Planck Length vs Size of objects that exhibit quantum behaviour

Question

This is one of those right brain questions where I would like to get an intuitive picture of something. Therefore it's going to be hard to express, and probably even harder to understand and answer. I do apologise in advance!

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I have always considered that the reason why tiny matter like electrons behave in the counter-intuitive ways of "the quantum world", is because of their proximity in size to the planck length. I can't explain why I thought this really, it's just been my intuition. Maybe:

E.g. the Heisenberg Uncertainty Principle way of looking at things implies that the reason why electrons behave this way is because the product of their momentum and position is closer to the planck constant compared to the large scale that we humans live on.

Ostensibly then, there is a connection between the planck length, and the counter-intuitive behaviour of electrons i.e. the planck constant.

I've known for a while that the difference between the size of the diameter of the electron is vastly larger than the planck length, but I never quite considered that that vastness is so extreme, that it brings the above intuitive understanding I've had into serious question, until today.

Is my intuition correct, if so how to resolve? If not, where am I going wrong exactly?

So, can someone help my right brain get an intuition on this? How is it possible that the electron is so vastly bigger than the planck length (it's way way closer to our size than the planck length!), yet exhibits such extreme quantum behaviour?

I am just trying to imagine, if the quantum effects are that extreme on the electron scale, what is there to say about all the orders of magnitude of scale between that and the planck length?

Answer 1

The quantum effects are not due exactly to proximity with the Planck scale, and they are not continuous in the sense that the world gets "more quantum" as you consider more intense scales. Your theory either is quantum or classical, there is no spectrum.

However, part of your intuition is correct. "Quantumness" is related to how the scales of the problem compare with $\hbar$. This has nothing to to with the Planck scale because $G$ and $c$ are nowhere to be seen, but $\hbar$ is an important quantity. Namely, Heisenberg's uncertainty principle states that $\Delta x \Delta p \geq \frac{\hbar}{2}$. This gives a notion of "graininess" to physical quantities. When the physical quantities in consideration are much larger than $\hbar$ (such as the products of typical lengths and momenta, or the typical values of angular momenta) this "graininess" is irrelevant, and a classical description will do just fine. As you get closers to the scales dictated by $\hbar$, quantum effects start to kick in and you've got to consider quantum mechanics in order to get a good description.

Answer 2

The idea that the electron is mach larger than the Planck scale is not a scientific notion. Science involves knowledge that we obtained with the aid of experimental observations in support of out theoretical understanding. As far as the size of the electron is concerned to my best knowledge we still consider it as a dimensionless particle, because all the high energy experiments involving electrons show a scale invariance far beyond the mass of the electron (Bjorken scaling), from which we conclude that it is a point particle. By contrast, when the energy in scattering experiments involving protons reach a scale corresponding to the proton mass, the proton is resolved into sub-particles.

Moreover, the notion of a Planck scale is also a non-scientific hypothesis because there is no experimental evidence (and there probably never will be) that such a scale actually exists. It is a theoretical idea that comes from the combination of fundamental constants that gives a very tiny mass/energy.

So any argument that the electron size is large compared to the Planck scale does not have any scientific support.