The essence of quantum tunneling is that the particle "appears" to jump-off the potential barrier wall with an energy lesser than the "potential" of the wall! However, in effect, it actually penetrates through the barrier.

Based on your question it can happen in either case, although the definition applies better to the second case "electron leaves its electronic cloud to become the part of other during interaction".

Quantum tunneling occurs if a particle has less energy than the potential well that the particle has to negotiate. 

According to classical physics, a particle with less kinetic energy than the potential of the potential well will not cross the potential well.

But according to quantum physics, there is a probability to crosse this barrier.

https://en.wikipedia.org/wiki/Quantum_tunnelling

In quantum physics, energy cannot be measured exactly due to Heisenberg's uncertainty relation. One can imagine quantum tunneling as follows:

A particle borrows energy from vacuum such that it has the energy to cross the barrier in a classical sense. But for don't violating energy conservation, this borrowed energy must be payed back. The result is that the barrier is crossed due to quantum uncertainty.

Now tunneling probability depends on the broadness of the potential barrier. The more broad the barrier is, the lower the tunneling probability. This is because most energy borrowing is payed back after a very very short time: The particle will not cross the barrier if it has gained energy only for a very very short time.

Tunneling is a purely quantum mechanical phenomenon as Remi has just mentioned above. It does not have a classical analog.

Consider the nuclear fusion process which fuels the sun and other stars. Essentially four protons fuse together to make a helium-4 and two protons. But because the protons are positively charged there is a repulsive coulomb barrier between them. As the protons come closer to each other, repulsion increases and the barrier height increases. So the question is how to overcome this barrier ?  Answer is quantum tunneling.

Core temperature of the sun is about 106 to 107 Kelvins which is lower than the Coulomb barrier which is about 109 Kelvins. Therefore in a classical model we cannot have fusion reaction in the Sun, and quantum mechanics thus becomes a very important theory.

Dear All,

the full story is more complex ! ...

Tunnel effects can be recognized in classical PDEs and in quantum PDEs. In fact a tunnel effect is a change of transverse sectional topology in a solution of PDEs or quantum PDEs. In the last case one talks of quantum tunnel effect. These phenomena are characterized by integral bordism groups for PDEs and quantum PDEs respectively. Of course to produce a fusion of four protons into two particles (... example by Biswajoy ...) it is necessary to work in a suitable quantum PDE. But it is also possible two obtain a fusion of four macroscopic particles into two only ones when their dynamics is described by a suitable commutative PDE.

My best regards

Agostino 

Dear Agostino,

Could you please provide a reference (articles or books) where one could find it in more detail?

Dear Vikash,

tunneling effects in commutative PDEs and quantum PDEs can be found in some my papers and works spread in many decades.

For a panorama about you can look to my research-homepage link:

https://docs.google.com/viewer?a=v&pid=sites&srcid=dW5pcm9tYTEuaXR8YWdvc3Rpbm9wcmFzdGFyby1lbmd8Z3g6MTdhODkwZmM1YTZiYTc1Yg.

In particular in the paper [29]

http://www.sciencedirect.com/science/article/pii/003448779190063S

I first introduced quantum tunnel effects in PDEs, and in the book [90]

https://books.google.it/books?hl=it&lr=&id=O5IbRdY7gqQC&oi=fnd&pg=PA307&dq=Geometry+of+PDE's+and+Mechanics+by+Prastaro+A.&ots=J5NLZFOWjH&sig=cBqqa4P-Td9UMqJQRebPlKCFFbY#v=onepage&q=Geometry%20of%20PDE's%20and%20Mechanics%20by%20Prastaro%20A.&f=false

I first introduced quantum tunnel effects in quantum PDEs too.

Good reading,

Agostino

Thanks Agostino. I always had the feeling that there is some underlying topology embedded in quantum phenomenon, but its not taught in main stream Physics courses. 

Right Vikash !

But it is also necessary to link algebraic topology with geometry of quantum PDEs. This new mathematics has allowed me to formulate a quantum gravity theory that reconciles Einstein's GR with QM.

My compliments and wishes .

Agostino

If each cloud is in its potential well with a potential barrier in between, quantum tunnelling is what has been explained. Solving the Schödinger equation, it is found that the wave function is not zero on the other side of the barrier. According to Born's probabilistic interpretation, that mean there is a tiny probability the electron passes the barrier.  That must be understood in the quantum mechanical context, that is, there is a preparation where the particle is put on one side, and a measurement on the other side.

@ Claude

Please clarify if the measurement made on the "other" side validates the presence of the particle on the respective side? 

And won't one will get the same measured value even if the measurement is made inside the potential wall? If the answer to this is "yes", then it does not matter which side the measurement has been made. The objective observation is that the particle has finite but low probability to escape the potential and so effectively it tunnels through (or, jumps-off) the barrier. The particle cannot be inside and outside the potential at the same time, I guess. Of-course it depends on the energy of the wall and also its width.