Question on uncertainty principle

[annie122]

If I understand correctly, the uncertainty principle works, because in order to measure the position accurately, you need a smaller wavelength. But observing with a particle with smaller wavelength means observing with a particle with larger momentum. Therefore, when the observation is made by hitting the thing you want to observe with the observing particle, the more precise the measurement of position is, the observed thing is hit with a larger momentum, making the momentum of the observed larger, or in other words, uncertain.
(I'm also not sure if my usage of the word "uncertain" is proper.)

But how can you make a position of measurement precise, if what is observed gets a big momentum kick?

Is it that although the observed thing's momentum becomes large/uncertain AFTER the observation is made, it is still the value before it is hit by the observing particle AT THE VERY INSTANT the observation is made?

 

[danR]

My understanding is that it is much more fundamental than measurement: that the simultaneous 'exact' position and momentum do not even exist.

 

[eaglelake]

You can measure the position accurately by passing it through a narrow slit. There is no wavelength involved in this position measurement, but, localizing the particle at the slit “causes” an uncertainty in momentum. This is a purely quantum mechanical effect with no classical explanation. Using the same narrow slit, if you repeatedly measure the momentum, you will get many different momentum values. Getting many different values of the momentum means that there is an uncertainty in the momentum. The only way to determine if there is an uncertainty in the momentum is to make many momentum measurements with an identical experimental apparatus. If you always get the same momentum result, then there is no uncertainty; $$\Delta p = 0$$

The uncertainty in momentum has nothing to do with how big the momentum value is. In fact it has nothing to do with how you make the measurement. It is determined by the experimental apparatus. Knowing the corresponding wavefunction, you can then calculate the momentum uncertainty for that particular experiment. What we call ‘uncertainty’ in quantum mechanics is called the ‘standard deviation’ in classical statistics.

You are correct: We know the momentum only at the instant the particle is detected. But, a single measurement tells us nothing about the uncertainty.

Best wishes