the problem of infinite divisibility and how QE sheds some light


by San K
Tags: divisibility, infinite, light, sheds
fzero
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Jun24-12, 03:48 PM
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Some of what you write represents some misconceptions about quantum mechanics that are common in popularized discussions.

Quote Quote by San K View Post
Could there be events/changes that require less than a quanta?
Quantum is a term that popularized and watered-down discussions of quantum mechanics often only incompletely explain.

In the primary usage of the term, where for instance we learn that energy is quantized in units which are called "quanta," we must be more specific and explain that this is most often the case for systems which are bound states. So for example, the electrons in an atom exist in orbitals which have a discrete amount of energy relative to the unbound system. Photons which are emitted when an electron in an excited state decays to a lower energy state will be found to have a discrete spectrum (with a caveat that I will address below).

However, the spectrum of free particles in quantum mechanics is not quantized: their energy and momentum lie in the continuous spectrum of values contrained by ##E = \sqrt{(pc)^2 + (m c^2)^2}##.

Furthermore, even higher-order transitions between bound states involve a continuous spectrum of photons. For example, the 2s to 1s transition in hydrogen occurs at lowest order by the emission of two photons. Only the sum of their energies is quantized, ##E_1+E_2 = E_{2s} - E_{1s}##. There is a kinematic distribution of energies ##E_1,E_2## that is peaked around ##E_1=E_2##.

The second use of the term quanta arises in quantum field theory, which is often itself called "second quantization." This notion of the term refers to particles states being discrete excitations of a classical field. For instance the photon in QFT is the "quantum" of the electromagnetic field. The excitations are quantized in the sense that a classical EM wave corresponds to a finite (though large) number of photons. However, in the free-particle case, the energy-momentum of these individual quanta are not themselves quantized and form a continuum.

None of the above addresses any notion of minimal length. That is the realm of gravity and QM alone doesn't address the issue. The uncertainty principle allows us to probe infinitesimally small scales at the expense of having an infinite uncertainty in energy-momentum.

perhaps one or more of the below:

- a photon transferring its energy to two other photons (..like a billiards ball striking two other balls)...where the total energy transferred was just 1 quantum....this quanta must however now must be split into two between the two entangled photons
Momentum conservation forbids a single photon from splitting into two photons without having a fourth object around to carry some momentum in the final state. If this object were present, it could carry a continuous amount of momentum away and the spectrum of final state photons would be continuously distributed. Only the sum of final state energies is required to equal the energy of the initial photon. And in the free particle case, the initial photon energy wasn't quantized to begin with for the reasons I gave earlier.

- a change in spin of a photon that requires less than a quanta of energy
Spin is quantized, even for free particles. How a change in spin translates to an amount of energy depends on the specific process being considered.

- a change in momentum or position that requires less than a quanta of energy
Yes, as I've been clarifying, a change in momentum for a free particle corresponds to a continuous change in energy. It is hard to how entanglement sheds any light on this.

- or consider a quanta of energy/momentum applied to a photon....the back part of the photon will have slightly more energy (compression) than the front part.......because the quanta cannot be considered perfectly rigid.....however quantum entanglement provide some clues in our quest to resolve this ......
Photons are point particles. They do not have a front or back part.
micromass
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Jun24-12, 05:34 PM
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This should belong in the philosophy forums. I'm moving it.
micromass
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Jun24-12, 05:34 PM
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This doesn't meet the guidelines of the philosophy forums. Thread locked.


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