1st Law of thermodynamics (conservation of energy) and quantum tunneling Doesn't quantum tunneling contradict the 1st Law of thermodynamics? As far as I remember my school physics teacher told us that quantum tunneling is the reason why we can observe the alpha decay, as the energy gained from the mass deficit is not enough to overcome the potential barrier of the nucleus. How can one explain this paradox without giving up the 1st Law of thermodynamics?
 A: 
Doesn't quantum tunneling contradict the 1st Law of thermodynamics?

Thermodynamics is an emergent theory, based on underlying statistical mechanics. Conservation of energy is a general law in classical mechanics and is carried over into the formulation of quantum mechanics in a consistent manner.

As far as I remember my school physics teacher told us that quantum tunneling is the reason why we can observe the alpha decay, as the energy gained from the mass deficit is not enough to overcome the potential barrier of the nucleus. How can one explain this paradox without giving up the 1st Law of thermodynamics?

So it is not the first law of thermodynamics, because tunneling is a quantum mechanical phenomenon on individual quantum mechanical particles/entities and not a collective effect. The question correctly stated is saying: is the law of conservation of energy violated by tunneling?
Here is tunneling for particles described by a quantum mechanical wavefunction in a potential well.

This is the quantum mechanical solution of "particle in a potential well". If the potential well does not have infinite walls, the energy state solutions carry through and out of the walls in quantum mechanics. Energy conservation is not violated because the particle occupies the same energy level, inside and out side. For the probability of it to exist at any place, the wavefunction must be non-zero and open- that is what "probability" means: Out of N such particles in the same potential well ( N alpha+ rest of nucleus) n alpha may be found outside the potential well is what the solution tells us.
That is why quantum mechanics was developed, to explain phenomena that cannot be reconciled with classical mechanics.
(Please note that if this were a classical barrier with the same parameters, lets say a car  climbing a hill and coming down to the exact same level, the energy spent to go to the top would be regained when reaching the same level on the other side of the hill, so again no energy would be spent in total. It is not energy conservation that classically is paradoxical, but the probabilistic disappearance of a barrier.)
A: There is no violation of conservation of energy. 
Quantum mechanics simply allows for a non-zero probability of a particle with energy $E$ to be detected in or behind a potential wall with potential $V>E$. The particle doesn't magically gain "enough energy" to go through the barrier, it's just that our classical idea of the region being completely forbidden for particles with smaller energy is false in quantum mechanics - quantumly, the wavefunction inside such a classically forbidden region decays exponentially, but does not immediately vanish and stops decaying after entering a classically allowed region again, so you get a non-zero probability to detect the particle outside the potential well or even inside the barrier.
