A Few Clear Questions on the Nature of Quantum Theory

[nonequilibrium]

Hello, I'm adding one or two words in bold in front of each question so you can browse for questions that interest you, please don't get turned off by the amount of text, I'll be very grateful even if you can answer just one. These are questions that popped up whilst following an introductory course.

1. Quantum-particle
   Is a photon and an electromagnetic wave and a "quantumwave"? Or is it one and the same? The thing that makes me thing they're the same: in QM we seem to interpret E² as related to the probability, hence the amplitude of the EM wave is equal to the amplitude of the quantum wave, suggesting both waves are equal. On the other hand, if an EM wave = quantum wave, why can't we see my electron quantum wave is also an EM wave?
   (side Q: is there also a generalized principle of Fermat ("light travels on the shortest path (in time)") for quantum waves?)
   
2. Radiation
   If I understand correctly: an atom can shine out light by electron excitation, this light is quantized => sharp spectrum lines (after being filtered by wavelength); a material can also shine out light by atom excitation? I mean atoms bumping into each other, called heat radiation, what Planck investigated. This is continuous. Why? Because there are no constraints on the speeds so every kind of collision can happen and thus every kind of photon? If so: what about the fact that photon-energy quantization was crucial in Planck's study of thermal radiation, doesn't that quantization contradict a continuous spectrum? Also, what were the oscillators--that Planck used--representing: it seems logical that they would be the atoms, right? (instead of electrons)
   
3. Mathematics
   I read in my book (Serway...) that for photons we can't use the Schrodinger equation. Is there a generalized mathematical formula that works for photons and the rest?
   
4. Energy Quantization
   Imagine we have a potential well and that the ground state of an electron corresponds with a minimum speed v_min. What if we let an electron enter the well with a speed less than the v_min? If it bounces back out before entering, please also read my question 5 about tunneling (more precisely part c)
   
5. Tunneling
   So the set-up is: potential is a constant U > 0 everywhere except in between x = 0 and x = L where it is zero.
   a) I quote Serway (p1198): "According to QM, however, a finite probability exists that the particle can be found outside the well even if E < U. ... This uncertainty allows the particle to be outside the well as long as the apparent violation of conservation of energy does not exist in any measurable way."
   What does it mean with "does not exist in any measurable way"? Either i) we cannot measure it outside, which would mean the probability of finding it there is effectively zero, but then QM would be wrong because it predicts a non-zero value (assuming "the probability of a particle being at position A" = "the probability of us measuring the particle at position A", which has to be true(?) since QM only talks about measurable quantities); or ii) we can measure it outside of the potential well, in which case we can measure the breach in energy conservation, but why can we not then keep the particle there and win the extra energy? What keeps us from doing that (since we can see it if ii) is true)?
   
   b) In the calculation of psi (with the time-independent eq.) we say that E < U even outside of the potential well, but doesn't the previous citation (+ physical sense) say that if the particle is outside of the well, its energy E must (temporarily...) be greater than (or equal to) U?
   
   c) Consider another particle with E > U and thus flying above the whole potential, approaching the potential well: this particle can reflect when reaching the potential well even though there's not an actual wall, it's like we're throwing a ball across a wall and it still bounces back. Okay. Now what is the explanation for this? Is it the following: due to the UP it's possible that just when flying above the potential barrier, the energy is suddenly E < U for a very short time in which the particle bounces into the wall; the energy is then "restored" but the particle of course continues in its reflected path. Or in our analogy: due to the UP it's possible that when the ball is flying across the wall, the ball is for a brief moment lower than the top of the wall, bounces, and then restores back to its initial height and it has been reflected.
   
6. Group Velocity
   Why is $$\lambda f = 2v$$ for a quantum particle instead of equal to the velocity of the particle?
   
7. E = hf more general than E = mc²
   E = mc² used to be always true, in the sense that whenever something had a total energy E, we associated with it a total mass m (relativistic mass); i.e. energy and mass were "the same". Now we don't use the idea of relativistic mass anymore, and E = mc² is actually the rest-energy, so now mass has become "one form" of energy. Now I was wondering, what about E = hf? It's a lot less popular than E = mc², but am I right in thinking E = hf is still always true in the sense that whenever something has total energy E, we associate with it that frequence f?
   
8. Compton experiment
   The Compton experiment is regarded as a great (early) proof of QM. I was wondering: why? What would we expect to be different in classical theory? Is the difference only quantitative? (in the sense that the numbers we calculate differ and experiment proved the quantum numbers were the correct numbers) Or is there a qualitative difference? I read in my book that clasically we would expect something having to do with the doppler effect, but it wasn't really clear. Is that implying that we wouldn't expect the light to give part of its impulse to the electron? Because Maxwell's theory is also classical and surely out of his equations follows that light exerts pressure?

 

[the_house]

I'll take you up on your offer to just answer one of the questions. :)

[*]Mathematics
I read in my book (Serway...) that for photons we can't use the Schrodinger equation. Is there a generalized mathematical formula that works for photons and the rest?

The Schrodinger equation applies to a non-relativistic system. In particular, it assumes a fixed number of particles, which is rarely, if ever, the case for massless particles like photons. To describe a relativistic system with creation and annihilation of particles, one needs the entire formalism of http://en.wikipedia.org/wiki/Quantum_field_theory" . In general, there's no way to boil this down to a simple evolution equation, so I'll just leave it at that for now.

 

[Chopin]

Is a photon and an electromagnetic wave and a "quantumwave"? Or is it one and the same? The thing that makes me thing they're the same: in QM we seem to interpret E² as related to the probability, hence the amplitude of the EM wave is equal to the amplitude of the quantum wave, suggesting both waves are equal. On the other hand, if an EM wave = quantum wave, why can't we see my electron quantum wave is also an EM wave?

Yes, they're all the same. A photon is just a name for a chunk of electromagnetic wave. As for electrons and photons, they are still different things, but both can be described using some of the same mathematical language (to be precise, both of them have a wavefunction, although the two are described by different rules.) So an electron is not an EM wave, although like an EM wave, it has some wavelike properties such as a wavelength.

(side Q: is there also a generalized principle of Fermat ("light travels on the shortest path (in time)") for quantum waves?)

Basically, yes. It's called Feynman's path integral. It's essentially applying Lagrangian mechanics to QM, and is the method by which modern Quantum Field Theory is formulated.