Notes on (abstract) quantum physics as related to classical
mechanics
Here are some remaks about the wave/particle duality and the classical
approximations of quantum physics by classical mechanics, that I first
wrote
in this forum. This is a mere draft, a few introductory
remarks, not any clear and coherent explanation.
For a really coherent explanation of the wave-particle duality, see the
next texts in this site, starting with the geometric
expression of Markov processes, followed by
some core structures of quantum physics
(quantum states and measurements), and continuing up to the double-slit experiment.
On the wave-particle duality : how
quantum physics is articulated with classical mechanics
The descriptions of light in terms of waves
on the one hand, in terms of particles (photons) on the other
hand, are but two approximations of one the quantum law, that are
each describable in the language of classical mechanics.
The description in terms of classical waves applies when the
photons number is large and its value is undetermined (its
undetermination is >> 1)
The description in terms of classical particles applies when the
scale of the considered phenomena is much larger than the
wavelength of the light involved.
These two domains of approximation have an intersection, a class
of phenomena where they both apply, where both explanations in
terms of particles or of waves are valid.
In fact this class of double approximation is very familiar to us
as it is the way light most often appears to us in everyday life:
it is the field of geometrical optics.
So, both "explanations" of the momentum of light, either in terms
of the momentum of photons, or in terms of the force that the
magnetic field applies to the electric current in a material
absorbing the light, are equally valid, as two classical
approximations of the same thing.
The same laws of geometrical optics can be "explained" in two
ways: either in terms of classical particles or of classical
waves. Moreover, the description in terms of particles can itself
be reduced to a sort of third possible "explanation" by a pure
space description getting rid of the time dimension, as photons
have zero mass and go at the speed of light (too fast for us to
track it): that is the laws of geometrical optics. A light ray
(say, a laser one, with fixed frequency), can be compared to an
elastic at equilibrium. An elastic that can be stretched without
modifying its tension, whose value (equal to its linear density of
energy, just like surface tension but with 1 dimension instead of
2) only depends on its environment (refractive index). The surface
(say, an horizontal one) separating two materials with different
refractive indices, exerts a vertical force on the ray; the point
of intersection remains at equilibrium with respect to its
horizontal movements slipping on the surface, so that the
horizontal components of the tension on each side must stay the
same. In other words, the amount of potential energy that each
side would provide to this point for any given small movement on
the surface, must cancel.
Now when the same question of geometrical optics is analyzed in
terms of waves, the role of the potential energy is played by the
oscillations number : for every small segment on the surface, the
number of oscillations of the wave along that segment must stay
the same whether it is measured on either side of the surface.
This is how we can see that different "explanations" in the
language of classical physics, correspond to the same mathematical
theory.
More comments in French
I wrote there :
De fait l'analyse en termes d'onde électromagnétique classique est rigoureusement une limite
classique correcte de l'électrodynamique quantique de sorte que les 2 manières d'analyser les
choses sont rigoureusement équivalentes et même mathématiquement identiques tant que
n'intervient pas la question du comptage des photons. Dès lors dans le cas de la présente
discussion, étant indépendante de la question du comptage des photons, tous les phénomènes
décrits étant parfaitement déductibles de l'électromagnétisme classique, toute invocation du
point de vue quantique et des outils associés est tout simplement hors sujet et ne fait qu'embrouiller
les explications: si quelqu'un s'imagine que le fait de prendre un seul photon pour parler de sa
"fonction d'onde" au lieu d'en étudier un gros paquet en termes des équations de Maxwell (le
paquet de photons implicitement contenu dans une onde électromagnétique classique) aurait
pour avantage de nous rapprocher des causes plus fondamentales et ainsi d'apporter une
compréhension plus correcte et plus approfondie, je suis au regret de vous décevoir :
ce n'est absolument pas le cas.
L'expression "onde électromagnétique induite par ce photon" est incorrecte:
il n'y a pas de sens à dire qu'un photon étant un objet quantique, peut induire
une onde électromagnétique qui est un objet classique. C'est plutot l'inverse:
grosso modo, tout onde électromagnétique classique est ré-analysable, via un
changement de base dans l'espace de Hilbert, sous forme d'une grosse
superposition quantique sur l'ensemble des entiers n, d'états à n photons...
Par ailleurs la vidéo donne bien des clés de compréhension essentielles mais
en sens inverse de celui supposé: ce n'est pas la mécanique quantique qui
explique l'optique mais c'est l'optique (et généralement la mécanique ondulatoire)
qui explique la mécanique quantique et la manière dont par passage à la limite elle
se réduit à la mécanique classique exprimée par le principe de moindre action.
On peut se demander comment il peut être cohérent de dire que la mécanique
classique par laquelle est formulée le concept d'onde, peut engendrer la mécanique
quantique. C'est qu'elle ne le fait pas complètement : elle explique seulement le
comportement de la particule dans le cas linéaire, c'est-à-dire de manière équivalente,
dans le cas où la loi est linéaire par rapport au champ, et linéaire par rapport au nombre
de particules, autrement dit que les différentes particules n'interagissent pas entre elles
de sorte que n particules se comportent exactement comme n fois une particule seule.
Sur tout cela on peut introduire la "seconde quantification" à savoir la version quantique
de la mécanique ondulatoire (où on remplace le principe de moindre action sur le champ
par l'intégrale de chemin sur tout les champs) qui par cette correspondance donne le
cas plus général de superpositions quantiques quelconques d'états à n particules intriquées.
How can frequency be equated with energy
or mass ?
This equality may seem difficult to
understand as long energy is assumed to be a sort of primary
physical substance before quantum mechanics is introduced and
gives it further properties.
Instead, the real situation is that energy is a mathematical
quantity that emerges from the laws of quantum physics, and that
the frequency of quantum processes stands as the very definition
of this quantity.
To understand how it happens that this quantity appears to us in
the form of "energy", we need to remember the classical
characterization of energy, that is a quantity which is conserved.
So, to make a short story of that, the point is to explain the
"law of conservation of frequency" in the same way as how it
happens in quantum physics to give the conservation of energy.
Again this "law of conservation of frequency" can be directly
experienced and understood as a daily life phenomenon, so to
speak. All our radio communication systems are directly based on
it. It simply says this: any radio wave emitted at a given
frequency, is received anywhere at the same frequency. At least as
long as all obstacles that the wave can bounce on, stand still.
But if some object moves, then the frequency is modified by the
Doppler effect. This corresponds to the case where a photon
bounces on the object, and exchanges energy with it. The change of
frequency has a precise value depending on the directions of the
wave before and after, and on the movement of the object; it does
not depend on how much of the wave bounces on it. Only two wave
frequencies can be detected: the initial one and the reflected one
(and possibly more obtained after bouncing on the object several
times or through different trajectories), but none in between. So
is the case for the photon: either it bounces on the object and
exchanges energy with it, or it does not, but as long as we
separately measure the different frequencies, there is no option
in between.
How does a photon move and is it real
while it goes until is it observed ?
A single photon cannot be detected at
different places because the only way to detect it is by absorbing
it. As the absorption of a radio wave cannot modify its frequency,
its detection at different places requires this same frequency to
be detected at different places. But if we start with only one
unit of a given amount of energy, then we cannot get several
copies of the same amount of energy at the end.
How is the indeterminacy of the
energy coherent with its conservation
In all physics, conservation of energy (once understood, avoiding
some naive mistakes ...) is an inviolable law.
Including in quantum physics.
But an Heisenberg inequality says there cannot be simultaneous
determination of energy and time beyond the accuracy defined by
Planck's constant.
In quantum physics, the indetermination of a quantity, means
(implies) that, when measured, its value has some probability of
being found to have a particular value, but this value does not
exist prior to the measurement. This value is created by the
measurement, which makes a choice with uncontrollable result, and is
physically indistinguishable from the discovery of a result in the
case where instead of being undetermined, the quantity was
previously determined to have the value that has been found. In fact
there are cases, especially with the EPR paradox, where the same
physical situation can be interpreted either way, which are
mathematically equivalent (giving the same effective predictions)
without any possible to find a natural (satisfactory) and systematic
way to decide which interpretation is correct.
So if an object has its time measured with some precision (the time
of reaching a certain state during an evolution) its energy
necessarily has a certain range of indeterminacy (there must coexist
several possible different values of the energy inside that range
from the viewpoint of a more accurate measure of the energy, for the
time measurement to have been possible). So then if we measure its
energy more precisely that that range, the result will be random and
created by this measure. But then we can ask: if a measurement
performed by a procedure that conserves the energy of the system,
has the effect of giving an arbitrary value to this energy, how can
it be said to respect the conservation of energy? However, the
energy is preserved: you can not say it is not preserved because the
value was found among the possible values.
This is explained as follows: starting with a specified energy, and
then a time measurement is carried out over it, the object acquires
a determination of time and therefore an indeterminacy of energy,
because the time measurement procedure necessarily came
with the exchange of an undetermined amount of energy between the
object and the measuring device. Under these conditions, even if the
energy of the object becomes undetermined, the energy of the whole
(object + measuring device) remains determined to be preserved with
respect to the situation previous (or rather, it preserves the shape
of its original indeterminacy).
Therefore, the final measure of the energy of the object has the
effect of "deciding" what was the amount of energy previously
exchanged during the time measurement. So the energy "found" or
"created" by the final measure in the object, actually comes from
somewhere. Namely, it comes from a modification of the form of the
energy indeterminacy in the prior measuring device. Because, a
measuring device whose energy would, both, have been determined
before, and be finally measured after, a measurement it carried out,
being itself subject to the Heisenberg inequalities, is
fundamentally incapable of having measured a time (more accurately
than these energy determinations allowed).
Electrons don't jump between orbits !
Some people give a very bad description of quantum theory claiming that
"an electron would go from one orbit to another orbit without ever passing
through the space in between". This is a ridiculously inaccurate description.
Instead of this, the fact is that both "orbits" are largely overlapping spatially,
and the distinction between these "orbits" before and after this "jump"
only makes sense as a change of energy levels, but then for both energy
levels to be distinguished from each other, according to an Heisenberg
inequality, there is a minimal necessary time uncertainty, which is the
uncertainty of the time at which the "jump" would "occur". The energy
difference is defined by the period of the photon, which can be understood
as the electromagnetic wave emitted by an oscillating electric dipole,
this oscillating dipole being classically understandable as that of the charge of
the electron in its orbital movement around the nucleus. So the uncertainty of
the time of "jump" cannot be defined at any better resolution than, roughly
speaking, the time the electron takes to revolve around the nucleus.
Classical orbital movement which only appears when the electron is not
on a definite orbital but in a quantum superposition between 2 of them.
And what gives this superposition the ability to produce that oscillating
dipole, is precisely the overlapping between their wavefunctions, by
which they can interfere, an interference that lets the presence
probability of the electron, "beat" its most likely presence (in the
sense of "beat" in acoustics) from one side of the atom to the other.
On the tunnel effect
My reply to a piece of article someone posted from "Science et vie" n° 1239, décembre 2020.
Réponses de la physique quantique aux questions que se pose l'auteur de cet article de S&V : la particule de disparait pas d'un côté pour se rematérialiser instantanément de l'autre. En effet, la particule commence à exister en amplitude de probabilité à la fois dans le mur et de l'autre côté avant de finir éventuellement d'exister du côté initial. Non la particule ne disparait pas spontanément d'un côté. Si on voulait parler de sa disparition il faudrait se référer à une mesure, puisque "disparaître" signifie "cesser de paraître" ce qui n'aurait pas de sens du point de vue de quelqu'un qui ne la regarderait pas, mais seulement pour quelqu'un qui continue de la regarder. Mais l'activité d'espionnage intensif que suppose un regard à peu près (mais jamais vraiment) continu nécessaire à cela contredirait par sa violence les hypothèses de l'effet tunnel qu'on voulait observer. Non la particule ne se rematérialise pas de l'autre côté, car elle n'a jamais cessé d'être matérielle. Encore moins instantanément, car tout le processus est progressif, tout est plus lent que la vitesse de la lumière. Oui la particule traverse le mur. Non la particule n'a pas de trajet dans ce phénomène, tout comme elle n'en a jamais dans aucune autre circonstance par ailleurs.
Mais la première idiotie était de parler d'un mur "supposément infranchissable". Quels ignorants le qualifient ainsi ? Ceux qui ne connaissent que la mécanique classique bien sûr. En effet, on a la fameuse loi de conservation de l'energie E = Ep + Ec où Ep est l'énergie potentielle et Ec l'énergie cinétique. Elle reste valable en physique quantique. Y compris dans le fameux mur de l'effet tunnel où Ep > E. Quoi, yaurait un problème ? Que si E = Ep + Ec et Ep > E alors Ec < 0 ? Oui en effet dans le mur, Ec < 0. Ce n'est pas pour autaunt un problème pouvant empêcher la particuler de s'y trouver. Car pour l'y trouver il faut effectuer une mesure qui du coup lui fournira l'énergie qui lui manquait pour actualiser cette présence dans le mur ainsi devenue concurrente à celles hors du mur dont la probabilité persistait en l'absence de mesure. Problème résolu.
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