# Is an electron a delocalized excitation before measurement?

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1. Nov 6, 2015

### ajv

When we observe an electron it is always a localized excitation in the electron field. But when it's not being observed, does the excitation begin to spread through space and become a delocalized excitation?

2. Nov 6, 2015

### Mentz114

My understanding is that the electron field carries the potential for excitations to exist when interactions happen. So one cannot ask about excitations that are not being observed ( viz. interacting).

But I could be wrong and I'm interested to hear an authoritative reply.

Last edited: Nov 7, 2015
3. Nov 7, 2015

### vanhees71

Why should it always be localized? If you don't measure its position by, e.g., puting a detector in its way, it is in general not well localized. For such a measurement there is, given the state of the electron, a probability distribution for its position, not more not less. An electron is the better localized the narrower the position probability distribution is. That's it. According to quantum theory you can't tell more about the electron's position, and it can never be exactly localized at one point, because the position operator has a continuous spectrum.

4. Nov 7, 2015

### ajv

So sum up what you are saying, the excitation is extended (delocalized) prior to measurement rather than being localized during measurement, right?

5. Nov 7, 2015

### zhanhai

I don't quite understand "When we observe an electron it is always a localized excitation in the electron field".

First, it would depend on what is measured. If you are measuring the position of the electron then the statement could be right. If on the other hand you are measuring the energy/momentum of an electron in a crystal, that electron should possibly still be non-localized after the measurement.

Second, it may basically depend on the interpretation of the wave function; is it a characterization of matter distribution of electron? In some mainstream interpretation, it seems that the electron is localized even before it is measured, and what is non-localized is its probability of distribution of measured position; so the statement "When we observe an electron it is always a localized excitation in the electron field" would be wrong by itself.

6. Nov 7, 2015

### Staff: Mentor

My understanding of QFT isn't as good as I would like it.

That said, when speaking of the electron field you are speaking of QFT where even the number of particles is not fixed when not observed. I think the situation is more complex than simple pictures like not localised suggests.

Thanks
Bill

7. Nov 7, 2015

### ajv

Ok so if an electron is not being measured in any way shape or form, is it an extended object/ excitation?

8. Nov 7, 2015

### Staff: Mentor

Its a quantum field which IMHO is not amenable to visualisation or a description in words - its only the math that's adequate.

The only deception I can point at is the harmonic oscillator from ordinary QM:
https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator

If you do a Fourier transform of a quantum field it mathematically is the same as the 'superposition' of a lot of harmonic oscillators and you get creation and annihilation operators as well as the number operator. This is all suggestive of particles. In fact it can be shown to be just an alternative formulation of standard QM:
http://susanka.org/HSforQM/[Styer]_Nine_Formulations_of_Quantum_Mechanics.pdf

Thanks
Bill

9. Nov 8, 2015

### A. Neumaier

Electrons (whether measured or not) are almost always extended objects. For the notion of a localized electron is a semiclassical concept that, strictly speaking, makes sense only at sizes bigger than its Compton wavelength. At smaller distances, the intuitive imagination associated with the particle concept breaks down completely, and the only right description is in terms of quantum field theory.

The theory of electrons is called QED (quantum electrodynamics). In QED, there is an electron field and an electromagnetic field. The electron field describes the properties associated semiclassically with electrons and positrons. But the correspondence is not simple. For example, there are no operators associated with single electrons. In particular, there is no position operator in QED. Thus one cannot meaningfully say that ''this electron moves from here to there', not even on the level one is used for a quantum mechanical particle (i.e., with a probability distribution on histories).

But there are operators for the charge density - so this is a concept that makes sense at any length scale. Therefore one can consistently talk about the charge density of the electron field, and indeed it is this charge density that is displayed by atomic microscopes. It is also the key quantity relevant in quantum chemistry, where the electron field is responsible for the binding of atoms.

Since the charge is quantized, there is also a notion of particle number (actually electron number minus positron number). But this is a global quantity, obtained by integrating the charge density over all of space at any particular time. (Which time does not matter as charge is conserved). It is this number that bridges the gap to the electron picture. For in case that a small volume contains an elementary charge, i.e., an electron field corresponding to the charge of a single electrons, and this small volume is reasonably closed in the sense that its interaction with the rest of the universe can be well approximated by an external potential then one can describe the system approximately by single-particle quantum mechanics (i.e., the Dirac or Schroedinger equation). This may apply to single electrons emitted by a source (cathode rays), to the outermost electron in an atom, or to a valence electron of an atom in an ion trap. These are the only cases where one may consider an electron to be (approximately) well localized.

Already the case of two elementary charges is much more complicated. It gives rise to delocalized 2-electron superpositions making up simple bonds between atoms. The only natural intuitive description is then that of the charge density, i.e., an electron cloud. Multiple bonds involve even more charges, and again the charge density and the electron cloud are then natural ways to think about these on an intuitive level.

Last edited: Nov 8, 2015
10. Nov 8, 2015

### vanhees71

"Electrons (whether measured or not) are almost always extended." I'd put this slightly differently: An electron's probability distribution, whether measured or not, is almost always broad. Of course, it's never described by a $\delta$ distribution in position space, because that's not a square integrable function.