Quantum Mechanics: Particles & Position Explained

[Runner 1]

I just started taking a Quantum Mechanics course at my university (only 3 days into it) and some of the topics got me thinking about stuff. Namely, about "particles". I searched on Physics Forums and found some other posts pertaining to my question, but a lot of the responses referenced topics I haven't learned yet or seemed like intentional obfuscation. So I'm looking for simple and easily understandable explanations.

Here's the question:Imagine an empty universe, with only space and time, but nothing within it.

Say an electron appears within then universe, and it has a velocity. Define a time t such that t = 0 upon the electron's appearance. Now for each location in the universe, that location will either be affected* by the electron's existence or it will not (i.e. some location trillion's of light years away from the electron will not be immediately affected). So for any location, and for a given t, there are two states possible: either it is affected by the electron, or it is not.

Now let's create some 3D plots with axes x, y, and z, and title the plots according to the value of t. For any x, y, or z that is affected by the electron, we make that point black. (We would have, for example, an empty plot for t < 0.)

If I were to look at these plots for different values of t, what would they look like? If I looked at the plot for t=0, would there be a single, zero-dimensional black dot? For t > 0, I assume you would have a continuous 3D region of the plot that is black, completely predicted as a function of t and the electron's velocity. Is this correct?

I know this may seem like a weird question, but the whole particle-wave nature of the electron confuses me. I hate that in most physics courses, macroscopic analogies are given to topics for which the analogies do not translate well. It forces my mind to try and imagine a small marble or an ocean wave, which I think is counterproductive to understanding how these things work.

It's easier for me to think "Here are some mathematical functions. These functions describe how things that exist affect other things that exist, and experiment confirms that they predict reality well". The problem is -- I never really understand to what extent or scale these functions have an effect -- how a particle really affects other particles within space and time. And for me, the easiest way to see that is with a plot.

One thing I wonder about the plot is if the "blob" of the electron's effect is solid black and has a well-defined edge. Or are parts of the blob shades of gray (superpositions?) And is it really solid gray, or is it gray like a computer monitor with alternating white and black pixels that makes the whole screen look gray when standing back from it?

*By "affected", I mean if there was a test particle at x, y, and z, would it behave differently than if no electron were present at all in the universe? If it would behave differently, then it is "affected". If it wouldn't, then it is "not affected".

 

[Bill_K]

Runner 1, These are good questions. The short answer is that introductory quantum mechanics does not know about relativity or the speed of light. It assumes that influences can propagate infinitely fast, and that an electron on Earth will have a small but immediate effect even on Alpha Centauri. In your course you'll be dealing with a wavefunction ψ(x,t) that is defined everywhere, literally everywhere, and you'll integrate things out to infinity.

To be more specific about your "blob" question, the wavefunction for an electron placed initially at the origin is a Gaussian that spreads with time. And Gaussians go out to infinity! If you want to see some math, it's called the Green's function, and can be found for example in Eq 6 of http://cdsweb.cern.ch/record/514621/files/0108083.pdf.

In relativistic quantum mechanics, of course, the region of influence spreads outward at the speed of light.

 

[jtbell]

for each location in the universe, that location will either be affected* by the electron's existence or it will not (i.e. some location trillion's of light years away from the electron will not be immediately affected). So for any location, and for a given t, there are two states possible: either it is affected by the electron, or it is not.

Now let's create some 3D plots with axes x, y, and z, and title the plots according to the value of t. For any x, y, or z that is affected by the electron, we make that point black. (We would have, for example, an empty plot for t < 0.)

QM gives us only probabilities that the electron "affects" a particular location, not a binary yes/no answer. A better picture uses shades of gray, with white (or blank) representing probability 0 and black representing probability 1. Your electron is a fuzzy gray ball that is darkest at the center and "fades out" towards white (but never quite reaching white) in all directions. As time passes, the center of the ball moves with speed v, the diameter of the ball increases gradually, and it becomes lighter and lighter gray overall, as the probabilities "spread out" over a larger volume.

 

[Runner 1]

QM gives us only probabilities that the electron "affects" a particular location, not a binary yes/no answer. A better picture uses shades of gray, with white (or blank) representing probability 0 and black representing probability 1. Your electron is a fuzzy gray ball that is darkest at the center and "fades out" towards white (but never quite reaching white) in all directions. As time passes, the center of the ball moves with speed v, the diameter of the ball increases gradually, and it becomes lighter and lighter gray overall, as the probabilities "spread out" over a larger volume.

Interesting. You say that "QM gives us only probabilities that the electron 'affects' a particular location, not a binary yes/no answer". Does this mean that there exists a binary yes/no for a given (x,y,z) but that information is unknowable to us, or does it mean that the lack of a binary yes/no is a fundamental property of the universe? Or is that question itself unsolvable?

 

[Runner 1]

Runner 1, These are good questions. The short answer is that introductory quantum mechanics does not know about relativity or the speed of light. It assumes that influences can propagate infinitely fast, and that an electron on Earth will have a small but immediate effect even on Alpha Centauri. In your course you'll be dealing with a wavefunction ψ(x,t) that is defined everywhere, literally everywhere, and you'll integrate things out to infinity.

To be more specific about your "blob" question, the wavefunction for an electron placed initially at the origin is a Gaussian that spreads with time. And Gaussians go out to infinity! If you want to see some math, it's called the Green's function, and can be found for example in Eq 6 of http://cdsweb.cern.ch/record/514621/files/0108083.pdf.

In relativistic quantum mechanics, of course, the region of influence spreads outward at the speed of light.

This is also very interesting. So I'm trying to understand what you're saying:

In particular, when the electron is created at t=0, it will have a quantum wave function whose domain is all of space. However, general relativity limits how fast the effects of the electron's existence spread from its origin. So the two theories are in conflict then? (If QM says there is a non-zero probability of the electron affecting a point two light years away, and general relativity says there is no way for that electron to affect that point until two years have passed, then the two theories appear to be in conflict -- or maybe I'm misunderstanding).

 

[jtbell]

Does this mean that there exists a binary yes/no for a given (x,y,z) but that information is unknowable to us, or does it mean that the lack of a binary yes/no is a fundamental property of the universe? Or is that question itself unsolvable?

There is no generally accepted answer to these questions. This is the subject of the various interpretations of QM, which all give the same (probabilistic) predictions for the results of experiments, and about which people argue a lot, sometimes with a passion normally reserved for religious debates.

 

[ibysaiyan]

This is also very interesting. So I'm trying to understand what you're saying:

In particular, when the electron is created at t=0, it will have a quantum wave function whose domain is all of space. However, general relativity limits how fast the effects of the electron's existence spread from its origin. So the two theories are in conflict then? (If QM says there is a non-zero probability of the electron affecting a point two light years away, and general relativity says there is no way for that electron to affect that point until two years have passed, then the two theories appear to be in conflict -- or maybe I'm misunderstanding).

Yes, EPR paradox is what you're pointing at.
There was an article I read sometime ago which introduced double split experiment and went into short detail about Schrodinger's cat in a box experiment,CI.Now I'm still in the process of understanding Q.M as such my knowledge may not be up to date but take the example of a box:

Say we place an electron inside it and then half the box,common sense guide us that the probability of finding electron is 50-50 which is to say it could be in either side of the partition however according to CI, without any observation made yet the probability distribution of the box is same in both sides.Once we have made an observation on side 'X' and an electron appears.. the probability distribution disappears on the adjoining side , so once we have closed the box again to stop observing,the distribution returns back to the 'X' side only.

P.S: What I have described above is C.I now I am not aware of the validity of this with our current knowledge.
-ibysaiyan