Epistemic status: Relating how this was explained to me in the hopes that somebody will either say “That’s right!” or “No, you’re still wrong, let me correct you!”
The way this was explained to me is that this is one of those things that is deceptively simple but always explained very poorly.
Knowing the position of the particle/excitation means reducing the width of \deltaX, which means summing more plane waves. Summing more plane waves means having less precision in the frequency/energy/momentum domain. Conversely, having less positional certainty (wider \deltaX) means you require fewer plane waves to describe the excitation, meaning you know the frequency decomposition (and therefor the energy/momentum description) very accurately, in a sense because the position is spread out.
The confusion enters because educators insist on talking about “knowing the position of a particle” when a particle literally is a wavelike excitation of a field and does not have a position in the sense that you think of a bowling ball having a position.
That sounds right to me, and I agree that this is sometimes explained badly.
Are you saying that this explains the perceived asymmetry between position and momentum? I don’t see how that’s the case, you could say exactly the same thing in the dual perspective (to get a precise momentum, you need to “sum up” lots of different position eigenstates).
If you were making a different point that went over my head, could you elaborate?
I doubt that I understand this very well. I thought there was a chance I might help and also a chance that I would be so obviously wrong that I would learn something.
Epistemic status: Relating how this was explained to me in the hopes that somebody will either say “That’s right!” or “No, you’re still wrong, let me correct you!”
The way this was explained to me is that this is one of those things that is deceptively simple but always explained very poorly.
Knowing the position of the particle/excitation means reducing the width of \deltaX, which means summing more plane waves. Summing more plane waves means having less precision in the frequency/energy/momentum domain. Conversely, having less positional certainty (wider \deltaX) means you require fewer plane waves to describe the excitation, meaning you know the frequency decomposition (and therefor the energy/momentum description) very accurately, in a sense because the position is spread out.
The confusion enters because educators insist on talking about “knowing the position of a particle” when a particle literally is a wavelike excitation of a field and does not have a position in the sense that you think of a bowling ball having a position.
That sounds right to me, and I agree that this is sometimes explained badly.
Are you saying that this explains the perceived asymmetry between position and momentum? I don’t see how that’s the case, you could say exactly the same thing in the dual perspective (to get a precise momentum, you need to “sum up” lots of different position eigenstates).
If you were making a different point that went over my head, could you elaborate?
I doubt that I understand this very well. I thought there was a chance I might help and also a chance that I would be so obviously wrong that I would learn something.