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I am wondering if there are any natural phenomenon in every-day life that cannot be explained by classical physics but can only be explained by quantum mechanics. By classical physics, I mean Newtonian mechanics and Maxwell's electromagnetic theory.
I know that there are macro-scale quantum phenomena such as superconductivity, but that isn't something that we can see in ordinary life.
In addition to the other answers, it is also true for ferromagnets, the strong magnetism is explained by the means of exchange interaction. With the machinery of QM one explains the hysteresis and formation of magnetic domains. The weak magnetism, paramagnetisim and diamagnetism can be explained on a classical level.
To add more, the energetical structure of levels in crystals and semiconductors is due to collective quantum phenomena.
I think Fluorescence could be an answer. You only see light beeing emitted at certain frecuencies, correlated to the quantized energy levels in atoms.
To someone familiar with quantum physics it becomes hard to think of something which is not quantum, because we need quantum physics to explain how atoms can be stable and why one solid material does not interpenetrate another when they are pushed against each other.
Let me expand a little on this.
Classical physics remains a wide-ranging and very important part of physics, but it does not provide the foundations. What I mean is, classical physics can say "if we have a body of such and such a mass, and so much charge, then here is how much it will push on another body, and here is the acceleration," and so on. But classical physics can't tell us what those bodies are made of. Look at the small parts of anything, and the concepts of well-defined energy and momenta described by ordinary numbers just doesn't work any more. You need the mathematics of operators and quantum amplitudes. So for me the main answer to the question "show me a quantum phenomenon" would be "how can ordinary stuff such as a table, a book, the floor, a rock, a cup, a spoon, be made of electrically charged things such as electrons and protons?" Classical physics says it would be impossible, because there is no way for a bunch of charged things to come to stable equilibrium according to classical physics. If the electrons were not moving initially then there will always be a direction to move such that they begin to accelerate. If they avoid hitting the protons by orbiting around them or by oscillating too and fro, then they will emit electromagnetic waves and rapidly (in a few nanoseconds) spiral in to the protons and the matter collapses. But this is not happening. Why not? Because quantum physics says electrons can spread out into wavy clouds and just settle around a proton without emitting energy.
Another nice example is the periodic table of the chemical elements. We see the elements laid out the way they are, because elements in each group (vertical line in the table) have similar chemical properties---they undergo similar chemical reactions. So why are there 8 main groups? And why are there 10 further groups that come in after the first three periods? And where do hydrogen and helium fit in? Quantum mechanics gives a very neat answer to all of these questions. You have states labelled by $n$, which takes integer values. Then the quantum physics of rotational energy says that for each $n$ you can have another integer $l$ taking values from $0$ to $n-1$. And the quantum physics of angular momentum says that for each $l$ you have $2l+1$ possible ways for the angular momentum direction to go. And for each state of motion of an electron there are 2 spin states. So now chemistry falls into place:
hydrogen and helium: n=1; l=0; 2 spin states so 2 possibilities all together.
Now the first period (horizontal line in the table): n = 2, l=0 or 1, 2 spin states, makes overall (1+3)*2 = 8 possibilities says quantum theory---and that is just what is observed.
Second period is similar,
Then in the third period we have an interlude where the $l=2$ states are involved. That's a further $5 * 2 = 10$ possibilities says quantum theory---which is just what is observed.
So when you look at a periodic table of the elements, you are looking at quantum physics "written" into the chemical properties of the ordinary things around you.
But I know you still want to see something "quantum" even more directly. Have you ever heard the crackle from a Geiger counter?
The way the question is phrased, specifically: what is not explained by Newton or Maxwell, there are 2 obvious candidates:
The stability of matter: atoms governed by Maxwell's equations and Newtonian mechanics decay immediately due to EM radiation.
The lack of an Ultraviolet Catastrophe: thermal radiation has infinite power, which we do not observe.
however, one would not generally list those as "quantum effects".
The whole "color temperature" notion and the finite speed of the radiative heat exchange.
A classical blackbody has an infinite power of electromagnetic radiation at any non-zero absolute temperature (see UV catastrophe). One needs a quantized light in order to understand the thermal radiation.
The whole "chemistry" thing is based on the fact that "atoms" (quantas of matter) do exist.
Atoms themselves and the substances as a whole have a finite volume because their electrons have quantized energy levels. Classical atoms will have decaying orbits of their electrons and these electrons will fall over their nuclei.
Shot noise - in any low-light photography, in sound processing and in a lot of other places. It wouldn't happen and the noise as a whole would have a different properties if it wasn't for the finite number of the signal carriers (electrons, photons).
Star twinkling...
Well, our world is quantum-based. I can add more and more.
Polarized light is a decent candidate for this, because it is relatively easy to produce and stays coherent over large distances when passing through air.
For a demonstration, you only need a laser or LCD monitor and three linearly polarizing filters. Let's say, the light emitted by the monitor is diagonally polarized. By filtering it horizontally, it will only be half as bright, because diagonally polarized light has a 50% horizontal component. The light passing through the filter will then be horizontally polarized.
If this light is then filtered by a vertical polarizer, no light will be visible, because all of it is filtered. This is expected by both classical and quantum mechanics, so no surprise yet. But then you introduce another filter in between the horizontal and vertical one. Classically this should not make a difference, because the wave has one polarization, that will at most be filtered more, so still no light will be visible.
But this is not what is observed. If the middle filter is set to a diagonal or antidiagonal position, light is transmitted through the whole setup. Quantum mechanics does explain this effect because the horizontal wave is projected onto the diagonal basis and gains a vertical component, that then is transmitted through the last filter.
According to me the photoelectric effect, is the simplest form of Quantum mechanics you will see in your daily life. Solar panels work by using the principle of photoelectric effect. But unfortunately you can only see its effect but not the photoelectrons leaving the metal surface. Most probably you wouldn't find any quantum effect that you can 'see' in your daily life. Another effect would be an element like Helium resembling a Harmonic Oscillator, which doesn't freeze even at very low temperatures which is not predicted by Classical mechanics. This is because the energy eigenstates can't be zero even in the ground state.
A good example of an every-day phenomenon that relies on quantum mechanics is the Sun !
The Sun and hence the sunlight we receive on Earth is powered ny nuclear fusion. In particular, it relies on the fusion of protons to produce helium nuclei. In order for fusion to proceed, protons need to approach each other at distances within the range of the strong nuclear force ($\sim 10^{-15}$ m), but the Coulomb repulsion barrier between protons at such distances is $\sim 1$ MeV.
Classically, in order for fusion to occur it would require a significant fraction of the protons to have kinetic energies that exceeded this barrier. Even if we allow for high energy protons with say $E \sim 10k_BT$ in the tail of the Maxwell-Boltzmann distribution, that would still require central temperatures $>10^9$ K at the core of the Sun; about 100 times higher than the Sun's actual core temperature.
Again classically, the virial theorem suggests that to get temperatures this high and be in equilibrium, the Sun would have to be 100 times smaller in radius - i.e. about the size of the Earth. Because of the much higher densities and pressures in the core, the fusion reactions would proceed at least $10^{6}$ times faster. The Sun would therefore be the size of the Earth and a million times more luminous. However, the Sun's lifetime would be measured in thousands of years and life on Earth could never exist.
Quantum mechanics to the rescue. The protons do not need to reach kinetic energies of 1 MeV in order to initiate fusion. Instead, protons with 100 times less energy are able to "tunnel" through the Coulomb barrier, enabling fusion to proceed at much lower temperatures and densities, and giving the Sun a lifetime of $\sim 10$ billion years.
I can sit on a chair and not fall through. Also, sapphires are blue.
