So, let’s completely abstract ourselves and forget any classical definitions. Because with pin is a concept unique to the quantum world. Let's try to figure out what it is.
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Spin and angular momentum
Spin(from English spin– rotate) – the intrinsic angular momentum of an elementary particle.
Now let's remember what angular momentum is in classical mechanics.
Momentum is a physical quantity that characterizes rotational motion, more precisely, the amount of rotational motion.
In classical mechanics, angular momentum is defined as the vector product of a particle’s momentum and its radius vector:
By analogy with classical mechanics spin characterizes the rotation of particles. They are represented in the form of tops rotating around an axis. If a particle has a charge, then, when rotating, it creates a magnetic moment and is a kind of magnet.
However, this rotation cannot be interpreted classically. All particles, in addition to spin, have an external or orbital angular momentum, which characterizes the rotation of the particle relative to some point. For example, when a particle moves along a circular path (an electron around a nucleus).
Spin is its own angular momentum , that is, characterizes the internal rotational state of the particle regardless of the external orbital angular momentum. Wherein spin does not depend on external movements of the particle .
It is impossible to imagine what is rotating inside the particle. However, the fact remains that for charged particles with oppositely directed spins, the trajectories of motion in a magnetic field will be different.
Spin quantum number
To characterize spin in quantum physics, it was introduced spin quantum number.
Spin quantum number is one of the quantum numbers inherent in particles. Often the spin quantum number is simply called spin. However, it should be understood that the spin of a particle (in the sense of its own angular momentum) and the spin quantum number are not the same thing. The spin number is denoted by the letter J and takes a number of discrete values, and the spin value itself is proportional to the reduced Planck constant:
Bosons and fermions
Different particles have different spin numbers. So, the main difference is that some have a whole spin, while others have a half-integer. Particles with integer spin are called bosons, and half-integer ones are called fermions.
Bosons obey Bose-Einstein statistics, and fermions obey Fermi-Dirac statistics. In an ensemble of particles consisting of bosons, any number of them can be in the same state. With fermions, the opposite is true - the presence of two identical fermions in one system of particles is impossible.
Bosons: photon, gluon, Higgs boson. - in a separate article.
Fermions: electron, lepton, quark
Let's try to imagine how particles with different spin numbers differ using examples from the macrocosm. If the spin of an object is zero, then it can be represented as a point. From all sides, no matter how you rotate this object, it will be the same. With a spin of 1, rotating the object 360 degrees returns it to a state identical to its original state.
For example, a pencil sharpened on one side. A spin of 2 can be imagined as a pencil sharpened on both sides - when we rotate such a pencil 180 degrees, we will not notice any changes. But a half-integer spin equal to 1/2 is represented by an object, to return which to its original state you need to make a revolution of 720 degrees. An example would be a point moving along a Mobius strip.
So, spin- a quantum characteristic of elementary particles, which serves to describe their internal rotation, the angular momentum of a particle, independent of its external movements.
We hope that you will master this theory quickly and be able to apply the knowledge in practice if necessary. Well, if a quantum mechanics problem turns out to be too difficult or you can’t do it, don’t forget about the student service, whose specialists are ready to come to the rescue. Considering that Richard Feynman himself said that “no one fully understands quantum physics,” it is quite natural to turn to experienced specialists for help!
Spin is the most simple thing which can demonstrate the differences between quantum mechanics and classical mechanics. From the definition it seems that it is associated with rotation, but one should not imagine an electron or a proton as rotating balls. As with many other established scientific terms, it has been proven that this is not the case, but the terminology is already established. An electron is a point particle (zero radius). And spin is responsible for magnetic properties. If an electrically charged particle moves along a curved trajectory (including rotation), then a magnetic field is formed. Electromagnets work like this - electrons move along the wires of a coil. But spin is different from a classical magnet. Here's a nice animation:
If magnets are passed through a non-uniform magnetic field (note different shape northern and south poles magnet that sets the field), then depending on the orientation of the magnet (its magnetic moment vector), they will be attracted (repelled) from the pole with a greater concentration of magnetic field lines (the pointed pole of the magnet). In the case of a perpendicular orientation, the magnet will not deviate anywhere at all and will land in the center of the screen.
By passing electrons we will only observe an upward or downward deviation at the same distance. This is an example of quantization (discreteness). The electron spin can take only one of two values relative to a given magnet orientation axis – “up” or “down”. Since an electron cannot be mentally imagined (it has neither color, nor shape, nor even a trajectory of movement), as in all such animations, colored balls do not reflect reality, but I think the essence is clear.
If the electron deviates upward, then its spin is said to be directed “upward” (+1/2 is conventionally designated) relative to the axis of the magnet. If down, then -1/2. And it would seem that spin can be described by an ordinary vector indicating the direction. For those electrons where it was directed upward, they will deflect upward in the magnetic field, and for those electrons downward, they will deflect downward, respectively. But not everything is so simple! The electron is deflected up (down) the same distance relative to any magnet orientation. In the video above, it would be possible to change not the orientation of the magnets being passed through, but to rotate the magnet itself, which creates the magnetic field. The effect in the case of ordinary magnets would be the same. What will happen in the case of electrons - unlike magnets, they will always deviate by the same distance up or down.
If, for example, you pass a vertically located classical magnet through two magnets perpendicularly oriented relative to each other, then deflecting upward in the first, it will not deviate in the second at all - its magnetic moment vector will be perpendicular to the magnetic field lines. In the video above, this is the case when the magnet hits the center of the screen. The electron must deviate somewhere.
If we pass through the second magnet only electrons with spin up, as in the figure, then it turns out that some of them also have spin up (down) relative to another perpendicular axis. Right and left are in fact, but spin is measured relative to the chosen axis, so "up" and "down" is common terminology along with the axis indication. The vector cannot be directed immediately up and to the right. We conclude that spin is not a classical vector attached to an electron like the vector of the magnetic moment of a magnet. Moreover, knowing that the electron’s spin is directed upward after passing through the first magnet (we block those that deviate downward), it is impossible to predict where it will deviate in the second case: to the right or to the left.
Well, you can complicate the experiment a little more - block the electrons that deviate to the left and pass them through a third magnet, oriented like the first.
And we will see that the electrons will be deflected both up and down. That is, the electrons entering the second magnet all had a spin up relative to the orientation of the first magnet, and then some of them suddenly became spin down relative to the same axis.
Strange! If you pass classical magnets through such a design, rotated at the same arbitrarily chosen angle, then they will always end up at the same point on the screen. This is called determinism. Repeating the experiment with full compliance with the initial conditions, we should obtain the same result. This is the basis of the predictive power of science. Even our intuition is based on repeatability of results in similar situations. In quantum mechanics, it is generally impossible to predict where a particular electron will deviate. Although in some situations there are exceptions: if you place two magnets with the same orientation, then if the electron deflects upward in the first, then it will definitely deflect upward in the second. And if the magnets are rotated 180 degrees relative to each other and in the first the electron deviates, for example, downward, then in the second it will definitely deviate upward. And vice versa. The spin itself does not change. This is already good)
What general conclusions can be drawn from all this?
- Many quantities that could take any value in classical mechanics can only have some discrete (quantized) values in quantum theory. Besides spin, the energy of electrons in atoms is a prime example.
- Objects of the microworld cannot be assigned any classic characteristics until the moment of measurement. We cannot assume that the spin had any particular direction before we looked at where the electron deviated. This general position and it concerns all measured quantities: coordinates, speed, etc. Quantum mechanics . She claims that the objective classical world, independent of anyone, simply does not exist. demonstrates this fact most clearly. (observer) in quantum mechanics is extremely important.
- The measurement process overwrites (makes irrelevant) information about the previous measurement. If the spin is directed upward relative to the axis y, then it does not matter that previously it was directed upward relative to the axis x, it may turn out to be spin down relative to the same axis x subsequently. Again, this circumstance concerns not only the back. For example, if an electron is detected at a point with coordinates ( x, y, z) this generally does not mean that he was at this point before. This fact is known as “wave function collapse.”
- There are such physical quantities the values of which cannot be known simultaneously. For example, you cannot measure spin relative to the axis x and at the same time relative to the axis perpendicular to it y. If we try to do this simultaneously, the magnetic fields of the two rotated magnets will overlap and instead of two different axes we will get one new one and measure the spin relative to it. It will also not be possible to measure consistently due to the previous conclusion No. 3. It is too general principle. For example, position and momentum (velocity) also cannot be measured simultaneously with great accuracy - the famous Heisenberg uncertainty principle.
- It is impossible in principle to predict the result of a single measurement. Quantum mechanics only allows us to calculate the probabilities of a particular event. For example, you can calculate that in the experiment in the first picture, when the magnets are oriented 90° to each other, 50% will deviate to the left and 50% to the right. It is impossible to predict where a particular electron will deviate. This general circumstance is known as the “Born rule” and is central to.
- Deterministic classical laws are derived from probabilistic quantum mechanical laws due to the fact that there are a lot of particles in a macroscopic object and probabilistic fluctuations are averaged out. For example, if in the experiment in the first picture a vertically oriented classical magnet is passed through, then 50% of its constituent particles will “pull” it to the right, and 50% to the left. As a result, he will not deviate anywhere. With other orientations of the magnet angles, the percentage changes, which ultimately affects the deflection distance. Quantum mechanics allows you to calculate specific probabilities and, as a consequence, you can derive from it a formula for the deflected distance depending on the orientation angle of the magnet, usually obtained from classical electrodynamics. This is how classical physics is derived and is a consequence of quantum physics.
Yes, the described actions with magnets are called the Stern-Gerlach experiment.
There is a video version of this post and an elementary introduction to quantum mechanics.
© Martyr of Science.
Accepted following designations:
- Vectors – in slightly bold letters bigger size than the rest of the text.W, g, A.
- explanations of the designations in the tables – in italics.
- integer indices – in bold, regular size.m, i, j .
- non-vector variables and formulas – in slightly larger italics:q,
r,
k,
sin,
cos .
Moment of impulse. School level.
The angular momentum characterizes the amount of rotational motion. This is a quantity that depends on how much mass is rotating, how it is distributed relative to the axis of rotation, and at what speed the rotation occurs.
Momentum of momentum of a rotating axisZdumbbells made from two mass ballsm, each of which is located at a distancelfrom the axis of rotation, with the linear speed of the ballsV, is equal to:
M= 2·m·l·V ;
Well, of course, the formula says 2 because the dumbbell has two balls.
Moment of impulse. University level.
MomentumL material point (angular momentum, angular momentum, orbital momentum, angular momentum) relative to some origin is determinedthe vector product of its radius vector and momentum:
L= [ r X p]
Where r- radius vector of the particle relative to the selected fixed reference point in the given reference system,p- momentum of the particle.
For several particles, angular momentum is defined as the (vector) sum of the following terms:
L= Σ i[ r i X p i]
Where r i ,
p i- radius vector and momentum of each particle entering the system, the angular momentum of which is determined.
In the limit, the number of particles can be infinite, for example, in the case solid with a continuously distributed mass or a generally distributed systemthis can be written as
L= ∫ r xd p
where d p- momentum of an infinitesimal point element of the system.
From the definition of angular momentum it follows that its additivity both for a system of particles in particular and for a system consisting of several subsystems is satisfied:
L Σ= Σ iL i
Experience of Stern and Gerlach.
In 1922, physicists performed an experiment in which it turned out that silver atoms have their own angular momentum. Moreover, the projection of this angular momentum onto the axisZ(see figure) turned out to be equal to either some positive value or some negative value, but not zero. This cannot be explained by the orbital angular momentum of the electrons in the silver atom. Because the orbital moments would necessarily give, among other things, a zero projection. And here there are strictly plus and minus, and nothing at zero. Subsequently, in 1927, this was interpreted as proof of the existence of spin in electrons.
In the experiment of Stern and Gerlach (1922), by evaporating atoms of silver or another metal in a vacuum furnace using thin slits, a narrow atomic beam is formed (Fig.).
This beam is passed through a non-uniform magnetic field with a significant magnetic induction gradient. Magnetic field inductionBin experiment it is large and directed along the axisZ. Atoms flying in the magnet gap along the direction of the magnetic field are acted upon by a forceF z, caused by the induction gradient of a non-uniform magnetic field and depending on the magnitude of the projection of the magnetic moment of the atom onto the direction of the field. This force deflects a moving atom in the direction of the axisZ, and during the passage of the magnet, the moving atom is deflected the more, the greater the magnitude of the force. In this case, some atoms are deflected upward and others downward.
From the standpoint of classical physics, silver atoms flying through a magnet should have formed a continuous wide mirror strip on a glass plate.
If, as quantum theory predicts, spatial quantization takes place, and the projection of the magnetic momentp Z M atom takes only certain discrete values, then under the influence of forceFZthe atomic beam must split into a discrete number of beams, which, settling on a glass plate, give a series of narrow discrete mirror strips of deposited atoms. This is exactly the result observed in the experiment. There was only one thing: there was no stripe in the very center of the plate.
But this was not yet the discovery of spin in electrons. Well, a discrete series of angular momentum of silver atoms, so what? However, scientists continued to think why is there no stripe in the center of the plate?
A beam of unexcited silver atoms split into two beams, which deposited two narrow mirror strips on a glass plate, shifted symmetrically up and down. Measuring these shifts made it possible to determine the magnetic moment of an unexcited silver atom. Its projection onto the direction of the magnetic field turned out to be equal to+ μ B or -μ B. That is, the magnetic moment of an unexcited silver atom turned out to be strictly Not equal to zero. There was no explanation for this.
However, it was known from chemistry that the valence of silver is equal to +1
. That is, there is one active electron in the outer electron shell. And the total number of electrons in an atom is odd.
Electron spin hypothesis
This contradiction between theory and experience was not the only one discovered in various experiments. The same difference was observed when studying the fine structure of the optical spectra of alkali metals (they, by the way, are also monovalent). In experiments with ferromagnets, an anomalous value of the gyromagnetic ratio was discovered, differing from the expected value by a factor of two.
In 1924 Wolfgang Pauli introduced a two-component internal degree of freedom to describe the emission spectra of the valence electron in alkali metals.
Once again, it’s striking how Western scientists easily come up with new particles, phenomena, and realities to explain old ones. In the same way, the Higgs boson was introduced to explain mass. Next will be the Schmiggs boson to explain the Higgs boson.
In 1927, Pauli modified the recently discovered Schrödinger equation to take into account the spin variable. The equation modified in this way is now called the Pauli equation. With this description, the electron has a new spin part of the wave function, which is described by a spinor - a “vector” in an abstract two-dimensional spin space.
This allowed him to formulate the Pauli principle, according to which in some system of interacting particles each electron must have its own non-repeating set of quantum numbers (all electrons are in different states at each moment of time). Since the physical interpretation of the electron's spin was unclear from the very beginning (and this is still the case), in 1925 Ralph Kronig (assistant of the famous physicist Alfred Lande) suggested that the spin was a result of the electron's own rotation.
All these difficulties of quantum theory were overcome when, in the fall of 1925, J. Uhlenbeck and S. Goudsmit postulated that the electron is the carrier of its “own” mechanical and magnetic moments, not related to the electron’s motion in space. That is, it has a spinS = ½
ћ
in units of the Dirac constantћ
, and a spin magnetic moment equal to the Bohr magneton. This assumption was accepted by the scientific community, since it satisfactorily explained known facts.
This hypothesis is called the electron spin hypothesis. This name is related to the English wordspin, which translates as “circling”, “spinning”.
In 1928, P. Dirac further generalized the quantum theory to the case of relativistic particle motion and introduced a four-component quantity - the bispinor.
Relativistic quantum mechanics is based on the Dirac equation, originally written for a relativistic electron. This equation is much more complex than the Schrödinger equation in its structure and the mathematical apparatus used to write it. We will not discuss this equation. Let's just say that from the Dirac equation the fourth, spin quantum number is obtained as “naturally” as the three quantum numbers when solving the Schrödinger equation.
In quantum mechanics, the quantum numbers for spin do not coincide with the quantum numbers for the orbital momentum of particles, which leads to a non-classical interpretation of spin. In addition, the spin and orbital momentum of particles have a different relationship with the corresponding magnetic dipole moments that accompany any rotation of charged particles. In particular, in the formula for spin and its magnetic moment, the gyromagnetic ratio is not equal 1
.
The concept of electron spin is used to explain many phenomena, such as the arrangement of atoms in periodic table chemical elements, fine structure of atomic spectra, Zeeman effect, ferromagnetism, and also to substantiate the Pauli principle. A newly emerging field of research called "spintronics" deals with the manipulation of charge spins in semiconductor devices. In nuclear magnetic resonance the interaction of radio waves with the spins of nuclei is used, allowing spectroscopy of chemical elements and obtaining images internal organs in medical practice. For photons as particles of light, spin is related to the polarization of light.
Mechanical spin model.
In the 20-30s of the last century, many experiments were carried out that proved the presence of spin in elementary particles. Experiments have proven the reality of spin as a moment of rotation. But where does this rotation come from in an electron or proton?
Let's assume in the simplest way that an electron is a tiny solid ball. We assume that this ball has a certain average density and certain physical parameters close to the known experimental and theoretical values of a real electron. We have experimental values:
Electron rest mass:m e
Electron spin S e = ½
ћ
As the linear size of the object, we take its Compton wavelength, confirmed both experimentally and theoretically. Compton electron wavelength:
Obviously this is the diameter of the object. The radius is 2 times smaller:
We have theoretical quantities obtained from mechanics and quantum physics.
1) Calculate the moment of inertia of the objectI e
. Since we do not know its shape reliably, we introduce correction factorsk e, which, depending on its shape, can theoretically range from almost 0,0
(needle rotating around the long axis) until 1,0
(with the exact shape of a long dumbbell as in the picture at the beginning of the article or a wide but thin donut). For example, a value of 0.4 is achieved with the exact shape of a ball. So:
2) From the formula S = I· ω , we find the angular velocity of rotation of objects:
3) This angular velocity corresponds to linear velocityV"surface" of the electron:
Or
V = 0,4 c;
If we take, as in the figure at the beginning of the article, an electron shaped like a dumbbell, then it turns out
V = 0,16 c;
4) We perform the calculations for the proton or neutron in a completely similar way. The linear velocity of the “surface” of a proton or neutron for the ball model is exactly the same, 0.4c:
5) Draw conclusions. The result depends on the shape of the object (coefficientkwhen calculating the moment of inertia) and from the coefficients in the formulas for electron or proton spins (½). But, whatever one may say, on average it turns outabout, close to the speed of light. Both the electron and the proton. No more than the speed of light! A result that can hardly be called accidental. We made “meaningless” calculations, but got an absolutely meaningful, highlighted result!
It's not like that, guys! - said Vladimir Vysotsky. This is not a signal, this is a dilemma: either - or! Either something in half, or something in pieces. Einstein and Schrödinger make these arguments meaningless, since according to Einstein, at speeds on the order of the speed of light, mass grows to infinity, and according to Schrödinger, they have neither shape nor size. However, everything in the world is “relative” and it is unknown what and who deprives whom of meaning. The theory of Gukuum has the answer according to which wave vortices - electrons, in Gukuum spin at the linear speed of light! Actually, mass - it always moves and always exclusively at the speed of light. An electron and a proton, every element in them, every point moves along its own closed trajectory and at no other than the speed of light. This is precisely the real and simple meaning of the formula:
This is practically double the formula for the kinetic energy of the wave. Why doubled? – Because in an elastic wave, half of the energy is kinetic, and the second half of the energy is hidden, potential, in the form of deformation of the medium in which the wave propagates.
Phrases explaining electron spin.
What is the physical nature of the presence of spin in an electron if it is not explainable from a mechanical point of view? There is no answer to this question not only in classical physics, but also in the framework of non-relativistic quantum mechanics, which is based on the Schrödinger equation. Spin is introduced in the form of some additional hypothesis necessary to harmonize experiment and theory.
Reasoning about form or internal structure elementary particles, such as the electron, are easily classified as “meaningless” in modern physics. Since you can’t see them with your eyes, then there’s nothing to ask! Microbes were born with the invention of the microscope (Mikhail Genin). Attempts at such reasoning always end with the words that,
Phrase No. 1.
The laws and concepts of classical physics cease to apply in the microworld.
If the location of the object itself is unknown, thisΨ
-function, then what can we say about its structure? Smeared - and that's it. There is no device.
The same is said about the physical meaning of angular momentum - the spin of the electron (proton). There seems to be rotation, there is also spin, but
Phrase No. 2.
Asking what this rotation looks like “does not make sense.”
There are analogies in the macro world. Let's say we want to ask an oligarch: how did you earn your billions? Or where do you store the stolen goods? - And they answer you: your question makes no sense! A secret sealed with seven seals.
Phrase No. 3.
The electron spin has no classical analogue.
That is, the spin seems to have some kind of analogue, but it does not have a classical analogue. It seems to characterize the internal property of a quantum particle associated with the presence of an additional degree of freedom. The quantitative characteristic of this degree of freedom is spinS= ½
ћ
is the same value for an electron as, for example, its massm 0
and charge -
e. However, spin is actually rotation, it is the moment of rotation and is manifested in experiments.
Phrase No. 4.
Spin is introduced in the form of an additional hypothesis that does not follow from the basic principles of the theory, but is necessary to harmonize experiment and theory
.
Phrase No. 5.
Spin is some internal property, like mass or charge, that requires a special, as yet unknown justification
.
In other words. Spin (from the English spin - spin, rotation) is the intrinsic angular momentum of elementary particles, which has " quantum nature"and not related to the movement of the particle as a whole. Unlike orbital angular momentum, which is generated by the motion of a particle in space, spin is not associated with any motion in space. Spin is an allegedly internal, exclusively quantum characteristic that cannot be explained within the framework of mechanics.
Phrase No. 6.
However, despite all its mysterious origin, spin is an objectively existing and completely measurable physical quantity.
At the same time, it turns out that spin (and its projections onto any axis) can only take integer or half-integer values in units of the Dirac constantħ =
h/2π. Where h– Planck’s constant. For those particles that have half-integer spins, the spin projection is not equal to zero.
Phrase No. 7.
There is a space of states that are in no way related to the movement of a particle in ordinary space. A generalization of this idea in nuclear physics led to the concept of isotopic spin, which operates in a "special isospin space".
As they say, just grind and grind!
Subsequently, when describing strong interactions, internal color space and the quantum number “color” is a more complex analogue of spin.
That is, the number of mysteries increased, but they were all solved by the hypothesis that there is a certain space of states that are not associated with the movement of a particle in ordinary space.
Phrase No. 8.
So, in the most in general terms we can say that the electron’s own mechanical and magnetic moments appear as a consequence of relativistic effects in quantum theory.
Phrase No. 9.
Spin (from the English spin - twirl, rotation) is the intrinsic angular momentum of elementary particles, which has a quantum nature and is not associated with the movement of the particle as a whole.
Phrase No. 10.
The existence of spin in a system of identical interacting particles is the cause of a new quantum mechanical phenomenon that has no analogue in classical mechanics: exchange interaction.
Phrase 11.
Being one of the manifestations of angular momentum, spin in quantum mechanics is described by the vector spin operator ŝ, the algebra of whose components completely coincides with the algebra of orbital angular momentum operators
l
. However, unlike orbital angular momentum, the spin operator is not expressed in terms of classical variables, in other words, it is only a quantum quantity.
A consequence of this is the fact that spin (and its projections onto any axis) can take not only integer, but also half-integer values.
Phrase 12.
In quantum mechanics, the quantum numbers for spin do not coincide with the quantum numbers for the orbital momentum of particles, which leads to a non-classical interpretation of spin.
As they say, if you repeat something often, you begin to believe it. Now they say, democracy, democracy, rule of law. And people get used to it and begin to believe.
Also implicitly used is the translation from English word"spin" - from English. rotate. They say the English know the meaning of the spin, it’s just that the translators can’t translate it sensibly.
Electron structure.
As an attempt to google the size of an electron shows, this is also the same mystery for all physicists as the nature of the electron’s spin. Try it, and you won’t find it anywhere, neither in Wikipedia nor in the Physical Encyclopedia. A variety of figures are being put forward. From fractions of a percent the size of a proton, to thousands of proton sizes. And without knowing the size of the electron, or even better, the structure of the electron, it is impossible to understand the origin of its spin.
Now let’s approach the explanation of spin from the position of a structural electron. From the perspective of the elastic universe theory. This is what an electron looks like.
What is shown here are not hard rings or bagels, but wave rings. That is, waves running in a circle, mathematics gives such a solution. Spinning in circlesat the speed of light, and (!) adjacent rings move in opposite directions. Actually, this figure is an illustration of the formula for energy distribution inside an electron:
Those interested can easily check this formula.
Hereq– radial coordinate.
It is this rotation of the component rings that creates the total non-zero internal angular momentum - the spin of the electron. This is the key to the appearance of spin, which still remains a mystery in conventional science. True, no one actually seeks to solve this riddle, but this is a separate question.
It is this rotation of neighboring rings in opposite directions that, firstly, gives the convergence of the integral over the moment of rotation, and secondly, creates a discrepancy between the magnetic moment and spin.
This (approximate) picture shows only the main, closest rings; there are an infinite number of them. The entire object is a single whole, very stable, no part of it can be removed. And this whole is elementary particle, electron This is not fiction, not fantasy, not adjustment. This is, once again, strict mathematics!
Let those who believe that in a hydrogen atom (the simplest case) have an electron rotating around the nucleus do not be frightened by surprise. No, it does not rotate as a whole around the core. It’s just that an electron is a cloud, a real wave cloud, and it is such even when it is single and free. It's just that the nucleus of a hydrogen atom is inside an electron.
Explanation of the spin phenomenon.
And then all that remains is to calculate the angular momentum of this complex structure of wave donuts.
The angular momentum of an electron is determined as follows.
- There are energy distributions in the electron. When moving from layer to layer, the direction of energy movement changes to the opposite.
Thus, a plausible general formula for the projection of the angular momentum of all particles isMz, has the form:
R- previously determined value.
Under the integral sign there are four elements, which are highlighted in square brackets for clarity. The first square bracket contains elements of the electron mass density (difference from energy -c 2 in the denominator), taking into account the “layering” of the traveling wave on itself (r 2 in the denominator) and also taking into account the sign with which this mass will enter the angular momentum formula (functionsign). That is, depending on the direction of rotation of this element. The second square bracket is the distance from the axis of rotation - the axisZ. The third square bracket is the speed of movement of the mass element, the speed of light. The fourth is the element of volume. That is, this is the moment of impulse in its classical sense.
This equation for angular momentum is not declared to be quantitatively accurate, although this is not excluded. But it gives a correlation picture of the distribution of angular momentum. And as will become clear from the final results, such a definition of angular momentum also gives a good quantitative value of angular momentum (up to sign).
Full moment electron momentum after numerical integration:
Where L 1
And L 2
- Lame Gukuum coefficients (elasticity characteristics). They are provided on the specified website.
As analysis shows, this formula fits perfectly into known physical results. But its analysis is too voluminous to post here.
Comparison of theoretical and experimental particle sizes.
This is what this procedure is done for. Their known experimental spins and masses are substituted into the found theoretical formulas for the connection between particle sizes, their masses and spins. Then the (semi-)theoretical particle sizes are calculated and compared with the known experimental ones. It turned out to be more convenient.
The notations are introduced: loki (0,0), (1,0) and (1,1) are, respectively, electron, neutron and proton.
Theoretical values.
What is the relationship between the quantitiesλ 0.0, λ 1.0, λ 1.1to actual particle sizes? If you look at the theoretical distributions of particle density (or at the electron pattern), you can see that they are distributed in waves, with a decrease. The effective radius of each particle, up to the radius covering the bulk of the mass (this is 3-4 waves of density) is approximately equal to:
R 0,0 ≈ 2,5 π units q ;
R 1,0 ≈ 2 π units q ;
R 1,1 ≈ 2 π units q .
Where h- the usual, not crossed out Planck's constant.
Let him who has eyes see: the effective theoretical radii of locks (0,0), (1,0) and (1,1) are equal to almost exactly half the Compton wavelength of the electron, neutron and proton. That is, the Compton wavelength of a particle acts as its diameter.
The Compton wavelength is a linear size, and the mass of a particle characterizes the volume of the particle, that is, the linear size in a cube. As you can see, in the formula mass is in the denominator. For this reason, you shouldn’t take this formula too seriously. It would be, in our opinion, more correct to take the particle size as a value proportional to the following:
Where K– some coefficient of proportionality.
Initially, a proton is 12 times smaller (in size) than an electron and easily fits into the central hole of the electron. And then, when an electron interacts with a proton, the electron changes its state (in the proton’s field) and inflates another 40 times, which is not surprising.
This is how the hydrogen atom works (a yellow proton inside a gray electron).
As is known from official physics, the Compton size of an electron(R compt=1,21▪10 -10cm .) is approximately 40 times smaller than the size of a hydrogen atom (the first Bohr radius is:R boron=0,53▪10 -8cm .). This is an apparent contradiction with our theory, which needs to be eliminated and clarified. Or, when hydrogen is formed, the electron (like a wave cloud) changes its shape and stretches. At the same time, it envelops the proton. Or we need to reconsider what the Bohr radius is and what its physical meaning is. Physics in terms of particle sizes needs to be completely revised.
SPIN selling is a sales method developed by Neil Rackham and described in his book of the same name. The SPIN method has become one of the most widely used. Applying this method You can achieve very high personal sales results, Neil Rackham was able to prove this by conducting extensive research. And despite the fact that recently many have begun to believe that this sales method is becoming irrelevant, almost all large companies use the SPIN sales technique when training salespeople.
What is SPIN sales
In short, SPIN selling is a way of leading a client to a purchase by asking certain questions one by one; you are not presenting the product openly, but rather pushing the client to independently come to a decision to make a purchase. The SPIN method is best suited for so-called “long sales”, often these include sales of expensive or complex goods. That is, SPIN should be used when it is not easy for the client to make a choice. The need for this sales methodology arose primarily due to increased competition and market saturation. The client has become more discerning and experienced and this has required more flexibility from sellers.
The SPIN sales technique is divided into the following blocks of questions:
- WITH situational questions (Situation)
- P problematic issues (Problem)
- AND compelling questions (Implication)
- N guiding questions (Need-payoff)
It’s worth noting right away that SPIN sales are quite labor-intensive. The point is that in order to put this technique into practice, you need to know the product very well, have good experience sales of this product, such a sale itself takes a lot of time from the seller. Therefore, SPIN sales should not be used in the mass segment, for example in, because if the purchase price is low and the demand for the product is already high, then there is no point in spending a lot of time on long communication with the client, it is better to spend time on advertising and.
SPIN sales are based on the fact that the client direct offer the seller often includes a defense mechanism of denial. Buyers are pretty tired of being constantly being sold something and reacting negatively to the very fact of the offer. Although the product itself may be needed, it’s just that at the time of presentation the client thinks not that he needs the product, but that why is he being offered it? Using the SPIN sales technique forces the client to accept independent decision about the purchase, that is, the client does not even understand that his opinion is being controlled by asking the right questions.
SPIN sales technique
The SPIN sales technique is a sales model based not only on, but on theirs. In other words, to successfully use this sales technique, the seller must be able to ask the right questions. To begin with, let’s look at each group of SPIN sales technique questions separately:
Situational questions
This type of question is needed to fully identify his primary interests. The purpose of situational questions is to find out the client’s experience of using the product you are going to sell, his preferences, and for what purposes it will be used. As a rule, about 5 open questions and several clarifying questions are required. Based on the results of this block of questions, you should liberate the client and set him up for communication, which is why you should pay attention open questions, and also use . In addition, you must collect all the necessary information to pose problematic questions in order to effectively identify key needs worth using. As a rule, the block of situational questions takes the longest time. When you have received the necessary information from the client, you need to move on to problematic issues.
Problematic issues
By asking problematic questions, you must draw the client's attention to the problem. It is important at the stage of situational questions to understand what is important to the client. For example, if the client is always talking about money, then it would be logical to ask problematic questions regarding money: “Are you satisfied with the price you are paying now?”
If you haven't decided on your needs and don't know what problematic questions to ask. You need to have a number of prepared, standard questions that address various difficulties that the client may encounter. Your main goal is to identify the problem and the main thing is that it is important to the client. For example: a client may admit that he is overpaying for the services of the company he is using now, but he does not care about this, since the quality of services is important to him, not the price.
Probing Questions
This type of question is aimed at determining how important this problem is for him, and what will happen if it is not solved now. Extractive questions should make it clear to the client that by solving the current problem, he will benefit.
The difficulty with elicitation questions is that they cannot be thought through in advance, unlike the others. Of course, with experience, you will develop a pool of such questions, and you will learn to use them depending on the situation. But initially, many sellers who are mastering SPIN selling have difficulty asking such questions.
The essence of elicitation questions is to establish for the client the investigative connection between the problem and its solution. Once again, I would like to note that in SPIN sales, you cannot tell the client: “our product will solve your problem.” You must formulate the question so that in response the client himself says that he will be helped to solve the problem.
Guiding Questions
Guiding questions should help you; at this stage, the client should tell you for you all the benefits that he will receive from your product. Guiding questions can be compared to a positive way to close a transaction, only the seller does not summarize all the benefits that the client will receive, but vice versa.
Contrary to popular belief, spin is a purely quantum phenomenon. Moreover, spin has nothing to do with the “rotation of a particle” around itself.
To understand correctly what spin is, let's first understand what a particle is. From quantum field theory we know that particles are those of a certain type of excitation of the primary state (vacuum) that have certain properties. In particular, some of these excitations have mass that reminds us very much of the traditional mass from Newton's laws. Some of these excitations have a non-zero charge, which is very similar to the charge from Coulomb's laws.
In addition to the properties that have their analogues in classical physics (mass, charge), it turns out (in experiments) that these excitations must have one more property that has absolutely no analogues in classical physics. I will emphasize this again: NO analogues (this is NOT particle rotation). During the calculations, it turned out that this spin is not a scalar characteristic of the particle, like mass or charge, but another (not vector).
It turned out that spin is internal characteristic such excitation, which in its mathematical properties (transformation law, for example) is very similar to the quantum moment.
Then it went on and on. It turned out that the properties of such excitations, their wave functions, very much depend on the magnitude of this very spin. Thus, a particle with spin 0 (for example, the Higgs boson) can be described by a one-component wave function, and for a particle with spin 1/2 there must be a two-component function (vector function) corresponding to the projection of the spin onto a given 1/2 or -1/2 axis. It also turned out that spin carries with it a fundamental difference between particles. Thus, for particles with an integer spin (0, 1, 2), the Bose-Einstein distribution law holds, which allows as many particles as desired to be in one quantum state. And for particles with half-integer spin (1/2, 3/2), due to the Pauli exclusion principle, the Fermi-Dirac distribution operates, which prohibits two particles from being in the same quantum state. Thanks to the latter, atoms have Bohr levels, because of this, connections are possible and, therefore, life is possible.
This means that spin specifies the characteristics of a particle and how it behaves when interacting with other particles. A photon has a spin equal to 1 and many photons can be very close to each other and not interact with each other, or photons with gluons, since the latter also have spin = 1, and so on. And electrons with a spin of 1/2 will repel each other (as they teach in school - from -, + from +.) Did I understand correctly?
And another question: what gives the particle itself the spin or why does the spin exist? If spin describes the behavior of particles, then what does the spin itself describe and make possible (any bosons (including those existing hypothetically) or so-called strings)?
