# What's Really so Quantum About Heisenberg's Uncertainty?

What's Really so "Quantum" About Heisenberg's Uncertainty?

I've never really understood what was so interesting and strange about Heisenberg's Uncertainty (or Robertson's Inequalities). If we take as axiom that particles exist as wave functions that satisfy schrodinger's equation then what is Heisenberg's Uncertainty other than a quip about the nature of a Fourier Transform? To make a Fourier Series of a wave that is very localized in position then one must have a broad range of k values with large coefficients in their series. Why is this so mind blowing?

A. Neumaier

I've never really understood what was so interesting and strange about Heisenberg's Uncertainty (or Robertson's Inequalities). If we take as axiom that particles exist as wave functions that satisfy schrodinger's equation then what is Heisenberg's Uncertainty other than a quip about the nature of a Fourier Transform? To make a Fourier Series of a wave that is very localized in position then one must have a broad range of k values with large coefficients in their series. Why is this so mind blowing?

Today it is trivial. In Heisenberg's time it was revolutionary.

I've never really understood what was so interesting and strange about Heisenberg's Uncertainty (or Robertson's Inequalities). If we take as axiom that particles exist as wave functions that satisfy schrodinger's equation then what is Heisenberg's Uncertainty other than a quip about the nature of a Fourier Transform? To make a Fourier Series of a wave that is very localized in position then one must have a broad range of k values with large coefficients in their series. Why is this so mind blowing?
If particles were wavefunctions, it would all be very simple. But they aren't, so why a particle's position is uncertain when its momentum is certain? If an electron's position uncertainty is 1000km, it means the electron is "dispersed" in 1000km? As you see, it's not so simple.

A. Neumaier

If particles were wavefunctions, it would all be very simple. But they aren't, so why a particle's position is uncertain when its momentum is certain? If an electron's position uncertainty is 1000km, it means the electron is "dispersed" in 1000km? As you see, it's not so simple.

Whatever electrons ''are'', it is completely determined by their state (wave function or density matrix). In an often quite meaningful sense, electron's ''are'' the charge and matter distribution determined by their state. For example, this is what atom microscopes ''see'' when they look at matter, what chemist compute when they do quantum molecular computations to predict a molecule's properties, and what other matter responds to in the (often good) mean field approximation.

Whatever electrons ''are'', it is completely determined by their state (wave function or density matrix). In an often quite meaningful sense, electron's ''are'' the charge and matter distribution determined by their state. For example, this is what atom microscopes ''see'' when they look at matter, what chemist compute when they do quantum molecular computations to predict a molecule's properties, and what other matter responds to in the (often good) mean field approximation.
But would you honestly say that the electron of my previous post is a 1000km long electronic cloud?

A. Neumaier

But would you honestly say that the electron of my previous post is a 1000km long electronic cloud?

If your assumptions were actually prepared in reality, yes. But in practice, one cannot prepare electrons in plane waves, to an accuracy that your setting would make sense.

Like for photons, the shape of a single electron can have the shape of (the squared modulus of) an arbitrary solution of the Dirac equation in which only positive energies occur.

If an electron is prepared in a device, its positional uncertainty is no bigger than the size of the relevant part of the preparing device. Of course, later manipulations can delocalize the electron, but I cannot imagine machinery for turning it into a state as described by you.

Electrons in a typical electron beam are localized quite well orthogonal to the beam direction, and are delocalized to some extent in the direction of the beam, corresponding to the uncertainty in the time when the electron was produced. In any case, this is very far from a plane wave, which is uniformly distributed over 3-space. (Plane waves are primarily used in introductory quantum mechanics, mainly for didactical reasons.)

Now suppose your electron beam has a very high speed so that the uncertainty in the time where the electron is produced translates into a spatial uncertainty of 1000km, and suppose also that the electron can move 1000km without significant external interactions. Then it is easy to imagine how its position is uniformly delocalized along the beam and across the 1000km.

This results in a very long and thin cloud - but we call that a (very low intensity) beam, not a cloud.

Thanks, Arnold.