In 1922, German physicists O. Stern and W. Gerlach set up experiments aimed at measuring magnetic moments Pm atoms of various chemical elements. For chemical elements that form the first group of the periodic table and have one valence electron, the magnetic moment of the atom is equal to the magnetic moment of the valence electron, i.e. one electron.
The idea of the experiment was to measure the force acting on an atom in a highly inhomogeneous magnetic field. The inhomogeneity of the magnetic field must be such that it affects distances on the order of the size of an atom. Only in this way it was possible to obtain a force acting on each atom separately.
The scheme of experience is shown in fig. 7.9. In a flask with a vacuum, 10–5 mm Hg. Art., heated silver ball TO, up to the evaporation temperature.
Rice. 7.9 Fig. 7.10
Silver atoms flew at a thermal velocity of about 100 m/s through slit diaphragms IN and, passing through a sharply inhomogeneous magnetic field, fell on a photographic plate A.
If the angular momentum of an atom (and its magnetic moment) could take arbitrary orientations in space (i.e., in a magnetic field), then one could expect a continuous distribution of hits of silver atoms on a photographic plate with a high hit density in the middle. But in experience, completely unexpected results were obtained: on a photographic plate, two sharp stripes - all atoms deviated in a magnetic field in a twofold way, corresponding only to two possible orientations of the magnetic moment (Fig. 7.10).
This proved the quantum nature of the magnetic moments of electrons . Quantitative analysis showed that the projection of the magnetic moment of the electron is equal to Bohr magneton :
.
Thus, for silver atoms, Stern and Gerlach obtained that magnetic moment projection atom (electron) to the direction of the magnetic field numerically equal to the Bohr magneton.
Recall that
.
The experiments of Stern and Gerlach not only confirmed the spatial quantization of angular momenta in a magnetic field, but also gave experimental confirmation that the magnetic moments of electrons Same consist of a certain number of "elementary moments", i.e. are of a discrete nature. The unit of measurement for the magnetic moments of electrons and atoms is Bohr magneton (ħ - unit of measurement of the mechanical moment of impulse).
In addition, a new phenomenon was discovered in these experiments. The valence electron in the ground state of a silver atom has an orbital quantum number l = 0 (s-
state). But at l = 0 
(the projection of the angular momentum on the direction of the external field is equal to zero). There was a question, spatial quantization what angular momentum was found in these experiments and the projection of which magnetic moment is equal to the Bohr magneton.
In 1925, Gottingen University students Goudsmit and Uhlenbeck suggested the existence own mechanical angular momentum of the electron (back ) And, respectively, the intrinsic magnetic moment of the electron P ms .
The introduction of the concept of spin immediately explained a number of difficulties existing by that time in quantum mechanics. And first of all - the results of the experiments of Stern and Gerlach.
The authors gave this interpretation back: electron - spinning top. But then it follows that the "surface" of the top (electron) must rotate with a linear velocity equal to 300 With, Where With is the speed of light. This interpretation of spin had to be abandoned.
In the modern view – spin , like charge and mass,is a property of the electron.
P. Dirac subsequently showed that the existence of spin follows from the solution of the relativistic Schrödinger wave equation.
It follows from the general conclusions of quantum mechanics that spin must be quantized
: 
, Where s – spin quantum number
.
Likewise, back projection per axle z (L sz) (axis z coincides with the direction of the external magnetic field) must be quantized and the vector can have (2 s+ 1) different orientations in a magnetic field.
It follows from the experiments of Stern and Gerlach that there are only two such orientations: , and hence s= 1/2, i.e. the spin quantum number has only one value.
For atoms of the first group, the valence electron of which is in s- state ( l = 0), the angular momentum of an atom is equal to the spin of the valence electron . Therefore, the spatial quantization of the angular momentum found for such atoms in a magnetic field is proof that the spin has only two orientations in an external field. (Experiments with electrons in p- state confirmed this conclusion, although the picture turned out to be more complex) (the yellow line of sodium is a doublet due to the presence of spin).
Numerical value back electron :
By analogy with the spatial quantization of the orbital momentum, the spin projection is quantized (similarly, as , then and ). The projection of the spin on the direction of the external magnetic field, being a quantum quantity, is determined by the expression.
SPIN selling is a selling 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. Using this method, you can achieve very high results in personal sales, Neil Rackham was able to prove this through 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 selling
In short, SPIN selling is a way of leading a customer to a purchase by asking certain questions one by one, you are not presenting the product openly, but rather pushing the customer to come to the decision to make a purchase on their own. The SPIN method is best suited for the so-called "long sales", often these are 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 technique arose primarily due to increased competition and market saturation. The client has become more selective and experienced, and this has required greater flexibility from the sellers.
The SPIN sales technique is divided into the following blocks of questions:
- WITH situational questions (Situation)
- P problematic questions (Problem)
- AND enticing questions (Implication)
- H directing questions (Need-payoff)
It should be noted right away that SPIN sales are quite labor intensive. The thing is to put this technique into practice, you need to know the product very well, have good experience in selling this product, in itself such a sale takes a lot of time for the seller. Therefore, SPIN sales should not be used in the mass segment, for example, because if the purchase price is low, and the demand for the product is already high, then it makes no sense to spend 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, when the seller directly offers the goods, often includes a protective mechanism of denial. Buyers are pretty tired of the fact that they are constantly selling something and react negatively to the very fact of the offer. Although the product itself may be needed, it’s just that at the time of the presentation, the client doesn’t think that he needs the product, but that why is he being offered it? The use of the SPIN sales technique forces the client to make an independent purchase decision, that is, the client does not even understand that his opinion is controlled by asking the right questions.
SPIN Sales Technique
The SPIN sales technique is a sales model based not only on but on their. In other words, to successfully apply this sales technique, the salesperson must be able to ask the right questions. To begin with, we will analyze separately each group of questions of the SPIN sales technique:
situational questions
This type of questions is necessary for the full and definition of its primary interests. The purpose of situational questions is to find out the experience of using the product that you are going to sell, his preferences, for what purposes it will be used. As a rule, about 5 open-ended questions and a few clarifying questions are required. As a result of this block of questions, you must liberate the client and set him up for communication, which is why you should pay attention to open questions, as well as use. In addition, you must collect all the necessary information to pose problem questions in order to effectively identify key needs worth using. As a rule, the block of situational questions is the longest in time. When you have received the necessary information from the client, you need to move on to problematic issues.
Problematic issues
When asking problematic questions, you should 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 a 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 have not decided on the needs, and do not know what problematic questions to ask. You need to have a set 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 for the client. For example: a client may admit that he overpays for the services of the company that he uses now, but he does not care, because the quality of services is important for him, not the price.
Extraction questions
This type of questions 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 extraction questions lies in the fact that they are not thought out in advance, unlike the others. Of course, with experience, you will form a pool of such questions, and you will learn how to use them depending on the situation. But initially, many sellers who master SPIN sales have difficulty asking such questions.
The essence of extractive questions is to establish for the client of causes an investigative link between the problem and its solution. Once again, I would like to note that in SPIN sales, you cannot say to the client: "our product will solve your problem." You should formulate the question in such a way that in response the client himself will say that he will be helped to solve the problem.
Guiding questions
Guiding questions - should help you, at this stage the client should speak for you all the benefits that he will receive from your product. Guiding questions can be compared to a positive way to complete the transaction, only the seller does not summarize all the benefits that the client will receive, but vice versa.
In this regard, one speaks of an integer or half-integer particle spin.
The existence of spin in a system of identical interacting particles is the cause of a new quantum mechanical phenomenon that has no analogy in classical mechanics, the exchange interaction.
The spin vector is the only quantity characterizing the orientation of a particle in quantum mechanics. From this position it follows that: at zero spin, a particle cannot have any vector and tensor characteristics; vector properties of particles can be described only by axial vectors; particles may have magnetic dipole moments and may not have electric dipole moments; particles may have an electric quadrupole moment and may not have a magnetic quadrupole moment; a nonzero quadrupole moment is possible only for particles with a spin not less than unity.
The spin moment of an electron or another elementary particle, uniquely separated from the orbital moment, can never be determined by means of experiments to which the classical concept of the particle trajectory is applicable.
The number of components of the wave function that describes an elementary particle in quantum mechanics grows with the growth of the elementary particle spin. Elementary particles with spin are described by a one-component wave function (scalar), with spin 1 2 (\displaystyle (\frac (1)(2))) are described by a two-component wave function (spinor), with spin 1 (\displaystyle 1) are described by a four-component wave function (vector), with spin 2 (\displaystyle 2) are described by a six-component wave function (tensor) .
What is spin - with examples
Although the term "spin" refers only to the quantum properties of particles, the properties of some cyclically operating macroscopic systems can also be described by a certain number that indicates how many parts the rotation cycle of some element of the system must be divided in order for it to return to a state indistinguishable from the initial one.
It's easy to imagine spin equal to 0: this is the point - it looks the same from every angle no matter how you twist it.
An example spin equal to 1, most ordinary objects without any symmetry can serve: if such an object is rotated by 360 degrees, the item will return to its original state. For example - you can put the pen on the table, and after turning 360 ° the pen will again lie in the same way as before the turn.
As an example spin equal to 2 you can take any object with one axis of central symmetry: if you rotate it by 180 degrees, it will be indistinguishable from the original position, and in one full turn it becomes indistinguishable from the original position 2 times. An ordinary pencil can serve as an example from life, only sharpened on both sides or not sharpened at all - the main thing is that it be without inscriptions and monophonic - and then after turning 180 ° it will return to a position indistinguishable from the original one. Hawking cited an ordinary playing card such as a king or queen as an example.
But with a half integer back equal to 1 / 2 a little more complicated: it turns out that the system returns to its original position after 2 full revolutions, that is, after turning 720 degrees. Examples:
- If you take a Möbius strip and imagine that an ant is crawling along it, then, after making one revolution (traversing 360 degrees), the ant will end up at the same point, but on the other side of the sheet, and in order to return to the point where it started, you will have to go through all 720 degrees.
- four-stroke internal combustion engine. When the crankshaft is rotated 360 degrees, the piston will return to its original position (for example, top dead center), but the camshaft rotates 2 times slower and will complete a full revolution when the crankshaft rotates 720 degrees. That is, when the crankshaft rotates 2 revolutions, the internal combustion engine will return to the same state. In this case, the third measurement will be the position of the camshaft.
Such examples can illustrate the addition of spins:
- Two identical pencils sharpened only on one side (“spin” of each is 1), fastened with their sides to each other so that the sharp end of one is next to the blunt end of the other (↓). Such a system will return to an indistinguishable from the initial state when rotated by only 180 degrees, that is, the “spin” of the system has become equal to two.
- A multi-cylinder four-stroke internal combustion engine ("spin" of each of the cylinders of which is 1/2). If all cylinders operate in the same way, then the states in which the piston is at the beginning of the stroke in any of the cylinders will be indistinguishable. Consequently, a two-cylinder engine will return to a state indistinguishable from the original one every 360 degrees (total "spin" - 1), a four-cylinder engine - after 180 degrees ("spin" - 2), an eight-cylinder engine - after 90 degrees ("spin" - 4 ).
Spin properties
Any particle can have two kinds of angular momentum: orbital angular momentum and spin.
Unlike orbital angular momentum, which is generated by the motion of a particle in space, spin is not related to motion in space. Spin is an intrinsic, purely quantum characteristic that cannot be explained within the framework of relativistic mechanics. If we represent a particle (for example, an electron) as a rotating ball, and the spin as a moment associated with this rotation, then it turns out that the transverse velocity of the particle shell must be higher than the speed of light, which is unacceptable from the standpoint of relativism.
Being one of the manifestations of angular momentum, spin in quantum mechanics is described by the vector spin operator s → ^ , (\displaystyle (\hat (\vec (s))),) whose component algebra completely coincides with the algebra of operators of orbital angular momentum ℓ → ^ . (\displaystyle (\hat (\vec (\ell ))).) However, unlike the 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 the spin (and its projections on any axis) can take not only integer values, but also half-integer values (in units of the Dirac constant ħ ).
The spin experiences quantum fluctuations. As a result of quantum fluctuations, only one spin component, for example, can have a strictly defined value. At the same time, the components J x , J y (\displaystyle J_(x),J_(y)) fluctuate around the mean. The maximum possible value of the component J z (\displaystyle J_(z)) equals J (\displaystyle J). At the same time the square J 2 (\displaystyle J^(2)) of the entire vector, the spin is equal to J (J + 1) (\displaystyle J(J+1)). Thus J x 2 + J y 2 = J 2 − J z 2 ⩾ J (\displaystyle J_(x)^(2)+J_(y)^(2)=J^(2)-J_(z)^(2 )\geqslant J). At J = 1 2 (\displaystyle J=(\frac (1)(2))) the root-mean-square values of all components due to fluctuations are equal J x 2 ^ = J y 2 ^ = J z 2 ^ = 1 4 (\displaystyle (\widehat (J_(x)^(2)))=(\widehat (J_(y)^(2)))= (\widehat (J_(z)^(2)))=(\frac (1)(4))).
The spin vector changes its direction under the Lorentz transformation. The axis of this rotation is perpendicular to the momentum of the particle and the relative velocity of reference systems .
Examples
Below are the spins of some microparticles.
| spin | common name for particles | examples |
|---|---|---|
| 0 | scalar particles | π mesons , K mesons , Higgs boson , 4 He atoms and nuclei , even-even nuclei, parapositronium |
| 1/2 | spinor particles | electron, quarks, muon, tau lepton, neutrino, proton, neutron, 3 He atoms and nuclei |
| 1 | vector particles | photon, gluon, W and Z bosons, vector mesons, orthopositronium |
| 3/2 | spin vector particles | Ω-hyperon, Δ-resonances |
| 2 | tensor particles | graviton, tensor mesons |
As of July 2004, the baryon resonance Δ(2950) with spin 15/2 has the maximum spin among the known baryons. The spin of stable nuclei cannot exceed 9 2 ℏ (\displaystyle (\frac (9)(2))\hbar ) .
Story
The term "spin" was introduced into science by S. Goudsmit and D. Uhlenbeck in 1925.
Mathematically, the theory of spin turned out to be very transparent, and later, by analogy with it, the theory of isospin was constructed.
Spin and magnetic moment
Despite the fact that the spin is not related to the actual rotation of the particle, it nevertheless generates a certain magnetic moment, and therefore leads to an additional (compared to classical electrodynamics) interaction with the magnetic field. The ratio of the magnitude of the magnetic moment to the magnitude of the spin is called the gyromagnetic ratio, and, unlike the orbital angular momentum, it is not equal to the magneton ( μ 0 (\displaystyle \mu _(0))):
μ → ^ = g ⋅ μ 0 s → ^ . (\displaystyle (\hat (\vec (\mu )))=g\cdot \mu _(0)(\hat (\vec (s))).)The multiplier entered here g called g-particle factor; the meaning of this g-factors for various elementary particles are being actively investigated in particle physics.
Spin and statistics
Due to the fact that all elementary particles of the same kind are identical, the wave function of a system of several identical particles must be either symmetric (that is, does not change) or antisymmetric (multiplied by −1) with respect to the swapping of any two particles. In the first case, the particles are said to obey Bose-Einstein statistics and are called bosons. In the second case, the particles are described by the Fermi-Dirac statistics and are called fermions.
It turns out that it is the value of the particle's spin that tells what these symmetry properties will be. Formulated by Wolfgang Pauli in 1940, the spin-statistics theorem states that particles with integer spin ( s= 0, 1, 2, …) are bosons, and particles with half-integer spin ( s\u003d 1/2, 3/2, ...) - fermions.
Spin generalization
The introduction of the spin was a successful application of a new physical idea: the postulation that there exists a space of states that have nothing to do with the motion of a particle in the ordinary
(English) spin – spindle)- a fundamental characteristic of a microscopic particle (for example, an atomic nucleus or an elementary particle), which is in some respects analogous to the "proper angular momentum of the particle." Spin is a quantum property of particles and has no analogues in classical physics. While the classical angular momentum arises due to the rotation of a massive body with finite dimensions, the spin is inherent even in particles that are today considered to be point particles and is not associated with any rotation of masses inside such a particle. (The spin of non-point particles, such as atomic nuclei or hadrons, is the vector sum of the spins and the orbital angular momentum of its components, i.e. in this case, too, the spin is partially associated with rotational motion inside the particle.)Spin can only take certain (quantized) values:
Goals: 0,1,2,3…
half integer: 1/2, 3/2, …
Spin is an important characteristic of elementary particles.
Discovery history
The spin of an electron was discovered in 1925 by Uhlenbeck and Gouldsmith, who were conducting experiments on the splitting of an electron beam in an inhomogeneous magnetic field. The scientists hoped to see how the electron beam split into several, in the distance from the quantized orbital momentum. If the angular momentum of the electrons were equal to zero, then the beam would not split; if the angular momentum were , then the beam would split into three, etc., into 2L +1 beams at an angular momentum . The result exceeded all expectations: the beam split into two. This could be explained only by assigning an intrinsic moment to the electron. This intrinsic moment of the electron is called the spin. At first, it was thought that the spin corresponds to some internal rotation of the electron, but soon Paul Dirac derived a relativistic analogue of the Schrödinger equation (the so-called Dirac equation), which automatically explained the existence of the spin from completely different principles.
The concept of spin made it possible to construct a theory of the periodic system, to elucidate the structure of atomic spectra, to explain the nature of covalent bonds, i.e. to explain the nature of covalent bonds.
spin operator
Mathematically, a spin is described by a spinor - a column with 2S + 1 wave functions, where S is the value of the spin. So particles with zero spin are described by one wave function or a scalar field, particles with spin 1 / 2 (for example, electrons) - by two wave functions or a spinor field, particles with spin 1 - by three wave functions or a vector field.
Spin operators are matrices of dimension (2S +1) x (2S +1). In the case of particles with spin 1/2, the spin operator is proportional to the Pauli matrices
Since the Paulo matrices do not commute, only the eigenvalues of one of them can be determined simultaneously. Usually choose? z. Therefore, the projection of the spin onto the z-axis for an electron can have the following values.
The state c is often referred to as the state with the spin directed upwards, the state c is spoken of as the state with the spin directed downwards, although these names are quite arbitrary and do not correspond to any directions in space.
The values of the other spin components are undefined.
