Explanation of uncertainty principle

[TheBlackNinja]

Is this explanation of uncertainty principle from wikipedia correct? Is this "compressing" the Fourier series/integral? Because the function is not periodic, and with infinite frequencies some of then would put make the speed greater than c

"According to the de Broglie hypothesis, every object in our Universe is a wave, a situation which gives rise to this phenomenon. Consider the measurement of the position of a particle. The particle's wave packet has non-zero amplitude, meaning that the position is uncertain – it could be almost anywhere along the wave packet. To obtain an accurate reading of position, this wave packet must be 'compressed' as much as possible, meaning it must be made up of increasing numbers of sine waves added together. The momentum of the particle is proportional to the wavenumber of one of these waves, but it could be any of them. So a more precise position measurement – by adding together more waves – means that the momentum measurement becomes less precise (and vice versa).

The only kind of wave with a definite position is concentrated at one point, and such a wave has an indefinite wavelength (and therefore an indefinite momentum). Conversely, the only kind of wave with a definite wavelength is an infinite regular periodic oscillation over all space, which has no definite position. So in quantum mechanics, there can be no states that describe a particle with both a definite position and a definite momentum. The more precise the position, the less precise the momentum."

 

[TheBlackNinja]

Guys, I have only calculus and those "engineer level" physics and mathematics, if my question is wrongly formulated or just too nonsense no be answered someone please tell me

 

[BruceW]

Wikipedia seems correct. The frequency of the wavefunction is proportional to the momentum of the particle, but in special relativity, $p=mv$ is not true, it actually becomes
$$\displaystyle p= \frac{mv}{\sqrt{1- \frac{v^2}{c^2}}}$$
So this means the particle's velocity is restricted to below the speed of light, even if its momentum approaches infinity.

 

[zonde]

Is this explanation of uncertainty principle from wikipedia correct? Is this "compressing" the Fourier series/integral? Because the function is not periodic, and with infinite frequencies some of then would put make the speed greater than c

If you mean if it's incorrect because it involves speeds greater than c then no.
Otherwise it seems controversial. And it seems to suggest that non commuting variables can be attributed to single particle. But that would lead to EPR paradox.

Maybe uncertainty principle can be better explained from perspective of ensembles.
You can put ensemble of particles trough sieve that corresponds to certain interval of positions. You get reduced ensemble. If you use the same sieve second time ensemble gets through without being reduced.

The same goes if you use two sieves that correspond to certain interval of momentum.

But if you use position sieve then you use momentum sieve and then position sieve again then ensemble gets reduced after each sieve. So it "forgets" that it went trough position sieve when it goes through momentum sieve. As a result when it goes through position sieve second time it is reduced again.

This happens not just because sieves alter parameters of individual particles randomly but because there are stable configurations of ensembles that correspond to position sieve and momentum sieve. And these configurations are different i.e. there are no stable configuration that correspond to both position sieve and momentum sieve.

 

[BruceW]

This sieve analogy for the quantum mechanical operators isn't a great analogy, since it would also require particles being created as the ensemble went through a sieve. (Not just particles being 'filtered' out). But I can't think of any good analogy for the quantum operators, unfortunately...

 

[zonde]

This sieve analogy for the quantum mechanical operators isn't a great analogy, since it would also require particles being created as the ensemble went through a sieve. (Not just particles being 'filtered' out).

No.
Why do you think that it requires particles being created?
As far as I know quantum measurements always work as filters.

 

[BruceW]

If, for example, there was only one particle, and you sent it through a position sieve, then a momentum sieve.

 

[mquirce]

Please, can somebody tell me that if (G*M / R)^0.5 < or= C is or not right ? I think that this has to do with uncertainty principle. Because it has to do with a limit in frequency ( plank frequency ) so evited the notion of infinity.

 

[SpectraCat]

This sieve analogy for the quantum mechanical operators isn't a great analogy, since it would also require particles being created as the ensemble went through a sieve. (Not just particles being 'filtered' out). But I can't think of any good analogy for the quantum operators, unfortunately...

How about the triple Stern-Gerlach experiment? Best for the OP to look it up himself, but here is a brief summary:

1) We have beam of randomly oriented spin-1/2 particles propagating along y-axis
2) We pass beam through magnetic field gradient along z-axis
3) Particles with different spin-projections on z-axis (Sz=+1/2 and Sz=-1/2) are deflected in opposite directions, splitting the beam.
4) We select one of these beams (say Sz=+1/2), and pass it through a magnetic field gradient along x-axis
5) Particles with different spin-projections on x-axis (Sx=+1/2 and Sx=-1/2) are deflected in opposite directions, once again splitting the beam.
6) Again we select one of these beams (say Sx=+1/2) and pass it through third magnetic field gradient, this time oriented along the z-axis again. What happens?

Classically, one might expect to observe only Sz=+1/2, since you selected that component after the first field gradient. However this is not what is observed .. in fact, you get two beams with equal intensity in Sz=+1/2 and Sz=-1/2. This is because the system is quantum mechanical, and the projections of the spin along different axes do not commute. Therefore, when the beam of Sz=+1/2 particles is passed through the x-oriented magnetic field gradient, the x-components of the spin-projection are selected, but the z-components become mixed. Therefore the particles behave "as if" they "lost memory" of the first interaction with the z-oriented field gradient.

I don't know if there is a way to do the analogous experiment with position and momentum, since I am not sure there's a way to realistically measure the momentum of a particle in a non position-sensitive way. Perhaps a series of extremely fast shutters with very wide openings, or a series of choppers with precisely defined relationships between their phases and angular velocities?

 

[BruceW]

Good explanation. You're right, a series of Stern-Gerlach apparatus is probably the simplest way of explaining quantum measurement.