'I daresay you haven't had much practice,' said the Queen. Protons are much more massive and occupy very little volume.Absolute zero does not imply absolute rest. There are experimental differences between “hidden variables” where the uncertainties are just from bad measurements, and quantum mechanics where the uncertainty is an inextricable part of the system. Unlike the marbles that were used to make the particle-like interference pattern, where marbles starting in different places ended in different places, photons are identical.We have stated that this “matter wave” will give rise to an interference pattern.

For matter waves, the wave velocity is much more complicated and at this level you should avoid thinking about the frequency of matter waves.

They do not measure the value of any physical quantity.

Which are characteristics of electromagnetic waves?

This gives us the graph below:This graph peaks at \(x = 0\). Electrons diffracted after being reflected off a single crystal of nickel.A person entering a room with more than one entrance will always enter through one of them, not all of them at the same time.


So far all of this is true, but we have glossed over one important detail: we do not know \(v_{wave}\) for a matter wave!

If the light bounces directly back, the momentum of the particle must be\[\sigma_x \sigma_p \approx (\lambda) \left( \dfrac{h}{\lambda} \right) = h\]The estimates presented above are rough, but this shows us that by using light of a long wavelength, we can measure momentum well (\(p_{light} =0\)) but know almost nothing about where the particle is. This may seem like a philosophical distinction; how could one show that these quantum particles were not really “marble-like," that all the uncertainty came from our inability to measure?

Nothing else.The conflict between these two aspects of microscopic reality results in the Uncertainty PrincipleMomentum and position are measured in the same direction.Formally, a function and its Fourier transform cannot both be compactly supported.Quantum mechanics is like a student who always gets the correct answer to the first question in an exam, but never the second.Thus it seems Einstein was doubly wrong when he said, God does not play dice.

That way we should be able to eliminate the possibility that the photon is “interfering with itself”. It interacts with its environment in a specific place at a specific time.In contrast, when we are not observing these entities interacting with their environment, they behave in a wavelike manner — extended in space, diffracting around obstacles and through openings, interfering with other elementary entities of the same type (that is, electrons interfere with electrons, and photons with photons). However, with some more mathematical and physical consideration, it is possible to calculate the spectrum \(E_n\) of other potentials.Recall how attempting to observe particle behavior in the double-slit experiment affected the experimental results. \(\lambda\) here is called the We learned when studying waves that interference patterns occur because the waves from each slit superpose with each other.

Not only does God definitely play dice, but He sometimes confuses us by throwing them where they can't be seen.It is often stated that of all the theories proposed in this century, the silliest is quantum theory. 'There's no use trying,' she said: 'one CAN'T believe impossible things.' Adopted or used LibreTexts for your course? What made one photon make a spot on the film, and another photon make a different spot?

We generalize this to a probability \(P\):\[P \left(\text{particle in region } x \pm \dfrac{\Delta x}{2} \right) \propto \text{ energy transmitted to that region}\]For light, we can find the energy transmitted to the area using \(I \Delta x\).

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