A full introduction to quantum physics expressing the quantum states and measurements with their paradoxes and indeterminism is there.

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.

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.

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 indeterminavy of energy, because the time measurement procedure was necessarily accompanied 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).

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