2. Understand the physics of particles: 2) quantum ball on the spring

3. Understand the physics of particles: 3) waves, the classical form

4. Understand the physics of particles: 4) waves, the classical equation of motion

5. We are sorting out the physics of particles: 5) quantum waves

In fact, we have already entered the territory of fields some time ago, but I did not warn you about it – I wanted to concentrate on the waves that arise in these fields. Describing how the waves behave, we expressed their shape and time dependence using the function Z (x, t). Well, Z (x, t) is a field. It is a function of space and time with the equation of motion that determines its behavior. A suitable function of the motion would be such that if Z increases or decreases at a certain point, then Z will decrease or increase at neighboring points a little later. This feature allows the waves to turn out to be among the solutions of the equation.

In this article we look at several examples of fields Z (x, t) whose equations of motion resolve the presence of waves. The physical interpretation of these fields will be very different. They describe different properties of different materials. But the equations that they satisfy, and the waves that they demonstrate, will satisfy very similar mathematics, and they will behave themselves, too, despite their different physical origins. This in the future will become a very important moment.

And then we will do something radical – consider the fields in the context of the special theory of relativity. As Einstein showed, if you correct space and time and make them behave differently than most people expect, you will get a new type of field, such that its physical interpretation will not be a property of anything but an independent physical object.

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## Ordinary fields describing ordinary things

The field Z (x, t) can represent a set of different physical quantities. For example:

• The height of the rope stretched across the room.

• The height of the water in the river.

• Density of a crystal or gas.

• Position of atoms in a magnet.

• Wind speed.

• Temperature, density or air pressure.

In any of these cases, there exists a field Z (x, t): the altitude field, the density field, the orientation field, the wind field, the temperature field. Its value in the form of a function of space and time informs us of the height, density, orientation, wind speed or temperature of any medium – rope, river, crystal, gas, magnet, air – in all places at any time. Its equation of motion shows how, in principle, Z (x, t) can behave. It also shows how to predict the behavior of Z (x, t) in the future, if we know exactly its behavior in the present and in the past.

Each example has a field and environment, and we must not confuse the field with the environment. The field simply describes and characterizes one of the many properties of the corresponding environment. In completely different environments, fields with very similar waves can be very similar – we'll see this again.

Once again I will explain the moment, often causing misunderstanding. In general, the field can have nothing in common with the physical distance in space. Yes, in articles 3 and 4 I used an example of oxen on a rope to illustrate what Z (x, t) can be, because it's beautiful and intuitive. I also often plotted Z (x, t) for waves. This can give you the false impression that Z (x, t) is always associated with waves that cause a physical object (such as a rope) to move a distance Z in a direction perpendicular to the x axis. But this is not so, and three of our four examples show us.

## The pitch field of the rope

First, consider our initial example of waves, the oscillating rope. In this case, in the role of Z (x, t) there is an altitude field, which we call H (x, t). It tells us the height of the rope at each point of space on the x-axis along the rope at any time t. If the rope is at the equilibrium height H _{ 0 }then H (x, t) = H _{ 0 }. The height field is a constant in space and time. If a simple wave moves through it, then the height field will be described by our famous wave formula from previous articles.

If we know H (x, t), we know the height of the rope at all points of space and time. If we know what the rope is doing now and what it has done quite recently, using the equation of motion, we can predict what it will do in the future. It does not tell us much about the rope itself. The height field gives us only what its name implies: the height of the rope. A rope is a physical medium whose height is represented as a field H (x, t); It does not tell us anything about the color of the rope, its thickness, tension, material, etc.

In Fig. 1 I made for you a wave animation in the height field going from left to right. It may seem that I painted the same thing twice, first green, and then orange. But this is not the same thing. The orange curve is the rope itself, moving in physical space. A green curve is a graph representing what is happening with H (x, t), regardless of what H (x, t) (ie, height) means, or what kind of environment it is. And only in this case the green graph looks exactly the same as what is happening in the physical world. But in all other cases it will not be so.

* Fig. 1 *

## The lattice displacement field

Suppose we have a medium consisting of a crystal with atoms uniformly distributed at equal distances from each other. I drew them in Fig. 2 in one dimension – one can imagine similar situations with three dimensions, but for now this would be an unnecessary complication. Also, I marked every 10th atom in red, so that it's easier for you to track their movement. And I too greatly exaggerated the distance between atoms-imagine that there are several million atoms between every two red dots, and not 10.

Let's look at the displacement field D (x, t), telling us how much at time t the atom, Usually located at the equilibrium point x, moved from this point in the lattice. This means that in the case of the static state from which the animation begins, the field is everywhere zero, D (x, t) = 0, since all atoms are in their normal position. Then, on the animation, individual atoms begin to oscillate back and forth, and their movement, in general, spreads in the form of a wave going from left to right. At the top of the figure, the graph of the displacement field of the lattice D (x, t) shows how the field behaves when the wave passes. Note that the fields in Fig. 1 and 2 behave in a similar way, although the physical interpretation of the fields is very different.

* Fig. 2 *

## The field of magnetic orientation

What is a permanent magnet? It consists of a set of atoms, each of which serves as a tiny magnet with a tiny magnetic field, and all of them are aligned so that together they create a large magnetic field. The magnet is shown in Fig. 3, and in it every atom is directed upwards. In this case, the orientation field Θ (x, t) tells us how far at the time t the atom at the point x deviates from the vertical. Θ, in short, is the angle between the magnet of each atom and the vertical. The animation in Fig. 3 shows a wave in a magnet in which the directions of the atomic magnets oscillate left and right. Above the magnet is a green graph Θ (x, t); And again it looks exactly the same as in the previous cases.

## Air pressure field

Consider a molecular gas in a long tube. A measurement of x will be located along the tube. The molecules of the gas will move almost randomly, colliding with the walls of the tube and with each other. In equilibrium, the density (the number of molecules in a certain volume) and the pressure P (x, t) (the force acting on the surface of a small ball appearing at the point x at time t) are constant. But sound waves passing through the gas will cause pressure and density to oscillate, as shown in Fig. 4. Density and pressure periodically increase and decrease. The molecules move forward and backward, although on average they do not move at all, but the wave and its energy move from left to right through the gas. The graph of P (x, t) again looks very similar to the previous ones.

## Important Lessons

What can we learn from four examples showing waves of class 0? (The equation of motion has one value, cw, and all the waves in the corresponding field move with the speed cw. Different fields of class 0 have different values of cw). We can find out that the same field behavior can appear from the physically different fields existing in physically different environments. Despite the different origin, the waves in the altitude field, in the lattice displacement field, in the field of magnetic orientation and in the gas pressure field look identical from the field viewpoint. They satisfy the same type of equation of motion and the same numerical relationship between frequency and wavelength.

Small font: strictly speaking, if you create waves of a fairly small length, you can still distinguish between the behavior of different media. Once the wavelengths are equal to the distance between the atoms of the rope, or the crystal or magnet, the wave equations that the waves will satisfy will be more complex than those recorded by us, and their details will make it possible to distinguish the environments from each other. But often in practical experiments we do not even come close to observing similar effects.

The result of this will be that the study of waves and their quanta associated with fields does not necessarily tell you what serves as an environment, Or what is the physical interpretation of the field – which of the properties of the medium it represents. And even if you somehow know that this field is of a certain type – say, the pressure field – you usually still can not say, based on its behavior, the pressure of what it represents. All that you can learn by studying waves is whether the field belongs to class 0 or class 1, and what is its value cw; Or find out that the field belongs to another class.

In some cases this is very bad; The field only transmits partial information about the environment. Sometimes it's quite convenient; Field is a more abstract and universal thing than the physical material described by him.

Therefore the field does not define the environment, and its behavior often does not depend on the details and properties of the environment. Because of what the question arises.

Can a physical field exist – with waves consisting of quanta moving in space and transferring energy – without any supporting medium?

## A field without medium?

You can not hear a song without a singer. But the song has some kind of independent existence; It sounds different, depending on who sings it, but there is something inherent in the song, some kind of abstract quality, thanks to which it can always be recognized. This abstract essence is the melody of the song. We can discuss, study, learn a melody, record it with a musical record, without even hearing it in the performance of the singer. Many of us can even hum it in our heads. Somehow the melody exists even if no one sings the song.

If in all the examples and all examples I give and that are intuitively understandable to you, the field describes the property among, whereas there can be a field without Environment? But somehow the fields do not depend on their environment, because many different fields can behave identically, despite the fact that they describe a variety of different properties of completely different media. So, probably, it is possible to abstract the field from the environment.

In fact, it is not only possible, it seems to be necessary. At the very least, it is necessary either to have no environment, or to have an environment that can not be created from ordinary matter, which is fundamentally different from all the media we have examined-in that it functions in such a way (according to all experiments) that it does not exist at all

One of several radical elements of Einstein's special theory of relativity from 1905 was the idea that for light waves – for decades considered waves in electric and magnetic fields (electromagnetic waves), and moving with one Other speed in empty space – there is no medium. There are only fields.

The hypothetical environment was called "ether"; Einstein argued that such a thing does not exist, and wrote down a set of equations for which it was not required. I note that there is still a debate (often more philosophical than physical) about whether or not you need to imagine the existence of some strange environment that is very different from ordinary matter. The key elements of the Einstein version of relativity (as opposed to the versions of Galileo and Newton) were the following:

• Space and time are not what you think of them . Different observers, evenly moving relative to each other, will disperse in their estimates of the length of objects and time intervals between events (and these discrepancies can be accurately measured).

• There is a universal speed limit, with; Any observer measuring the speed of an object with respect to it will find that this velocity is less than or equal to с.

• In such a world, certain fields – "relativistic fields" – can exist without a medium, and they satisfy special equations of motion. The simplest relativistic field satisfies the equations of motion of the classes 0 or 1, with the velocity of the wave cw mentioned in the equation of motion taking the value c.

In short, there exist relativistic fields of class 0 satisfying the equation

$$

Light, and all electromagnetic waves, is the most famous, but not unique example – therefore, "c" is often called the "speed of light". And there are relativistic fields of class 1 that satisfy the equation

$$

We will see their examples in the next article. Note that relativity does not impose restrictions on μ (except for the need for μ ^{ 2 } to be positive) or on Z _{ 0 }. For relativistic fields, there are more complicated equations, but most of them in the description of simple physical processes are reduced to one of these two.

The relativistic fields are physically real and have a physical meaning in the Universe, that is:

Waves transfer energy and information from one place to another.

• Waves in one field can affect another field and change the physical processes that would take place in their absence.

But, unlike the fields, examples of which are given in this article, relativistic fields do not describe the property of any The usual physical environment consisting of something resembling ordinary matter, and, as far as is experimentally known, they do not describe the properties of anything at all. These fields, as far as we know today, are one of the fundamental elements of the universe.

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