Tunneling in classical mechanics

Question

According to a classical mechanics a particle that do not have enough energy will not be able to surmount a potential barrier.

But what if we have a lot of particles(e.g. electrons)? They can interact with each other(Coulomb's law) and transfer enough energy for some of them to surmount a barrier.

Does it explain a tunneling phenomenon without quantum theory?

No, because by adding additional interactions you are, of course, changing the entire system, including the 'barrier' potential. It is like saying that inert objects fall when I drop them, but what if I put a rocket motor on it - then it will rise! — Jon Custer, Feb 17 '16 at 18:55
@JonCuster I interpret the question like this: if the particle in question can be broken up into many, and with an unequal energy distribution between them, could some sub-particals cross the wall using only classical theory? (And then magically call the other sub-particals over to join.) — Gyro Gearloose, Feb 17 '16 at 19:01
@GyroGearloose - you mean if you had, say, a thermal distribution could some get over a finite potential? In that case, of course, there will be some tail to the thermal distribution with enough energy. But, if charged, then one particle leaving the well will change the potentials in question, altering the problem. But in no way could it be considered 'tunneling'. Personally, I interpret it as an attempt to get around having to deal with quantum mechanics, just like all the attempts to fiddle with single/double slit diffraction. — Jon Custer, Feb 17 '16 at 19:07
@JonCuster "Personally, I interpret it as an attempt to get around having to deal with quantum mechanics" me2, but that's OK. The other, more problematic part is why all the other sub-particles should magically cross the barrier, and of what I understand, this makes the difference between classical physics + thermodynamics as opposed to quantum mechanics. I'm just not firm enough to give a solid answer on this. — Gyro Gearloose, Feb 17 '16 at 19:14
@GyroGearloose - I don't think there is a solid answer - there is some source of confusion on the part of the poster, but I can't decide exactly what it is or I would attempt to straighten it out. Hence a downvote. — Jon Custer, Feb 17 '16 at 19:17
@GyroGearloose : in ref to your comment, I understand instead : if there are many particles, and with an unequal energy distribution between them, could some particles cross the wall using only classical theory? — , Feb 17 '16 at 21:05
@igael my understanding is different, but what is really meant, only the OP can tell (but even only if to distinguish this difference was not part of the original problem). Let's wait. — Gyro Gearloose, Feb 17 '16 at 21:15

score 2 · Accepted Answer · answered Feb 18 '16 at 05:26

This will help:

Please note that the energy is the same , inside and outside the barrier. Also it works on one particle. Your "model" would give a statisticacl gaussian distribution for the probability of finding the tunneling particle whereas alpha decays of a nucleus, for example, have an exponential fall off distribution.

No, tunneling is a quantum mechanical phenomenon.

score 1 · Answer 2 · answered Feb 18 '16 at 05:14

Well as all the comments they have given,the answer should be no.

Tunneling in quantum mechanics state that even you are given only one electron now, it still has some probability to surpass the potential and finally detected in which the classical physics can ever and never gives an explanation.

And in the case you take as an example, you should first notice that since the particles are electron,they are so small that you can't ignore the wave characteristics,thus the particle shouldn't be deal with classical physics in anyway. And then if they interact just like what you say, they have "different" energy now even if you maintain the potential barrier! So this is a totally different case from the former one and you can't say this is a quantum tunneling if you have now a system of larger energy, so large that it might surpass the potential barrier and even in the classical case they will go through the barrier.

And just like Jon said: you might even change the barrier since at reality you are changing the total system.

So tunneling can only be explained in quantum mechanics, and has no similar case in classical view

score 1 · Answer 3 · answered Feb 18 '16 at 07:34

You are quite correct that interactions between electrons can cause them to jump a potential barrier. For example thermionic emission is a good example of this. However this is a quite separate process from quantum tunneling.

Consider some collection of electrons - we'll use the electrons in the conduction band of a metal as an example. To get an electron out of the metal requires an energy called the work function, and this energy is typically a few electron volts. The thermal energy available to the electrons is $kT$, but at room temperature $kT$ is only about 0.025 eV. That suggests we would require 40 times room temperature or about 12000K to get thermionic emission, but actually we get emission at only a few hundred degrees C.

The reason for this is that $kT$ is the average energy of the electrons and some electrons have energies lower than this while some have energies higher than this. Electrons scatter off each other in a basically random way and as a result some can get very high energies.

The probability that an electron will have an energy $E$ is given by the Boltzmann distibution:

$$ P \propto e-{E/kT} $$

For energies a lot higher than $kT$ the probability is very small, but then there are a lot of electrons in a metal. The result is that even at relatively low temperatures some electrons will escape.

So interactions between electrons can indeed allow a few of them to jump a potential barrier. However this happens because those few electrons have an energy higher than the barrier. Quantum tunneling is a completely different process and allows an electron to jump a barrier even when it doesn't have enough energy to jump the barrier.

Tunneling in classical mechanics

3 Answers3