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I understand, generally and vaugely, how the uncertanty principle works, but not on a technical or conceptual level. Can someone help me with some of that?

*collaborates with a small part of the math behind it >_>*

Basically, you have two amounts of standard deviations. This in statistics means how spread out the extreme values of a measurement are. The more standard deviations we have, statistics tell us we have less certainty of something.
In here, we use the letter delta (Δ) to represent this. We have the deltas of position, and momentum (Δx and Δp).
The uncertainty principle says that the multiplication of these values has to be greater than a constant, which basically means that no matter how hard you try, you won't get more precision in both things at the same time.
The mathematical notation of this is

ΔxΔp ≥ ħ/2

Where ħ (h-bar) is the reduced Planck's constant, and is ~ 1.054E-34 Joules/second.
Since this is a multiplication with a fixed result, the lower one goes (the less standard deviation and thus more precision), the higher the other one goes (the less we know about the other one) to maintain that value.


As a thought experiment, picture an atom whose electron you want to measure both the location and momentum of. You use a microscope for this.
Say you want to know the exact position of the electron, you have to have a "high resolution". This is accomplished by using high-energy gamma rays (photons), since the max resolution you can get from a ray of wavelength λ is

sin θ = 1.22λ/D

where D is the size of the object. Therefore the smaller the wavelength, the better, since you want θ (the angle) to be smaller (the light will be less spread out).

So you shoot a high-energy photon to it and see where it bounces back to your microscope length.
Problem is, now that it has bounced, you may get a very accurate picture of where the electron was, but you've changed its momentum by hitting it with a photon (Δx gets small, Δp gets big). This is because in QM the momentum of this gamma ray would be

p = h/λ

Where p is the momentum, and h is Planck's constant. So for a high energy photon, λ is smaller (shorter wavelength = more frequency), so p gets bigger, and you just disturbed the electron a LOT.

Similarly, if you wanted to go the opposite way and not affect its momentum so much, you'd use a lower energy means, but the image wouldn't be as "high resolution", so you'd have less knowledge of the position (Δx gets big, Δp gets small).

Unfortunately, the best description and explanation of HUP is the one that's most mathematical, and least intuitive. I mean, the mathematical formulation is the one that's *correct*; any other words or physical intuition that we attempt to attach to it can cloud our judgment or understanding of it.

This is where the analogies with the macro world stop sweatdrop
Billiard balls are well-defined things. You don't have any variation on x since they're pretty un-changing in position. Momentum does affect them, yes, but at the subatomic scale, everything is a particle and a wave. It's not only the momentum equation that applies to the electron, it's the position one too.

You can freely observe the ball since a photon bouncing off it will have no impact on the ball's momentum, but this is not so for an electron. Simply observing the electron collapses its wavefunction - that is, makes it now stay in a specific state of position (the square of the values of the wave gives the probability distribution that the system will be in any given state). This is the intrinsic "uncertainty" in all measurements Heisenberg talks about - we modify an object when we measure it. At the macro scale, it's stupid to consider the effect of a photon bouncing off a billiard ball, but at the micro scale, "watching" an electron will disturb its states, since things aren't deterministic, but probabilistic.

The same equations won't apply (sin θ = 1.22λ/D) to observing a ball since the energy supplied by a single (or, hell, a shower of) photons by p = h/λ is VERY small at the macro scale, and almost any electromagnetic wave can detect a billiard ball, there's no need to use an "ultra energy" photon to pinpoint its location. But this is all stretching the microscope analogy a bit too thin sweatdrop .

Momentum would be m.v, the fact that v is "5 km/h at 30º" doesn't give you the position of it, just where (relative to itself) it will be at a given time, assuming constant velocity. Also the different particles travel at different speeds in vacuum - we all know light travels at c, but electrons travel at ~2.42 x 10^8 cm/s in the first atomic orbit, while it can move at 99% c when it's a beta ray.

But honestly, the actual algebra that controls this thing isn't something I'm familiar with (for Christ's sake I haven't even seen group theory yet D:!), I only know the concepts behind it (you have two related variables that can't be measured at the same time with infinite precision), maybe Layra or Iggy could explain it a bit better? *non-subtle wink at them* heart

Quantum mechanics: The dreams stuff is made of.

Upon rereading the post I made above, I feel like someone ought to make an Intro to QM thread.
Unfortunately, the very phrase "Intro to QM" screams of incompleteness and incomprehensibility. QM is nothing without the mathematics, and with the mathematics it is too much for a simple "intro"

For the TeX -> http://math.b3co.com and type there. You get the URL of an image-version of your LaTeX code. domokun

*spends the next 3 days understanding the math in Layra's post*
Yum knowledge. 3nodding

There's the IMG button up in the list of post-fun buttons.

And as they say, "Great minds think alike." Which actually means that I don't think at all, I just steal other people's thoughts and pretend they are my own. In this case, your thoughts.