# Probability of quantum tunnelling

## Main Question or Discussion Point

I'm continuing to read Jim Al-Khalili's book Quantum: A Guide for the Perplexed. I'm on page 190; today I just read a few paragraphs and page 176 about quantum tunnelling.

My question is this: what does it take to say that an electron, for example, has successfully and definitely tunnelled through a barrier? Or is there only the probability function of having done so?

Jim Al-Khalili explained quantum tunnelling in terms of the wavefunction. Does the wave function need to be collapsed by way of observation (a la Copenhagen interpretation) for the electron to definitely exist on the other side of the barrier?

I was thinking that if quantum tunnelling only occurs in terms of probability, then what happens after tunnelling becomes likely? Must we keep making our calculations using probability of quantum tunnelling to determine what the electron does on the other side of the barrier to interact with whatever is there? Is this to be added to and added to successively with each further interaction that takes place?

I would think perhaps not: One of the consequences of quantum tunnelling is decay of a nucleus, such as what happens in a fission bomb. Does the decay actually take place for sure, leading to the chain reaction that then detonates the fission bomb, or is it simply likely to occur? If so, then how does the fission bomb definitely detonate?

I bring this up because I know the fission bomb does really detonate -- for sure. (Of course, sometimes fission bombs are duds, which I think might have to do with the probability of nuclear decay.)

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ZapperZ
Staff Emeritus
2018 Award
My question is this: what does it take to say that an electron, for example, has successfully and definitely tunnelled through a barrier? Or is there only the probability function of having done so?

Jim Al-Khalili explained quantum tunnelling in terms of the wavefunction. Does the wave function need to be collapsed by way of observation (a la Copenhagen interpretation) for the electron to definitely exist on the other side of the barrier?

I was thinking that if quantum tunnelling only occurs in terms of probability, then what happens after tunnelling becomes likely? Must we keep making our calculations using probability of quantum tunnelling to determine what the electron does on the other side of the barrier to interact with whatever is there? Is this to be added to and added to successively with each further interaction that takes place?
These are quite confusing, and I have worked in tunneling!

Tunneling is described by (i) the tunneling matrix element that describes the probability of going from one state from one side of the barrier to the state on the other side of the barrier and (ii) the density of states of the initial state and the final state. Both of these factors govern the total probability of tunneling.

Note that tunneling isn't that exotic, and occurs more often than the examples you cited for nuclear processes. Tunneling is a diagnostic technique to study solids such as superconductors (I've highlighted an article on the progress in tunneling spectroscopy in superconductors in the Solid state forum). It is also use for imaging as in STM.

Zz.

as a simple example let's take in the place of the electron a bullet and in case of the barrier there is abox of wood... the bullet may just hit and fall on the box ...or it may hit and get into the box or hit the box and get through and be out ...there is probabilities for eeach case and the same with the electron... and because of the wave duality the barrier effect on its wave propereties that's why the wave decay when it tunnels till infinity...am i right?

ZapperZ
Staff Emeritus
2018 Award
as a simple example let's take in the place of the electron a bullet and in case of the barrier there is abox of wood... the bullet may just hit and fall on the box ...or it may hit and get into the box or hit the box and get through and be out ...there is probabilities for eeach case and the same with the electron... and because of the wave duality the barrier effect on its wave propereties that's why the wave decay when it tunnels till infinity...am i right?
Try something even simpler. Use the common example done in undergrad QM text - a free particle encountering a square potential barrier. Now, use that and ask your question.

Zz.

I used the example of an electron and a fission bomb

Therefore, my question is firmly placed in the realm of QM.

o| Hiram