2) quantum ball on the spring
3) waves, classical view
4) waves, classical equation of motion
5) Quantum waves
Here we finally reached our goal: to understand what are actually the things that we call "particles", namely, electrons, Photons, quarks, gluons and neutrinos. All this, of course, applies to modern science. It is worth remembering that in science there are no guarantees that the current understanding will not be further deepened.
The previous article described what fields are objects that have a value at any point of space and at any time (functions from Space and time) that satisfy the equation of motion and are physically meaningful in that they are able to transfer energy from one place to another and influence the physical processes of the universe.
We learned that most of the fields we know describe the environment , T Such as the height of the cord or the pressure in the gas. But we also learned that in Einstein's theory of relativity there is a special class of fields, relativistic fields that do not require a medium. Or, at least, if they have a medium, it is very unusual. Nothing in the field equations requires the presence of any medium and does not say what property of this medium is described by relativistic fields.
So while we will consider relativistic fields as elementary physical objects of the universe, and not as certain properties of an unknown medium. Whether such a point of view will be maintained among physicists will continue to be shown – time will tell.
We considered two classes of relativistic fields, and now we will study them in more detail. They satisfy either the equation of motion of Class 0, where c w = c (where c is the universal velocity limit, often called the "speed of light").
Or the equations of motion of Class 1, where cw = c
In the previous article it was shown that μ is the minimum wave frequency in such fields. In this paper we will denote it ν min .
Why is the universal velocity limit often called the speed of light? Waves with an equation of class 0 move with speed c w . Light (a general term for electromagnetic waves of any frequency), moving through an empty space, satisfies a relativistic equation of class 0, therefore waves of light (and waves of any relativistic fields satisfying a relativistic equation of class 0) move with velocity c.
Moreover, in the same article we saw that if a field of class 1 has a wave with amplitude A, frequency ν, wavelength λ and equilibrium state Z 0 then the equation of motion requires that the frequency and wavelength Were related to the magnitude Μ = ν min appearing in the equations, the formula
This is a Pythagorean formula – it can optionally be represented as a triangle, like in rice . 1. The minimum frequency of any wave is ν min and the assignment ν = ν min (and, therefore, as λ → ∞) corresponds to the contraction of the triangle to the vertical line (Figure 1, at the bottom). It is also possible to obtain a similar relation of class 0, making μ = ν min zero. Then you can extract the square root, and get
This is already Triangle, compressed to a horizontal line (Figure 1, right). In this case, the minimum frequency is zero. The field can oscillate as slowly as you like.
There are no restrictions on A. But this is because we ignore quantum mechanics. It's time to study relativistic quantum fields.
Relativistic quantum fields
The real world is quantum-mechanical, so the amplitude A can not be any. It takes discrete values proportional to the square root of n, a non-negative integer denoting the number of quanta of oscillations in the wave. The energy stored in the wave is
Where h – Planck's constant, necessarily appearing where quantum mechanics matters. In other words, the energy associated with each vibration quantum depends only on the frequency of the wave oscillations, and is equal to
This relationship was first proposed, specifically for light waves, by Einstein in 1905, in his explanation of the photoelectric effect.
But remember our Pythagorean ratio of frequency and wavelength. If we multiply it by h 2 we get that for a field quantum of class 1
It looks familiar. We already know that any object in Einstein's theory of relativity must satisfy an equation describing its energy, momentum and mass:
Another Pythagorean relationship. The minimum energy of the object is mc 2 which recalls the statement about the minimum frequency that a wave of class 1, ν min can possess. We may be tempted to assume that, probably, for a quantum of a relativistic field
The first equation first appeared in the work of Louis De Broglie in 1924 – almost 20 years after Einstein. Why did it take so long? I do not know.
Does this make sense? As we noted, relativistic fields of class 0 include electric fields, and their waves are electromagnetic waves, that is, light. The version of the formula (*) that we obtain for quantums of class 0 is the same as for fields of class 1, in which μ = ν min equals zero – that is, m = 0. Extract the square root, And we get
Or Einstein's equation for massless particles. And quanta of electromagnetic waves (including all kinds of light: visible, ultraviolet, infrared, radio waves, gamma radiation, etc., differing only in frequency, and therefore quantum energy) and truth will be massless particles – as soon as we apply the indicated Above a pair of equations (**) and (***). These are photons.
From equation (***) we can finally calculate the mass of a particle. Each particle that has a mass is a field quantum of class 1. The minimum frequency of such waves is ν min . The minimum energy of one quantum of such a wave is h, multiplied by the frequency. And the mass of the particle is just the minimum energy divided by c 2 .
If we want to understand where the mass of a particle comes from, we need to understand what defines ν min and why there is a minimum frequency at all. For particles such as electrons and quarks, this is completely unclear, but it is known that the Higgs field plays an important role in this.
Conclusion: particles of nature are quanta of relativistic quantum fields. Massless particles are quanta of waves of fields satisfying an equation of class 0. Those possessing mass correspond to fields of an equation of class 1. There are many details, but this fact is one of the basic fundamental properties of our world.
Are these quanta behaving As a particle?
We imagine particles, like particles of dust or grains of sand. Quanta in this sense are not particles – they are waves that have a minimum energy and amplitude for a certain frequency. But they behave so similar to particles that we can be forgiven for using the word "particle" in their description. Let's see why this is so.
If you lift a wave in the water and let it pass through the stones that lie not deep beneath the surface, part of the wave will cross the line of stones, and the part will reflect, as shown in Fig. 3. Which part of the wave will cross the line depends on the shape of the stones, their proximity to the surface, and so on. But the point is that part of the wave is transmitted through the stones, and some will be reflected. Part of the wave energy will go in the same direction, part will go in the opposite direction.
But if you send one photon in the direction of the reflecting glass, this photon either passes through it, or is reflected (Figure 4). More precisely, if you measure the behavior of a photon, you will find out whether it is reflected or transmitted. If you do not measure it – it will be impossible to say what happened. Welcome to the swamp of quantum mechanics. A photon is a quantum. Its energy can not be divided into the part that went through the glass, and the part that was reflected – because then on each side there will be less than one quantum, which is forbidden. (Fine: glass does not change the photon frequency, so energy can not be divided between two or more quanta of lower frequencies). So a photon, although it's a wave, behaves like a particle in this case. It either reflects off the glass, or it does not. It is reflected, or not – this quantum mechanics does not predict. It gives only the probability of reflection. But she predicts that, whatever happens, the photon will travel as a single whole and retain its identity.
And what about the two photons? It depends. For example, if photons are emitted at different times from different places, the observer will see two quanta separated in space, and probably moving in different directions (Fig. 5). They may have different frequencies.
Fig. 5: independent quanta
In a special case, when two photons are emitted together and ideally synchronously (as in lasers), they behave as shown in Fig. 6. If we send a combination of two photons to the glass, then not two things can happen, but three things. Either both photons will pass through the glass, or both will be reflected, or one will pass, and the other will be reflected. Glass will reflect 0, 1 or 2 photons – there are no other options. In this sense, the quanta of light again behave like particles, like small balls – if you throw two balls into a lattice in which there are holes, then 0, 1 or 2 balls can be reflected from the grid, and 0, 1 or 2 balls pass through the holes . There is no possibility, in which 1,538 balls will be reflected from the grid.
But these are photons that, having no mass, are obliged to move at the speed of light and E = p c. What about particles with a mass, like electrons? Electrons are quanta of an electric field, and like photons, they can be emitted, absorbed, reflected or transmitted as a whole. They have a certain energy and momentum,
where m e is the mass of an electron. The difference between electrons and photons is that they move more slowly than light, so they can rest. A sketch of such an event (in quantum mechanics, because of the uncertainty principle, nothing can be truly static) of a stationary electron is given in Fig. 7. It is a wave of the minimum frequency, obtained by assigning to the wavelength of a very large, almost infinite, value. Therefore, the spatial waveform in Fig. Does not show any convolutions – it just fluctuates in time.
So, yes, in fact, quanta behave very much like particles, and therefore, electrons, quarks, neutrinos, photons, gluons, W particles and Higgs particles will not be called "particles" A catastrophic deception. But the word "quantum" is suitable for this better – because it is precisely quanta.
Than fermions and bosons differ from each other
• All elementary particles are divided into fermions and bosons.
• Fermions (including electrons, quarts and neutrinos) satisfy the Pauli exclusion principle – two fermions of the same type can not do the same thing.
• Bosons (including photons, W and Z particles, gluons, gravitons and Higgs particles) are different: two or more bosons of the same type can do the same thing.
That's why photons can be made from lasers – because they are bosons, they can Be in the same state and generate a powerful beam of one light. But the laser can not be made from electrons that are fermions.
How does this difference manifest itself in the language of mathematics? It turns out that the formulas given by me are suitable for bosons, and for fermions they need to be changed – slightly, but with great consequences. For bosons we will have:
Which means that the energy of each quantum is h ν. This implies that boson quanta can do the same thing; When n is greater than 1, for a bosonic field the wave will consist of several quanta, oscillating and moving together. But for fermions:
which means each quantum has energy h ν. That implies that boson quanta can be made to do exactly the same thing; When n is greater than 1, the boson field has a wave made of more than one. But for fermions
The energy of one quantum is still equal to h ν, so that the whole discussion of particles and their energies, momentum and masses remains in effect. But the number of quanta in an electron wave can be only 0 or 1. Ten electrons, unlike ten photons, can not be organized into one wave of greater amplitude. Therefore, there are no fermion waves consisting of a large number of fermions, oscillating and moving together