Probability function for the position of a static electron under no potential

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TL;DR

Is it possible to define an approximated Gaussian probability distribution for a free static electron position under null potential?

Found plenty of math expressions pointing to the equivalence of the wave function as a Gaussian wave packet, but always presenting the Gaussian parameters (~width, peak) as undefined. Is it possible to have a defined Gaussian for the electron position in this case? Thank you

 

[pines-demon]

TL;DR Summary: Is it possible to define an approximated Gaussian probability distribution for a free static electron position under null potential?

Found plenty of math expressions pointing to the equivalence of the wave function as a Gaussian wave packet, but always presenting the Gaussian parameters (~width, peak) as undefined. Is it possible to have a defined Gaussian for the electron position in this case? Thank you

Find a random free electron and collapse its position. Let us assume, the device is somehow good at finding it within a certain uncertainty (so that the electron can be fairly described by a Gaussian wavepacket). The uncertainty in the Gaussian wavepacket comes from the accuracy of your detector, it is not a fundamental quantity that can be calculated or derived from postulates of quantum mechanics alone. Furthermore, the width of the Gaussian wavepacket increases with time.

 

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"The uncertainty in the Gaussian wavepacket comes from the accuracy of your detector, it is not a fundamental quantity that can be calculated or derived from postulates of quantum mechanics alone. "

Does not position uncertainty exist inherently for an electron, even in a static state? So, there is a wave function Ψ associated, that results in a probability distribution |Ψ|^2, that can be represented by a Gaussian representation, no? Thanks

 

[PeroK]

Does not position uncertainty exist inherently for an electron, even in a static state? So, there is a wave function Ψ associated, that results in a probability distribution |Ψ|^2, that can be represented by a Gaussian representation, no? Thanks

How do you know an electron is in a "static" state? You'd have to trap an electron and prepare it in some way. The initial wave function depends on that preparation procedure. The Gaussian is a nice mathematical function (with a variance parameter) that represents a particle with a spread of position that can be highly localised or spread out - and likewise with a spread of momentum.

I'm not sure to what extent you could prepare an electron in an exact Gaussian state. You could only estimate the wave-function from measurements on a large number of identically prepared systems. In general, a wave-packet can theoretically come in any shape. The Gaussian makes physical sense, as it's a probability distribution found extensively in nature.

 

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How do you know an electron is in a "static" state? You'd have to trap an electron and prepare it in some way. The initial wave function depends on that preparation procedure. The Gaussian is a nice mathematical function (with a variance parameter) that represents a particle with a spread of position that can be highly localised or spread out - and likewise with a spread of momentum.

I'm not sure to what extent you could prepare an electron in an exact Gaussian state. You could only estimate the wave-function from measurements on a large number of identically prepared systems. In general, a wave-packet can theoretically come in any shape. The Gaussian makes physical sense, as it's a probability distribution found extensively in nature.

Assuming the "static" state is pure theoretical, no trap procedure. The wave function can be only estimated by measurements? (can not be defined theoretically)

 

[PeroK]

Assuming the "static" state is pure theoretical, no trap procedure. The wave function can be only estimated by measurements? (can not be defined theoretically)

There is a key difference here between the mathematical model and experiment.

Mathematically, we can say: assume a free electron has an initial Gaussian wave-function with these parameters ... Then, we make predictions on the basis of that - perhaps to do with reflection and transmission probabilities at a potential barrier.

In terms of an experiment, we cannot put the electron physically into a Gaussian wave-packet. It's not like we can pour plastic into a Gaussian mould! Instead, we produce a free electron by some preparation procedure. You could search, for example, for "how does an electron gun work".

https://en.wikipedia.org/wiki/Electron_gun

We can do experiments with these electrons and if they are close enough to the theoretical/mathematical predictions, then we are getting somewhere. But, actually proving that our gun produces a Gaussian wave-packet may not be part of the experiment at all. The best you could do, perhaps, is to say that a Gaussian wave-packet provides a working model for the free electrons produced in our experiment.

 

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So, the electron particle position uncertainty math model is not something that is fixed. I can not say, just as an example, that every electron, with low velocity, potentially free, has a standard deviation position of 5nm (or so) ?

 

[pines-demon]

So, the electron particle position uncertainty math model is not something that is fixed. I can not say, just as an example, that every electron, with low velocity, potentially free, has a standard deviation position of 5nm (or so) ?

Nope. Even the Gaussian wavefunction spreads indefinitely with time. This is called dispersion, the Gaussian wavepacket is not a valid eigenfunction of a free electron so it will spread. A valid stationary solution is a plane wave, where the electron has a definite momentum that you can verify but its position is completely delocalized in space.

Note that you can prepare the particle in many other ways to get different wavefunction (measure angular momentum for example, or some complicated quantity). There is no unique fundamental wavefunction to a free electron.

 

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Would not be even possible to say that the electron (initial) position is inside of the plane wave wavelength? I guess that the Gaussian time dependency result is like a "random walk" for the electron.

 

[Nugatory]

Would not be even possible to say that the electron (initial) position is inside of the plane wave wavelength?

No, unless we’ve done something to make it that way. Where did we find this electron? Bound in an atom? Product of beta decay? Boiled off a hot filament in an electron gun? Released by a chemical reaction in a battery? Is it in vacuum or surrounded by matter?

The uncertainty in the “initial position” of an electron depends on how we’ve set things up (“state preparation” in the lingo), just as the “initial length” of a piece of string depends on how we cut it from the roll.