# Tunnelling of alpha particles - explanation

As it is now known because of Gamow, how alpha decay occurs. We have taught this theory in our class in last week. There we have been introduced with a strange phenomenon happening because of which alpha particles decay and the phenomenon was named as Quantum Tunnelling. In the theory proposed by Gamow it was assumed that alpha particles as a complete entity reside in the nucleus and after certain amount of collisions and time, the alpha particle will tunnel through the potential barrier. The professor who taught us was reasonably satisfied that there is no analog of this phenomenon which comes from classical physics.

I do believe on his statement but later on thinking more on it, I came across certain explanation of this phenomenon which I am about to discuss.

The prior assumption in my explanation is that, the nuclear potential barrier satisfies La Place’s equation.

If the nuclear potential satisfies La Place’s equation, a very curious property of the equation is that, the potential at a given point is actually coming from the average of potential surrounding that point.

Now, suppose after many number of collisions of alpha particles with the potential barrier, the alpha particles tunnels through the barrier. When it tunnels through, it seems as if from the point it has tunnel, the barrier is defined at that point. Now, since the barrier has to satisfy under any circumstances, the average property of potential which mentioned above leads to shrinking of the overall size of potential barrier of nucleus after alpha decay as just happened.

This can be understood by an analogy with a cotton ball, in which if you made a whole, and now you want to fill it with same number of materials used for that ball, you sacrifice the size of ball and fill it.

For nucleus, as per my understanding the main property which must be driving this shrinking of size of barrier is because the collective potential of nucleus assumed is satisfying La Place’s equation. Now if $$V$$ satisfies La Place’s equation at a point say $$x$$ and in neighbourhood of $$x$$ there are $$x_1$$ and $$x_2$$ then,

$$V(x) = \frac{V(x_1) + V(x_2)}{2}$$

Now because the V has become undefined at a point momentarily because of the Tunnelling effect, but still the overall potential satisfies La Place’s equation, it tries to fill this point of indeterminacy by shrinking the size of potential barrier.

Now, the claim made here that it reduces the size of potential barrier is actually trivial, as the nuclear potential energy of daughter nuclide is less than that of parent nuclide.

I would be thankful if someone validate my understanding and explanation of alpha decay.

## 1 Answer

I don't think that the nuclear potential generally satisfies the Laplace equation.

I also think that there may be some analog of quantum tunneling in classical physics: for example, high jumpers can (at least theoretically) bend in such a way that their center of mass will pass under the bar.

• If we assume finite well potential which is generally assumed for alpha decay explanation, La Place’s equation is satisfied.
– Aziz
Oct 15, 2021 at 6:43
• @Aziz: Finite well potential is not even differentiable everywhere, How could it satisfy the Laplace equation? Oct 15, 2021 at 7:21
• It satisfies la place’s equation on either side of barrier. Which is exactly what is required. Since when alpha particle leaves the barrier or enters, la places equation satisfies which then follows the arguments given in my question.
– Aziz
Oct 15, 2021 at 8:11
• the “overall potential” in question means that on either side of barrier, V satisfies la place’s equation but at barrier during Tunnelling it becomes indeterminate at one point.
– Aziz
Oct 15, 2021 at 8:15
• @Aziz : With all due respect, as of now, your reasoning contains obvious mistakes, and it is not my duty to improve your question. Oct 15, 2021 at 13:51