the true effects of quantum physics are matter-field interactions of some types... the Meissner effect and the quantum Hall effect are just two specific matter-field interaction (magnetic field applied on low temperature collective excitations of electrons).
It should in principle be possible to measure the spectrum of some atoms in a table-top experiment (after all, it's a late 19-th experiment), but it's less impressive than levitation I believe. All spectroscopy properties can only be perfectly understood using quantum mechanics, and can well be "seen" by naked eyes, like fluorescence (it sometimes requires IR glasses, but still I believe it's naked eyes since you can really see the fluorescence using this glasses).
Generically, all condensed matter problems require quantum mechanics to be perfectly understood: band theory (including band gap and crystal symmetry leading to the huge field of semi-conductor for instance), electronic  propagation in disordered system (including Mott insulator for instance, ... (see also Kaz's answer https://physics.stackexchange.com/a/65464/16689 on this point). 
Tunnel effect can be thought of a striking effect of quantum mechanics, even if it is hard to see with naked eyes. See nevertheless jinawee's answer https://physics.stackexchange.com/a/65416/16689 on this page.
A: Use a prism (or a diffraction grating if you have one) to break up the light coming from a florescent bulb. You'll see a bunch of individual lines rather than a continuous band of colors. This comes from the discrete energy levels in atoms and molecules, which is a consequence of quantum mechanics.
If the audience you have in mind is more advanced, you can present the ultraviolet catastrophe of classical mechanics. Classically, something with finite temperature would tend to radiate an infinite amount of energy. Quantum mechanics explains the intensity vs. wavelength curves that we actually see.
A: Reflections on Everyday Quantum Events
In one sense, it's hard not to see quantum mechanics in everyday life. For example, the existence of complex chemistry and the volume occupied by ordinary matter are both direct consequences of something called Pauli exclusion. That's a quantum rule that requires that every electron in the universe maintains a unique address that consists of its location in space (three numbers), its momentum (more-or-less velocity for same-mass electrons, and also three numbers), and one more odd one called spin orientation (binary, either up or down). When negatively charged electrons are packed tightly together by, say, the positive attraction of an atomic nucleus, these unique-address rules cause the electrons to take up unique positions and orientations around atoms (chemistry), and to resist being squeezed together beyond a certain point (volume).
Atomic bonding in chemistry -- without which we would not be here to discuss this! -- would largely disappear without that last odd address rule about up-down spins. The ability of two electrons to share the same space by having opposite spins gives some atoms the ability to steal an electron from other atoms by providing a cozy shared-address home away from home, an effect that in chemistry is called ionic bonding. In other cases the up-down pairing rule enables a pair of electrons to be shared equally by two atoms, which is also called covalent bonding.
Seeing is Believing
However, I think your question was really focused more on finding "a phenomenon visible to the naked eye that requires quantum mechanics," and that what you wanted was something a bit more profound and large than simply summing up the large-scale impacts of many very small quantum events. I suspect you were hoping for something you can see without your unaided eye, without the need for a lab filled with exotic equipment.
Such things really do exist. In fact, you very likely looked straight into an example just this morning. They are called mirrors.
That is, it the ability of polished metals manage to reflect beautifully accurate images of the worlds around them while most (not all!) other substances are dark, dull, or transparent, is a type of large-scale quantum event that is every bit as odd as exotica such as laboratory Bose condensates. It’s a classic example of familiarity generating indifference: Metallic reflection is so common and easy to observe that we forget just how profoundly odd and non-classical it is.
Spacing Out, For Real
So why is metallic reflection deeply quantum?
It's quantum in multiple ways, actually. The first step is that you have to send a huge number of electrons into a sort of curious alternative form of space, one in which the coordinate systems for finding the an electron no longer consists of three directions of space, but instead must be expressed in three directions of momentum.
How can an electron possibly get "lost" in ordinary space? The way they get there is surprisingly uncomplicated and ordinary sounding: In metals, certain electrons are given the freedom to run around freely throughout the entire volume of the metal. That is, metal atoms are firm believers in a sort of community-wide sharing of some of their electron children, caring not in the least if their own electron ends up very far away indeed, as long as other electrons stay close enough to cancel out their positive charges.
A roaming electron does not sound all that unusual until you realize that electrons are so very light that quantum mechanics cannot be ignored. What quantum mechanics does to very light objects is cause their quantum descriptions to start taking up space across the entire volume of the metal over which they roam. That is, instead of an electron moving back and for the across a crystal as an massive classical object would, an undisturbed and freely roaming electron is most accurately represented as being equally located at all locations in the metal at the same time.
Try to pull that trick with your car!
What is Your Address, Please?
However, since  in any given piece of metal the super-light shared electrons are all doing the same "I am everywhere!" trick at the same time, there arises a problem with that address issue I mentioned earlier: Every electron in the universe must have a completely unique address.
If these lost electrons are all sharing space it the same chunk of metal, it means that they also are sharing essentially identical (even if odd) locations in ordinary space... and that simply will not do. It means that each such electron in the metal must find some new way to maintain a unique "address" within the universe. The up-and-down option helps, but only allows two electrons to share the same address. So, the only option left is for the electrons to start climbing into the only remaining set of coordinates, which is the diverse range of speeds and directions (velocities) called momentum space.
Now I should point out that when observing this process from our perspective of ordinary space with XYZ coordinates, electrons climbing into momentum space just looks like they are all acquiring different speeds, which doesn’t sound all that exotic. But to electrons moving into momentum space, the view is very different indeed. Here's the main reason why: The electrons can actually bump into each other once they enter into momentum space, just like water molecules filling up a container in ordinary space. All of that bumping and jostling for momentum space forces the electrons to spread out and take up more room there, again in a way that is strikingly similar to how water molecules pile up in ordinary XYZ space.
Quantum Splish-Splash
In fact, the process of electrons jostling around and spreading out in momentum space is so similar to the way water molecules fill a container that such collection of electrons in metals are called Fermi sea. (An aside: Enrico Fermi must have had a really good press agent working for him, given all the cool stuff that is named after him in physics.) This type of momentum-space liquid even has a well-defined surface, just like an ordinary liquid.
However, recall that from our perspective in ordinary XYZ space, the electrons piled up in momentum space just appear to be moving at different velocities. This equivalence means that electrons closer to the surface of the Fermi sea in momentum space must necessarily also be moving faster in ordinary XYZ space. In fact, for a good conductor such as silver the electrons at the surface of the Fermi sea end up moving very fast indeed. Since speed for a small object is the same thing we call heat, just how hot (how hot) do these electrons end up being?
We're Feeling Hot Hot Hot
Well, if the electrons at the top of the Fermi sea in a large piece of silver suddenly lost all of their energy, it would be emitted in the form of X-rays. The burst would be so energetic than anyone standing nearby would be killed. That's pretty hot! Fortunately for jewelry wearers, this flatly cannot happen because all of those electrons lower in the Fermi sea refuse to budge. They really like their much cooler locations in momentum space, and they are not about to give them up!
Mirror Mirror On the Wall
Now it's time to bring all of this back around to your question of whether you can "see" quantum effects on the scale of ordinary life.
The quantum magic begins whenever you look into an ordinary mirror. As soon as you do, you are already gazing into a sea of electrons that from a quantum mechanical perspective don't quite exist in ordinary space. They are "lost" in the XYZ space we know best, a space in which their accurate quantum representations are in some cases as large as the entire surface of the mirror.
And most of those lost electrons are also hidden! That’s because light that we see bouncing off a mirror comes from only a very tiny percentage of the Fermi sea electrons, specifically only the extremely hot ones at the very top of the Fermi sea. This is because they are the only electrons that have any "wiggle room" left to accept a photon and play catch with it.
What happens is this: An electron at the Fermi sea surface can accept a particle of light, a photon, and by doing so speed itself up just a little more. But unlike the electrons further down in the sea, when an electron at the surface speeds up it creates an "empty spot" in the Fermi sea. The process is closely akin to the ways a splash of water can rise up into air, but then realizes it no longer has any water below it to keep it supported. Unlike the water in the see, the splash above the surface is not stable: It has to fall back to the surface.
Very much like such a splash of water, an electron at the Fermi surface that has been “splashed up” by an incoming particle of light (photon) has no support underneath it to keep it there. So, it must fall back to the surface of the Fermi sea. As it does so, it gives up the photon energy that it held ever so briefly by re-emitting a nearly identical version of the photon it just absorbed. This re-emission of a photon from an electron at the Fermi surface is the smallest and most fundamental unit of reflection, the event from which larger-scale reflections are composed.
Simplicity from Complexity
Now the really neat thing about such re-emissions is that if your metal is smooth and consistent and polished on the surface, each such re-emission effect ends up being directed by the high symmetry of both the flat metal surface and of its smooth Fermi sea of electrons, causing the emitted photon (or more precisely, many photons interacting over the entire surface) to emerge in a very precise fashion that we call the angle of reflection. It's a case where a lot of complicated physics guided by even more complicated math ends up having a gorgeously simple outcome, and event we simply call reflection.
And most amazingly, that simplicity is deeply dependent on quantum effects that cross the entire mirror. It requires electrons that has collectively lost their way in ordinary space, and taken up refuge in a space that is not like the space we usually see, yet still allows them to bump into each other. They form a liquid in this peculiar momentum space, a sea that flips upside down our very understanding of what an "object" or "liquid" is and how it should behave. The tiniest sliver of these hidden electrons then wave back to us as they surf the surface of their hidden, showing off the incredible speeds they have reached by tossing photons back at us in a coordinate juggling act that we see as in the vivid brightness of a mirror, or the beauty of a sparkling ornament, or in a bit of bright silver or gold.
Finale: Take a Moment to Reflect
So, metallic reflection is a deeply quantum event, one that takes place on a human scale, and one that is uniquely beautiful and useful. If you find your universe a bit dull some mornings, take a moment to say hello to this lovely bit of quantum weirdness when you look into a morning mirror! And reflect a bit on your reflection to remind you how a remarkable universe we live in.
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Addendum 2015-06-20: Vision as Quantum Physics
I must add an example of large-scale quantum phenomena that is much closer to home than a mirror. It is the fact that you can see at all.
Lenses, including the ones in your eyes, are profoundly quantum devices. If it were not for quantum mechanical conversion of the large-scale shape of a lens into guidance for how microscopic particles of light (photons) travel, the lenses in your eyes would be as opaque as steel, and you would not be reading this text.
The problem is this: Since light is emitted and received as tiny particle-like units of energy, or photons, classical physics requires that these photons remain particles during their travels between those two points.
And that is a problem. After all, how does an electromagnetic photon travel through a lens full of electron-rich atoms that should bat it this way and that like a maze of mind-boggling complexity, let alone form an image? It might bounce around for a few moments in the outermost atomic layers of the lens, but it would have no chance of penetrating deeper before being lost or absorbed.
It is quantum mechanics that rescues us from the paradox of classical-photon blindness.
Mathematically, quantum mechanics allows a single photon to "explore" the entire shape of a lens through a process called the integration of all possible histories. This process makes no sense at all classically, since it is as if the photon had explored literally every possible path between its starting and ending points. Those virtual explorations are then added up in a special way to produce the wave function of the photon, which tells which bundle of paths is the most likely to contain the actual photon.
It is this infinite array of virtual photon paths that allows a single photon to "sniff out" the overall shape and form of a lens such as the ones in your eyes. Given the incredibly tiny amount of energy contained in a single photon in comparison to a huge, human-scale lens, that is a rather remarkable feat. It is roughly like taking a small penlight into orbit and "seeing" the shape of the entire earth by shining it onto the night side. Remarkably, every photon must do this on its own, since the result of shining every photon in a beam of light one-by-one through a lens is the same as what you get by shining them all at once.
The bottom line is this: Every single form of reflection, refraction, or transparency that you see using ordinary light is pretty much a miracle of quantum mechanics. None of those effects can exist without the photons "sniffing out" the large-scale shape of a mirror, lens, or window (which is really just a flat lens) in a way that allows them to ignore the incredible complexity of those objects, and focus instead on their overall shape and optical properties.
So how far do you need to go to see profoundly quantum effects in everyday life? Not far at all, for the very act of using your eyes to look for such effects is itself deeply quantum.
A: You also need the tunnel effect in order to explain many aspects of electrical conductivity. For example, why oxidized copper wires are still well conductors instead of insulators.
Another interesting quantum mechanical effect is produced in photosynthesis. The process is called "hopping" and it happens when a chlorophyll absorbs a photon and then it emits an exciton that will propagate until it reaches a special type of chlorophyll molecule, which produces an electron transfer. There are some references like: http://www.chemphys.lu.se/old/Archive/annual_96/primarynew.htm .
There is also the hypothesis that quantum entanglement is produced in certain birds to allow navegation. See: http://prl.aps.org/abstract/PRL/v106/i4/e040503 .
A: In some sense all chemical reactions are fundamentally quantum mechanical, but in the case of chemiluminescence and related atomic light-emitting phenomena like the aurora, quantum physics enters another way: excited states of the molecule or ion can only decay and emit a photon because the electron in that state is being continually shaken by vacuum fluctuations, microscopic fluctuations in electric fields due to quantum uncertainty.
A: The sun is visible to the naked eye. The only reason the sun shines is quantum-mechanical tunneling. Without tunneling, fusion reactions would be impossible at the temperature of the sun's core.
A: It's funny that you mention "the naked eye", because all you have to do is to close your eyes. As it turns out, the reason why we don't see anything when we close our eyes is quantum mechanics.
Sean Carroll explains it nicely: There is a lot of black body radiation in the infrared range inside your eyes. Even though the total energy of this infrared light is much higher than visible light that enters through our lenses, it isn't absorbed by the receptors, because according to quantum mechanics it can only be absorbed in quantized packages (photons). And each individual photon doesn't have enough energy to be absorbed.
A: Magnetism is a nice example, you can explain the spin-spin alignment only with quantum mechanics (see exchange interaction), it is even possible to prove the Bohr-Van leeuwen theorem, which states that no classical theory can explain how a magnet works.
Reference: Feynman's Lectures on Physics
A: You and your environment still exists! If it was not for quantum mechanics everything would spontaneously disintegrate, as atoms are not stable in Classical mechanics, due to radiation emitted by an accelerating electron.
A: Stuff it explains --


*

*Normal force

*Electric conduction

*Why some atoms are stable (motivating issue)

*Zeeman effect (motivating issue)

*Why the universe isn't just a continuous cloud of matter

*Why are masses discretised (the difference between the masses of two particles)

*Much of chemistry

