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I know what this mathematically is and that we can only predict a particle's momentum and position with certain accuracy. But it left me with more confusions. To easily understand my query let's say we are measuring these values for an electron.

Now my question is that we measured these parameters and we found certain uncertainty in electron's position. Does it mean that electron was not at a certain point at that time or it is just that we couldn't predict it. Was electron coexisting at more then one point on same time?

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  • $\begingroup$ Your additional question at the end is unclear. I suggest just deleting it from the post. $\endgroup$ May 29, 2021 at 13:03

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I suspect you have mashed different concepts. In relation to the position and momentum of an electron, what the uncertainty principle means is that the more precisely we pin down one of those quantities, the less certain becomes the other.

The effect arises because the position and momentum are determined by the characteristics of the electron's wave function. For an electron to have a very well defined position, its wave function must be very localised in space. On the contrary, if the electron is to have a very well defined momentum, its wave function must have a very well defined frequency.

If you study the mathematics of waves you will find that the more localised a wave, the less well-defined its frequency. You cannot have a wave that has both a well-defined frequency and a well-defined position- the uncertainty principle follows directly from that property of waves.

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I know what this mathematically is and that we can only predict a particle's momentum and position with certain accuracy.

The HUP is not something that predicts a particle's momentum and position with certain accuracy. The HUP is not a statement about individual measurements; the accuracy of an individual measurement is still dependent on the method of measurement.

The HUP is a statement about many single measurements of similarly prepared systems (conceptually you can think of them as identical systems). Let's say you made a bunch of position measurements (one for each of the similarly prepared states) and found their standard deviation $\Delta x$, and then did another experiment making a bunch of momentum measurements and found their standard deviation $\Delta p$. Then, no matter which state you prepared, you know that the product $\Delta x\Delta p$ will never be smaller than $\hbar/2$.

Now my question is that we measured these parameters and we found certain uncertainty in electron's position. Does it mean that electron was not at a certain point at that time or it's just that we couldn't predict it.

No. "Uncertainty" is a horrible name for this concept, in my opinion. In QM the "uncertainty" is just the amount of "spread" you would get in a set of measurements of many states. It is a statistical property; it is not something like experimental uncertainty where "the 'true value' could be within this range".

Furthermore, you can never predict what the measurements will give you; all we can predict are probabilities. How likely is it that we will find the particle here? Or how likely is it that we will measure the momentum to be this value? These are the questions that QM can answer; there is no way to predict what a single measurement will actually be. The closest you could come to this might be to say you can predict the most likely position, but this is slightly different from "prediction" in the classical sense.

Was electron coexisting at more then one point on same time?

No. Quantum mechanics has some weird things, but it is still a logically consistent theory. Particles cannot exist in two places at once, despite many pop-sci articles saying that "superposition" means "in all of the states at the same time".

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Now my question is that we measured these parameters and we found certain uncertainty in electron's position. Does it mean that electron was not at a certain point at that time or it is just that we couldn't predict it. Was electron coexisting at more then one point on same time?

It would be contradictory to claim that one is measuring something and simultaneously stating that we do not have a value for the measured quantity. Quantum Mechanics (QM) is weird, but it is not nonsense. If a position measurement provides some values for the coordinates, that measurement tells us that the electron was there, not elsewhere! We can also exclude a real coexistence of many positions: it would imply that more than one electronic charge would be present at the same time. Conservation of charge is one of the best verified physical laws up to now.

If one prepares many systems in the same microscopic state, all the possible outcomes of position experiments, even performed with the most accurate experimental protocol, will give results accordingly to a statistical distribution depending only on the microscopic state. Similar statistics on momentum experiments on equally prepared systems in the same state will give another statistical distribution for the momenta. The spreads of the two distributions as measured by the square root of the variance will satisfy HUP.

That's the physical content of HUP. Many other statements around are either PopSci or misconceptions.

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