# Is it possible to explain the Quantum tunneling with the uncertainty principle?

I'm a high school student in South Korea(It is my first English question ever). I found descriptions of quantum tunneling with the uncertainty principle in Korea. There are two kinds of descriptions to explain quantum tunneling; position-momentum and time-energy uncertainty principle.

First, position-momentum uncertainty principle. When two protons collide, the uncertainty of momentum is decreasing, so the uncertainty of position is increasing. Therefore, it is possible to exist the probability of discovery in potential barrier <- is it the right description?

Second, time-energy uncertainty principle. Classically hydrogen needs more than 100 million degrees of temperature for nuclear fusion, but it isn't really the temperature inside the sun, is it? So, we can't go beyond the potential energy, but the time uncertainty is reduced when we look at the time of nuclear fusion, so energy uncertainty is increased and nuclear fusion is possible.

I want to know above all about the second description. These explanations are often found in Korea. But I couldn't find it when I looked it up in English. I wonder whether the explanation is correct.

Is it possible to explain quantum tunneling with the uncertainty principle?

Quantum tunneling should definitely be consistent with uncertainty principles, but I don't think tunneling is because of them.

In the typical example of tunneling where we can measure a particle to have crossed an energy barrier into a region that is classically inaccessible, the increase in the probability amplitude of finding the particle on the other side of the barrier can be explained using Schrodinger's equation. The tunneling doesn't need to be explained in terms of "lowering the momentum uncertainty", it is just how the system evolves due to its Hamiltonian.

Certainly if you were to make many position measurements and many momentum measurements of similarly prepared systems like this you would find that $$\Delta x\Delta p\geq\hbar/2$$, but I am not sure if that means this relation is what caused the tunneling. I suppose the most you could do is use the HUP to make an argument that if you know $$\Delta p$$, then you could make an argument as to how small $$\Delta x$$ could be. If this smallest value (given the mean position $$\langle x\rangle$$) still allows for the possibility of finding a particle in an classically forbidden region, then you could predict that tunneling is possible for the system. But just because you are using the consistency of the HUP with the rest of quantum mechanics doesn't necessarily mean the HUP caused the tunneling.

Additionally, a decrease in $$\Delta p$$ does not necessarily mean an increase in $$\Delta x$$. The only time you can say this for sure is if your state is already at the limit $$\Delta x\Delta p=\hbar/2$$. Then decreasing $$\Delta p$$ necessitates an increase in $$\Delta x$$ because the uncertainty principle must apply.

I would just explain quantum tunneling as an effect of quantum superposition. The probability of finding a particle somewhere can be expressed as a linear combination of position states. Quantum tunneling occurs because, according to Schrodinger's equation (at least non-relativistically) certain position states in the superposition corresponding to classically inaccessible positions will pick up non-zero probability amplitudes, and hence there is a probability of observing tunneling.

Of course, I may be completely missing some other way to look at QM here. Since a lot of the intuition of QM comes from the mathematical formalism, sometimes you can look at things differently and it still be ok. So, I hope I have at least provided an additional way to look at things here.

• If a square integrable function has compact support then its Fourier transform may not be zero except for isolated points. A more concrete theorem by Hardy states that if $|f(x)| \le A exp[-\alpha x^2]$ and its FT satisfies $|FT[f(x)]| = |\hat f (\xi)| \le B exp[-\beta \xi ^2]$ then $\alpha \beta \le \pi^2$, in other words both functions, $f(x), \hat f(\xi)$ cannot decrease arbitrarily fast. Apply this to tunneling: note that by spatially confining the particle its momentum must be milder in its decrease to zero. Apr 23 '20 at 16:49
• @hyportnex tunneling is not the same thing as spatially confining a particle though Apr 23 '20 at 17:22
• I did not mean to imply that, buggered my last sentence but cannot change it now. Instead all I was trying to point out was that the "sharper" the confinement in one variable is the slower the asymptotic decline of the pdf is in the transform variable. This is a sharper result than the HUP. If you confine the particle by a potential wall, restrict its total energy and thus its momentum, the broader and asymptotically slower to zero its spatial distribution becomes. Apr 23 '20 at 17:30
• @hyportnex Ah ok, I think I understand. You should post an answer with that information. I think you would explain it better than I could, and it seems like a good point to make. Apr 23 '20 at 17:31