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From http://www.hawking.org.uk/the-beginning-of-time.html

This means, it doesn't take into account, the Uncertainty Principle of Quantum Mechanics, which says that an object can not have both a well defined position, and a well defined speed: the more accurately one measures the position, the less accurately one can measure the speed, and vice versa.

Why can't two people agree to measure the same particle? One measures the speed and the other the position.

I suppose it raises the question - how do you "agree" on which particle to measure. But that may be part of the answer to this question.


marked as duplicate by sammy gerbil, Kyle Oman, Alfred Centauri, Yashas, John Rennie special-relativity Jun 16 '17 at 5:28

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    $\begingroup$ How is two people observing two things different from one observing two things? It has nothing to do with people, really. $\endgroup$ – Hritik Narayan Jun 15 '17 at 18:53
  • $\begingroup$ There is the quantum eraser experiment with the double slit, wherein a person watches the back screen and a computer measures which slit the photon/electron goes through but erases the data prior to the person being able to access it. Can't remember the outcome off the bat, but that's two independent non- communicating observers. $\endgroup$ – R. Rankin Jun 15 '17 at 19:53

So the uncertainty principle is fundamentally a statement of wave mechanics and the issue is that a wave is distributed over space with a certain wavelength, which goes inversely as its momentum. For example for light we have known $p =E/c$ for its momentum and energy since Maxwell; and Planck and Einstein derived that this wave is only seen in lumps of energy $E = h f,$ and the fundamental relationship between frequency and wavelength is given by $f = c/\lambda$, giving us $p = h/\lambda,$ as the momentum of the particle.

Rotations as waves

Let me venture into wave mechanics for a second. It turns out that the simplest wavy thing to consider are the rotation matrices $$R(\theta) = \begin{bmatrix}\cos\theta & -\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}$$ which rotate a 2D plane about its origin. You might be surprised that rotations and waves are related, until you see the operation of a crankshaft: A rotating crankshaft and a piston move in harmony The piston moves back and forth in a standard wavy way, but this is seen to be just one projection of the 2D components of the crankshaft's motion in a uniform circular motion.

Rotations are especially simple because $R(\phi)~R(\theta) = R(\phi + \theta),$ so they form a nice little structure that we call a "commutative group". Unfortunately while multiplication is beautiful additions like $R(\phi) + R(\theta)$ do not give you a pure rotation matrix directly, but we turn out to be very lucky that it gives you a scaled rotation matrix, and it turns out that the space of scaled rotation matrices $s~R(\theta)$ has a bunch more useful axioms; it is a field just like the rational and real numbers are. You can even identify each point of the 2D space with a scaled rotation matrix if you ask "what scaled rotation takes the point $(1, 0)$ to this point?" and then you can imagine that the real numbers are the x-axis of this 2D space; it turns out that addition is just vector-addition and multiplication is this weird operation of "multiply the two scale factors and add the two angles." These "numbers" that you have discovered are also known as the "complex numbers."

In quantum mechanics we use these to get wavy behavior but eat our particles too. Basically for various situations we refer to a complex number as a "probability amplitude", and (because it's mathematically more easy than anything else) we use the square of the scale factor as a proxy for the probability of the thing, forgetting that the angle means anything at all.

Fourier decomposition

Since Fourier we have understood that we can describe any function $f(t)$ if we use a sum of a bunch of different rotations at different frequencies: we just need to associate with each rotation frequency an amplitude of oscillation and an offset phase. If "a combination of a scale and a phase" sounds like another complex number: congratulations, you might really want to do a degree or two in physics or mathematics. The claim is that for each function $g(t)$ we can view it as some scaled-rotation function of the frequency $g[f]$ consisting of these scaled offsets, so that$$g(t) = \int_{-\infty}^\infty df~R(2\pi~f~t)~g[f].$$It turns out that there is a really easy trick to figure out what these amplitude-and-phase terms are; take a rotation $R(-2\pi~f'~t)$ rotating the opposite way of some target frequency $f'$, for anything that's rotating at any frequency other than $f'$ the two rotations will combine into a new rotation which will average out to 0 as it goes around the circle, while the one at the target frequency $f'$ will perfectly sum to $g[f']$:$$g[f] = \int_{-\infty}^\infty dt~R(-2\pi~f~t)~g(t).$$

Just like time has a frequency space $f = T^{-1}$, physical space has a "inverse wavelength space," a $\lambda^{-1}$ space. The shocking claim of quantum mechanics with its $p = h/\lambda$ relationship, is that this is momentum-space up to some special scale-factor $h$. The amplitude in this inverse-wavelength-space, if squared, is the probability to measure a certain momentum.


Okay so what's the problem? Imagine a highly confined chirp in time: what is its frequency? It turns out that that's highly uncertain! If you want to have a wave with a certain frequency $f$, that takes time; more precisely it takes the time $f^{-1}$ or so.

Similarly in space, if you want to confine a particle, you lose all ability to manipulate its wavelength components at wavelengths longer than your confinement. Since wavelength goes inversely as momentum, this suggests that you totally lose control of what it looks like at low momentum. The ultimate statement of this is to consider the infinitely thin Gaussian peak $g(x)$ which is centered on some point $x_0$ but has been scaled infinitely tall so that its integral is still 1: the Fourier transform of this function turns out to have a constant scale factor and perfectly predictable phase factor, $g[\lambda^{-1}] = R(-2\pi~x_0/\lambda).$ You can use a trick that $R(\theta)$ can be treated as the natural base $e$ raised to the power of $i\theta$ to solve all Gaussian wavepackets, actually, and find that the Fourier transform of one Gaussian wavepacket is a Gaussian in $\lambda^{-1}$-space. And sure enough the two widths vary inversely, a thinner and thinner Gaussian that's more defined position-wise is less and less defined momentum-wise.


Now the fact that real particles like the photon and the electron diffract and act wavy, I don't necessarily expect you to immediately understand. The "wave-particle duality" feature takes a lot of experience with the mathematics before you realize "oh, I have been separating the world into these two different boxes but Nature doesn't see those boxes as different, maybe I can blur my brain a little to see both at the same time."

However I do want you to now appreciate: when we say with experiments "oh, the electron is wavy and can diffract off of a crystal lattice," then there is necessarily this uncertainty relation between $x$-space and $\lambda^{-1}$-space. That is a central mathematical feature that waves have, you cannot confine them in space without getting a spread of wavelengths; you cannot confine them to one definite wavelength without them oscillating over more and more space.

The fact that the momentum and the wavelength have this inverse relationship is something that I hope you can just take as an experimental reality. We know the momentum of incoming electrons; we see from the diffraction what their wavelength is; and we observe that these two seem to vary inversely. Well, if those observations are true, then we have to get the corresponding duality between position and momentum; it has to be the case that those two obey an uncertainty principle.


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