High School The electron field as values of what quantity?

[Green dwarf]

My very amateur knowledge of the nature of matter is that particles like electrons are local disturbances in a universe-wide field, like the electron field and that a field is a set of values for some quantity, one for each position in space.
What I'm wondering is: What is that quantity in the case of the electron field? Is it nothing more than electron-ness?
Is the quantity a scalar? Is its value everywhere greater than zero or can it be negative or imaginary?
On a 3-d plot with 'electron-ness' on the vertical axis and position in space on the horizontal axes, I imagine an electron as being an isolated round hill or spike. Or is it a wave shape.
If it's a hill, in what sense is the electron a wave?
If it's a wave shape, are the waves moving when the electron is standing still?

 

[houlahound]

Depends on the environment the electron finds itself in. Away from all external fields it is a Gaussian that spreads out in space and time.

If it is subject to measurement the Gaussian becomes sharper to the extent of a delta function.

At least that's what I get from the theory.

I suggest you look up intro level classical waves math as you appear to have missed some basics on wave behaviour and terminology.

 

[Green dwarf]

Thanks houlahound.
Just to give me an idea, what sort of standard deviation would the Gaussian have when the electron is a long way from any other electrons in inter-galactic space? Would it be nanometres, metres, light years or ... ?
Does it even have a defined location when it is not interacting with other particles?
As you might have guessed, I'm not a physicist, just a curious high school teacher.

 

[houlahound]

Linear spread in time, read following link, shows about 1km spread in milliseconds. I did not check the calculation;

https://en.m.wikipedia.org/wiki/Wave_packet

"..example, if an electron wave packet is initially localized in a region of atomic dimensions (i.e., 10−10 m) then the width of the packet doubles in about 10−16 s. Clearly, particle wave packets spread out very rapidly indeed (in free space):[6] For instance, after 1 ms, the width will have grown to about a kilometer."

 

[A. Neumaier]

the electron field and that a field is a set of values for some quantity, one for each position in space.
What I'm wondering is: What is that quantity in the case of the electron field?

The electron field is a spinor operator field. But at the level with which your question is marked, the best description of the field values is in terms of the electron charge density (a scalar field) and an electron current (a 3-dimensional vector field). These are measurable. Expressed in terms of the underlying spinor field, the latter are given as the expectation values of quadratic expressions in the operator field.

 

[Nugatory]

Depends on the environment the electron finds itself in. Away from all external fields it is a Gaussian that spreads out in space and time.

If it is subject to measurement the Gaussian becomes sharper to the extent of a delta function.

At least that's what I get from the theory.

One caution here: This is describing the non-relativistic quantum theory of a particle, the stuff that you'll learn in your intro QM classes as an undergraduate. The question in this thread is about relativistic quantum field theory in which the particles are treated as excitations of a quantum field, and that's something quite different.

 

[Green dwarf]

Away from all external fields it is a Gaussian that spreads out in space and time.

Thinking about the electron as a Gaussian, it makes it hard to understand how the interference patterns observed in two-slit experiments arise. My understanding is that these arise because the wave that is the electron consists of a regular sequence of crests and troughs moving through space. I wouldn't expect a single bump to produce the same interference.

Also, the figures you present suggest that the electron spreads out at about 1000 km/s. Does a photon "spread out" in the same way, but at the speed of light?

 

[houlahound]

Why wouldn't you expect a single bump to interfere on passing thru a slit?

What shape do you think a laser beam (intensity profile) presents to a slit?

The shape of a wave is quiet arbitrary, see Fourier's theorem, but a wave is a wave is a wave. The intensity profile does not change wave behaviour...and neither does QM.

ETA what go you mean that a wave spreads at X km/s, I think you are confusing concepts?? Can you define what your quoted speed refers to in relation to a wave.

 

[Nugatory]

Thinking about the electron as a Gaussian, it makes it hard to understand how the interference patterns observed in two-slit experiments arise. My understanding is that these arise because the wave that is the electron consists of a regular sequence of crests and troughs moving through space. I wouldn't expect a single bump to produce the same interference.

A Gaussian is the sum of sinusoidal waves (the regular sequences of peaks and troughs that you're thinking about) of various frequencies and amplitudes. The wave equation is linear, meaning that the behavior of a sum of waves is the sum of the behavior of the individual waves, so if the waves making up the Gaussian packet interfere, you can get interference. You can visualize this with water waves: if you push sharply against the water you can send a single wavecrest out across the surface and if this encounters a wall with two openings it will pass through both.

Also, the figures you present suggest that the electron spreads out at about 1000 km/s. Does a photon "spread out" in the same way, but at the speed of light?

You have to be careful here. This discussion of electron behavior is based on non-relativistic quantum mechanics so only works when the speeds involved are small compared with the speed of light and the energies are low enough that new particles aren't being created. Neither condition applies to photons so you need quantum electrodynamics instead of ordinary QM.