Wave function and its statistical meaning. Types of wave function and its collapse. Wave function

In the coordinate representation, the wave function depends on the coordinates (or generalized coordinates) of the system. The physical meaning is assigned to the square of its modulus, which is interpreted as the probability density (for discrete spectra - simply the probability) to detect the system in the position described by the coordinates at the moment of time:

Then, in a given quantum state of the system, described by the wave function, we can calculate the probability that a particle will be detected in any region of finite volume configuration space: .

It should also be noted that it is also possible to measure phase differences in the wave function, for example, in the Aharonov-Bohm experiment.

Schrödinger equation- an equation that describes the change in space (in the general case, in configuration space) and in time of the pure state specified by the wave function in Hamiltonian quantum systems. Plays the same important role in quantum mechanics as the equation of Newton's second law in classical mechanics. It can be called the equation of motion of a quantum particle. Installed by Erwin Schrödinger in 1926.

The Schrödinger equation is intended for spinless particles moving at speeds much lower than the speed of light. In the case of fast particles and particles with spin, its generalizations are used (Klein-Gordon equation, Pauli equation, Dirac equation, etc.)

At the beginning of the 20th century, scientists came to the conclusion that there were a number of discrepancies between the predictions of classical theory and experimental data on atomic structure. The discovery of the Schrödinger equation followed de Broglie's revolutionary assumption that not only light, but also any bodies in general (including any microparticles) have wave properties.

Historically, the final formulation of the Schrödinger equation was preceded by a long period of development in physics. It is one of the most important equations in physics that explains physical phenomena. Quantum theory, however, does not require a complete rejection of Newton's laws, but only defines the limits of applicability of classical physics. Therefore, Schrödinger's equation must be consistent with Newton's laws in limiting case. This is confirmed by a deeper analysis of the theory: if the size and mass of a body become macroscopic and the accuracy of tracking its coordinate is much worse than the standard quantum limit, the predictions of the quantum and classical theories coincide, because the uncertain path of the object becomes close to the unambiguous trajectory.

Time dependent equation

The most general form of the Schrödinger equation is the form that includes time dependence:

An example of a non-relativistic Schrödinger equation in coordinate representation for a point particle of mass moving in a potential field with potential:

Time-dependent Schrödinger equation

Formulation

General case

In quantum physics, a complex-valued function is introduced that describes the pure state of an object, which is called the wave function. In the most common Copenhagen interpretation, this function is related to the probability of finding an object in one of the pure states (the square of the modulus of the wave function represents the probability density). The behavior of a Hamiltonian system in a pure state is completely described by the wave function.

Having abandoned the description of the particle's motion using trajectories obtained from the laws of dynamics, and having instead defined the wave function, it is necessary to introduce an equation equivalent to Newton's laws and giving a recipe for finding the quotients physical problems. Such an equation is the Schrödinger equation.

Let the wave function be given in n-dimensional configuration space, then at each point with coordinates , at a certain moment in time t it will look like . In this case, the Schrödinger equation will be written as:

where , is Planck’s constant; - the mass of the particle, - the potential energy external to the particle at a point at the moment of time, - the Laplace operator (or Laplacian), is equivalent to the square of the Nabla operator and in the n-dimensional coordinate system has the form:

Question 30 Fundamental physical interactions. The concept of physical vacuum in the modern scientific picture of the world.

Interaction. The entire variety of interactions is divided in the modern physical picture of the world into 4 types: strong, electromagnetic, weak and gravitational. According to modern concepts, all interactions are of an exchange nature, i.e. are realized as a result of the exchange of fundamental particles - carriers of interactions. Each of the interactions is characterized by the so-called interaction constant, which determines its comparative intensity, duration and range of action. Let us briefly consider these interactions.

1. Strong interaction ensures the connection of nucleons in the nucleus. The interaction constant is approximately 10 0, the range of action is about

10 -15, flow time t »10 -23 s. Particles - carriers - p-mesons.

2. Electromagnetic interaction: constant of the order of 10 -2, interaction radius is not limited, interaction time t » 10 -20 s. It is realized between all charged particles. Particle – carrier – photon.

3. Weak interaction associated with all types of b-decay, many decays of elementary particles and the interaction of neutrinos with matter. The interaction constant is about 10 -13, t » 10 -10 s. This interaction, like the strong one, is short-range: the interaction radius is 10 -18 m. (Particle - carrier - vector boson).

4. Gravitational interaction is universal, but is taken into account in the microcosm, since its constant is 10 -38, i.e. of all interactions is the weakest and manifests itself only in the presence of sufficiently large masses. Its range is unlimited, and its time is also unlimited. The exchange nature of gravitational interaction still remains in question, since the hypothetical fundamental particle graviton has not yet been discovered.

Physical vacuum

In quantum physics, the physical vacuum is understood as the lowest (ground) energy state of a quantized field, which has zero momentum, angular momentum and other quantum numbers. Moreover, such a state does not necessarily correspond to emptiness: the field in the lowest state can be, for example, the field of quasiparticles in a solid or even in the nucleus of an atom, where the density is extremely high. A physical vacuum is also called a space completely devoid of matter, filled with a field in this state. This state is not absolute emptiness. Quantum field theory states that, in accordance with the uncertainty principle, virtual particles are constantly born and disappear in the physical vacuum: so-called zero-point field oscillations occur. In some specific field theories, the vacuum may have non-trivial topological properties. In theory, several different vacua may exist, differing in energy density or other physical parameters(depending on the hypotheses and theories used). The degeneracy of the vacuum with spontaneous symmetry breaking leads to the existence of a continuous spectrum of vacuum states that differ from each other in the number of Goldstone bosons. Local energy minima at different meanings any fields that differ in energy from the global minimum are called false vacua; such states are metastable and tend to decay with the release of energy, passing into a true vacuum or into one of the underlying false vacua.

Some of these field theory predictions have already been successfully confirmed by experiment. Thus, the Casimir effect and the Lamb shift of atomic levels are explained by zero-point vibrations electromagnetic field in a physical vacuum. Modern physical theories are based on some other ideas about vacuum. For example, the existence of several vacuum states (the false vacua mentioned above) is one of the main foundations of the Big Bang inflationary theory.

31 questions Structural levels of matter. Microworld. Macroworld. Megaworld.

Structural levels of matter

(1) - A characteristic feature of matter is its structure, therefore one of most important tasks Natural science is the study of this structure.

It is currently accepted that the most natural and obvious sign of the structure of matter is the characteristic size of an object at a given level and its mass. In accordance with these ideas, the following levels are distinguished:

(3) - The concept of “microworld” covers fundamental and elementary particles, nuclei, atoms and molecules. The macrocosm is represented by macromolecules, substances in various states of aggregation, living organisms, starting with the elementary unit of living things - cells, man and the products of his activities, i.e. macrobodies. The largest objects (planets, stars, galaxies and their clusters form a megaworld. It is important to realize that there are no hard boundaries between these worlds, and we are only talking about different levels of consideration of matter.

For each of the considered main levels, in turn, sublevels can be distinguished, characterized by their own structure and their own organizational features.

The study of matter at its various structural levels requires its own specific means and methods.

Question 32 Evolution of the Universe (Friedmann, Hubble, Gamow) and cosmic microwave background radiation.

Based on the idea that an electron has wave properties. Schrödinger in 1925 suggested that the state of an electron moving in an atom should be described by the equation of a standing electromagnetic wave, known in physics. Substituting its value from the de Broglie equation instead of the wavelength into this equation, he obtained a new equation relating the electron energy to spatial coordinates and the so-called wave function, corresponding in this equation to the amplitude of the three-dimensional wave process.

The wave function is especially important for characterizing the state of the electron. Like the amplitude of any wave process, it can take both positive and negative values. However, the value is always positive. Moreover, it has a remarkable property: the greater the value in a given region of space, the higher the probability that the electron will manifest its action here, that is, that its existence will be detected in some physical process.

The following statement will be more accurate: the probability of detecting an electron in a certain small volume is expressed by the product . Thus, the value itself expresses the probability density of finding an electron in the corresponding region of space.

Rice. 5. Electron cloud of the hydrogen atom.

To understand the physical meaning of the squared wave function, consider Fig. 5, which depicts a certain volume near the nucleus of a hydrogen atom. The density of points in Fig. 5 is proportional to the value in the corresponding place: the larger the value, the denser the points are located. If an electron had the properties of a material point, then Fig. 5 could be obtained by repeatedly observing the hydrogen atom and each time marking the location of the electron: the density of points in the figure would be greater, the more often the electron is detected in the corresponding region of space or, in other words, the greater the probability of its detection in this region.

We know, however, that the idea of ​​an electron as a material point does not correspond to its true physical nature. Therefore Fig. It is more correct to consider 5 as a schematic representation of an electron “smeared” throughout the entire volume of an atom in the form of a so-called electron cloud: the denser the points are located in one place or another, the greater the density of the electron cloud. In other words, the density of the electron cloud is proportional to the square of the wave function.

The idea of ​​the state of an electron as a certain cloud of electric charge turns out to be very convenient; it conveys well the main features of the behavior of the electron in atoms and molecules and will be often used in the subsequent presentation. At the same time, however, it should be borne in mind that the electron cloud does not have definite, sharply defined boundaries: even at a great distance from the nucleus, there is some, albeit very small, probability of detecting an electron. Therefore, by electron cloud we will conventionally understand the region of space near the nucleus of an atom in which the predominant part (for example, ) of the charge and mass of the electron is concentrated. A more precise definition of this region of space is given on page 75.

> Wave function

Read about wave function and probability theories of quantum mechanics: the essence of the Schrödinger equation, the state of a quantum particle, a harmonic oscillator, a diagram.

We are talking about the probability amplitude in quantum mechanics, which describes the quantum state of a particle and its behavior.

Learning Objective

• Combine the wave function and the probability density of identifying a particle.

Main points

• |ψ| 2 (x) corresponds to the probability density of identifying a particle in a specific place and moment.
• The laws of quantum mechanics characterize the evolution of the wave function. The Schrödinger equation explains its name.
• The wave function must satisfy many mathematical constraints for computation and physical interpretation.

Terms

• The Schrödinger equation is a partial differential characterizing a change in the state of a physical system. It was formulated in 1925 by Erwin Schrödinger.
• A harmonic oscillator is a system that, when displaced from its original position, is influenced by a force F proportional to the displacement x.

Within quantum mechanics, the wave function reflects the probability amplitude that characterizes the quantum state of a particle and its behavior. Typically the value is a complex number. The most common symbols for the wave function are ψ (x) or Ψ(x). Although ψ is a complex number, |ψ| 2 – real and corresponds to the probability density of finding a particle in a specific place and time.

Here the trajectories of the harmonic oscillator are displayed in classical (A-B) and quantum (C-H) mechanics. The quantum ball has a wave function displayed with the real part in blue and the imaginary part in red. TrajectoriesC-F – examples of standing waves. Each such frequency will be proportional to the possible energy level of the oscillator

The laws of quantum mechanics evolve over time. The wave function resembles others, such as waves in water or a string. The fact is that the Schrödinger formula is a type of wave equation in mathematics. This leads to the duality of wave particles.

The wave function must comply with the following restrictions:

• always final.
• always continuous and continuously differentiable.
• satisfies the appropriate normalization condition for the particle to exist with 100% certainty.

If the requirements are not satisfied, then the wave function cannot be interpreted as a probability amplitude. If we ignore these positions and use the wave function to determine observations of a quantum system, we will not get finite and definite values.

4.4.1. De Broglie's conjecture

An important stage in the creation of quantum mechanics was the discovery of the wave properties of microparticles. The idea of ​​wave properties was originally proposed as a hypothesis by the French physicist Louis de Broglie.

For many years, the dominant theory in physics was that light is an electromagnetic wave. However, after the work of Planck (thermal radiation), Einstein (photoelectric effect) and others, it became obvious that light has corpuscular properties.

To explain some physical phenomena, it is necessary to consider light as a stream of photon particles. The corpuscular properties of light do not reject, but complement its wave properties.

So, photon is an elementary particle of light with wave properties.

Formula for photon momentum

.

(4.4.3)

According to de Broglie, the movement of a particle, for example an electron, is similar to a wave process with a wavelength λ defined by formula (4.4.3). These waves are called de Broglie waves. Consequently, particles (electrons, neutrons, protons, ions, atoms, molecules) can exhibit diffraction properties.

K. Davisson and L. Germer were the first to observe electron diffraction on a nickel single crystal.

The question may arise: what happens to individual particles, how are maxima and minima formed during the diffraction of individual particles?

Experiments on diffraction of electron beams of very low intensity, that is, as if individual particles, showed that in this case the electron does not “spread” in different directions, but behaves like a whole particle. However, the probability of electron deflection in certain directions as a result of interaction with a diffraction object is different. The electrons are most likely to fall into those places that, according to calculations, correspond to diffraction maxima; they are less likely to fall into places of minima. Thus, wave properties are inherent not only to a collective of electrons, but also to each electron individually.

4.4.2. Wave function and its physical meaning

Since a microparticle is associated with a wave process that corresponds to its movement, the state of particles in quantum mechanics is described by a wave function that depends on coordinates and time: .

If the force field acting on the particle is stationary, that is, independent of time, then the ψ-function can be represented as a product of two factors, one of which depends on time, and the other on coordinates:

This implies the physical meaning of the wave function:

4.4.3. Uncertainty relationship

One of the important provisions of quantum mechanics is the uncertainty relations proposed by W. Heisenberg.

Let the position and momentum of the particle be simultaneously measured, while the inaccuracies in the determination of the abscissa and the projection of the momentum onto the abscissa axis are equal to Δx and Δр x, respectively.

In classical physics there are no restrictions that prohibit simultaneous measurement of both one and the other quantity, that is, Δx→0 and Δр x→ 0, with any degree of accuracy.

In quantum mechanics, the situation is fundamentally different: Δx and Δр x, corresponding to the simultaneous determination of x and р x, are related by the dependence

Formulas (4.4.8), (4.4.9) are called uncertainty relations.

Let us explain them with one model experiment.

When studying the phenomenon of diffraction, attention was drawn to the fact that a decrease in the slit width during diffraction leads to an increase in the width of the central maximum. A similar phenomenon will occur during electron diffraction by a slit in a model experiment. Reducing the slit width means decreasing Δ x (Fig. 4.4.1), this leads to greater “smearing” of the electron beam, that is, to greater uncertainty in the momentum and velocity of the particles.

Rice. 4.4.1 Explanation of the uncertainty relation.

The uncertainty relationship can be represented as

,

(4.4.10)

where ΔE is the uncertainty of the energy of a certain state of the system; Δt is the period of time during which it exists. Relationship (4.4.10) means that the shorter the lifetime of any state of the system, the more uncertain its energy value. Energy levels E 1, E 2, etc. have a certain width (Fig. 4.4.2)), depending on the time the system remains in the state corresponding to this level.

Rice. 4.4.2. Energy levels E 1, E 2, etc. have some width.

The “blurring” of the levels leads to uncertainty in the energy ΔE of the emitted photon and its frequency Δν when the system transitions from one energy level to another:

,

where m is the mass of the particle; ; E and E n are its total and potential energies (potential energy is determined by the force field in which the particle is located, and for a stationary case does not depend on time)

If the particle moves only along a certain line, for example along the OX axis (one-dimensional case), then the Schrödinger equation is significantly simplified and takes the form

(4.4.13)

One of the simplest examples of using the Schrödinger equation is solving the problem of particle motion in a one-dimensional potential well.

4.4.5. Application of the Schrödinger equation to the hydrogen atom. Quantum numbers

Description of the states of atoms and molecules using the Schrödinger equation is sufficient challenging task. It is most simply solved for one electron located in the field of the nucleus. Such systems correspond to a hydrogen atom and hydrogen-like ions (singly ionized helium atom, doubly ionized lithium atom, etc.). However, even in this case, the solution to the problem is complex, so we will limit ourselves to only a qualitative presentation of the issue.

First of all, potential energy should be substituted into the Schrödinger equation (4.4.12), which for two interacting point charges - e (electron) and Ze (nucleus) - located at a distance r in vacuum, is expressed as follows:

This expression is a solution to the Schrödinger equation and completely coincides with the corresponding formula of Bohr’s theory (4.2.30)

Figure 4.4.3 shows the levels of possible values ​​of the total energy of a hydrogen atom (E 1, E 2, E 3, etc.) and a graph of the potential energy E n depending on the distance r between the electron and the nucleus. As the principal quantum number n increases, r increases (see 4.2.26), and the total (4.4.15) and potential energies tend to zero. Kinetic energy also tends to zero. The shaded area (E>0) corresponds to the state of a free electron.

Rice. 4.4.3. The levels of possible values ​​of the total energy of the hydrogen atom are shown
and a graph of potential energy versus distance r between the electron and the nucleus.

The second quantum number is orbital l, which for a given n can take values ​​0, 1, 2, ...., n-1. This number characterizes the orbital angular momentum Li of the electron relative to the nucleus:

The fourth quantum number is spin m s. It can take only two values ​​(±1/2) and characterizes the possible values ​​of the electron spin projection:

.(4.4.18)

The state of an electron in an atom with given n and l is denoted as follows: 1s, 2s, 2p, 3s, etc. Here the number indicates the value of the main quantum number, and the letter indicates the orbital quantum number: the symbols s, p, d, f correspond to the values ​​l = 0, 1, 2. 3, etc.

· Quantum observable · Wave function· Quantum superposition · Quantum entanglement · Mixed state · Measurement · Uncertainty · Pauli principle · Dualism · Decoherence · Ehrenfest's theorem · Tunnel effect

Experiments
Davisson-Germer experiment · Popper experiment · Stern-Gerlach experiment · Young experiment · Bell's inequalities test · Photoelectric effect · Compton effect

Formulations
Schrödinger representation · Heisenberg representation · Interaction representation · Matrix quantum mechanics · Path integrals · Feynman diagrams

Equations
Schrödinger equation · Pauli equation · Klein-Gordon equation · Dirac equation · Schwinger-Tomonaga equation · Von Neumann equation · Bloch equation · Lindblad equation · Heisenberg equation

Interpretations
Copenhagen · Hidden parameter theory · Many-worlds · De Broglie-Bohm theory

Development of the theory
Quantum field theory · Quantum electrodynamics · Glashow-Weinberg-Salam theory · Quantum chromodynamics · Standard Model · Quantum gravity

Famous scientists
Planck · Einstein · Schrödinger · Heisenberg · Jordan · Bohr · Pauli · Dirac · Fock · Born · de Broglie · Landau · Feynman · Bohm · Everett

See also: Portal:Physics

Wave function, or psi function $\backslash psi$ is a complex-valued function used in quantum mechanics to describe the pure state of a system. Is the coefficient of expansion of the state vector over a basis (usually a coordinate one):

$\backslash left|\backslash psi(t)\backslash right\backslash rangle=\backslash int\; \backslash Psi(x,t)\backslash left|x\backslash right\backslash rangle\; dx$

Where $\backslash left|x\backslash right\backslash rangle\; =\; \backslash left|x\_1,\; x\_2,\; \backslash ldots\; ,\; x\_n\backslash right\backslash rangle$ is the coordinate basis vector, and $\backslash Psi(x,t)=\; \backslash langle\; x\backslash left|\backslash psi(t)\backslash right\backslash rangle$- wave function in coordinate representation.

Normalization of the wave function

Wave function $\backslash Psi$ in its meaning must satisfy the so-called normalization condition, for example, in the coordinate representation having the form:

$(\backslash int\backslash limits\_(V)(\backslash Psi^\backslash ast\backslash Psi)dV)=1$

This condition expresses the fact that the probability of finding a particle with a given wave function anywhere in space is equal to one. In the general case, integration must be carried out over all variables on which the wave function in a given representation depends.

Principle of superposition of quantum states

For wave functions, the principle of superposition is valid, which is that if a system can be in states described by wave functions $\backslash Psi\_1$ And $\backslash Psi\_2$, then it can also be in a state described by the wave function

$\backslash Psi\_\backslash Sigma\; =\; c\_1\; \backslash Psi\_1\; +\; c\_2\; \backslash Psi\_2$ for any complex $c\_1$ And $c\_2$.

Obviously, we can talk about the superposition (imposition) of any number of quantum states, that is, about the existence of a quantum state of the system, which is described by the wave function $\backslash Psi\_\backslash Sigma\; =\; c\_1\; \backslash Psi\_1\; +\; c\_2\; \backslash Psi\_2\; +\; \backslash ldots\; +\; (c)\_N(\backslash Psi)\_N=\backslash sum\_(n=1)^(N)\; (c)\_n(\backslash Psi)\_n$.

In this state, the square of the modulus of the coefficient $(c)\_n$ determines the probability that, when measured, the system will be detected in a state described by the wave function $(\backslash Psi)\_n$.

Therefore, for normalized wave functions $\backslash sum\_(n=1)^(N)\backslash left|c\_(n)\backslash right|^2=1$.

Conditions for the regularity of the wave function

The probabilistic meaning of the wave function imposes certain restrictions, or conditions, on wave functions in problems of quantum mechanics. These standard conditions are often called conditions for the regularity of the wave function.

1. Condition for the finiteness of the wave function. The wave function cannot take infinite values ​​such that the integral $(1)$ will become divergent. Consequently, this condition requires that the wave function be a quadratically integrable function, that is, belong to Hilbert space $L^2$. In particular, in problems with a normalized wave function, the squared modulus of the wave function must tend to zero at infinity.
2. Condition for the uniqueness of the wave function. The wave function must be an unambiguous function of coordinates and time, since the probability density of detecting a particle must be determined uniquely in each problem. In problems using a cylindrical or spherical coordinate system, the uniqueness condition leads to the periodicity of wave functions in angular variables.
3. Condition for the continuity of the wave function. At any moment in time, the wave function must be a continuous function of spatial coordinates. In addition, the partial derivatives of the wave function must also be continuous $\backslash frac(\backslash partial\; \backslash Psi)(\backslash partial\; x)$, $\backslash frac(\backslash partial\; \backslash Psi)(\backslash partial\; y)$, $\backslash frac(\backslash partial\; \backslash Psi)(\backslash partial\; z)$. These partial derivatives of functions only in rare cases of problems with idealized force fields can suffer a discontinuity at those points in space where the potential energy describing the force field in which the particle moves experiences a discontinuity of the second kind.

Wave function in various representations

The set of coordinates that act as function arguments represents a complete system of commuting observables. In quantum mechanics, it is possible to select several complete sets of observables, so the wave function of the same state can be written in terms of different arguments. The complete set of quantities chosen to record the wave function determines wave function representation. Thus, a coordinate representation, a momentum representation are possible; in quantum field theory, secondary quantization and the representation of occupation numbers or the Fock representation, etc., are used.

If the wave function, for example, of an electron in an atom, is given in coordinate representation, then the squared modulus of the wave function represents the probability density of detecting an electron at a particular point in space. If the same wave function is given in impulse representation, then the square of its module represents the probability density of detecting a particular impulse.

Matrix and vector formulations

The wave function of the same state in different representations will correspond to the expression of the same vector in different coordinate systems. Other operations with wave functions will also have analogues in the language of vectors. In wave mechanics, a representation is used where the arguments of the psi function are complete system continuous commuting observables, and the matrix representation uses a representation where the arguments of the psi function are the complete system discrete commuting observables. Therefore, the functional (wave) and matrix formulations are obviously mathematically equivalent.

Philosophical meaning of the wave function

The wave function is a method of describing the pure state of a quantum mechanical system. Mixed quantum states (in quantum statistics) should be described by an operator like a density matrix. That is, some generalized function of two arguments must describe the correlation between the location of a particle at two points.

It should be understood that the problem that quantum mechanics solves is the problem of the very essence of the scientific method of knowing the world.

see also

Write a review about the article "Wave function"

Literature

• Physical encyclopedic Dictionary/ Ch. ed. A. M. Prokhorov. Ed. count D. M. Alekseev, A. M. Bonch-Bruevich, A. S. Borovik-Romanov and others - M.: Sov. Encyclopedia, 1984. - 944 p.

Links

• Quantum mechanics- article from the Great Soviet Encyclopedia.

This letter had not yet been submitted to the sovereign when Barclay told Bolkonsky at dinner that the sovereign would like to see Prince Andrei personally in order to ask him about Turkey, and that Prince Andrei would appear at Bennigsen’s apartment at six o’clock in the evening. On the same day, news was received in the sovereign's apartment about Napoleon's new movement, which could be dangerous for the army - news that later turned out to be unfair. And that same morning, Colonel Michaud, touring the Dries fortifications with the sovereign, proved to the sovereign that this fortified camp, built by Pfuel and hitherto considered the master of tactics, destined to destroy Napoleon, - that this camp was nonsense and destruction Russian army. Prince Andrei arrived at the apartment of General Bennigsen, who occupied a small landowner's house on the very bank of the river. Neither Bennigsen nor the sovereign were there, but Chernyshev, the sovereign’s aide-de-camp, received Bolkonsky and announced to him that the sovereign had gone with General Bennigsen and the Marquis Paulucci another time that day to tour the fortifications of the Drissa camp, the convenience of which was beginning to be seriously doubted. Chernyshev was sitting with a book of a French novel at the window of the first room. This room was probably formerly a hall; there was still an organ in it, on which some carpets were piled, and in one corner stood the folding bed of Adjutant Bennigsen. This adjutant was here. He, apparently exhausted by a feast or business, sat on a rolled up bed and dozed. Two doors led from the hall: one straight into the former living room, the other to the right into the office. From the first door one could hear voices speaking in German and occasionally in French. There, in the former living room, at the sovereign’s request, not a military council was gathered (the sovereign loved uncertainty), but some people whose opinions on the upcoming difficulties he wanted to know. This was not a military council, but, as it were, a council of those elected to clarify certain issues personally for the sovereign. Invited to this half-council were: the Swedish General Armfeld, Adjutant General Wolzogen, Wintzingerode, whom Napoleon called a fugitive French subject, Michaud, Tol, not a military man at all - Count Stein and, finally, Pfuel himself, who, as Prince Andrei heard, was la cheville ouvriere [the basis] of the whole matter. Prince Andrei had the opportunity to take a good look at him, since Pfuhl arrived soon after him and walked into the living room, stopping for a minute to talk with Chernyshev. At first glance, Pfuel, in his poorly tailored Russian general's uniform, which sat awkwardly on him, as if dressed up, seemed familiar to Prince Andrei, although he had never seen him. It included Weyrother, Mack, Schmidt, and many other German theoretic generals whom Prince Andrei managed to see in 1805; but he was more typical than all of them. Prince Andrei had never seen such a German theoretician, who combined in himself everything that was in those Germans. Pfuel was short, very thin, but broad-boned, of a rough, healthy build, with a wide pelvis and bony shoulder blades. His face was very wrinkled, with deep-set eyes. His hair in front, near his temples, was obviously hastily smoothed with a brush, and naively stuck out with tassels at the back. He, looking around restlessly and angrily, entered the room, as if he was afraid of everything in the large room into which he entered. He, holding his sword with an awkward movement, turned to Chernyshev, asking in German where the sovereign was. He apparently wanted to go through the rooms as quickly as possible, finish bowing and greetings, and sit down to work in front of the map, where he felt at home. He hastily nodded his head at Chernyshev’s words and smiled ironically, listening to his words that the sovereign was inspecting the fortifications that he, Pfuel himself, had laid down according to his theory. He grumbled something bassily and coolly, as self-confident Germans say, to himself: Dummkopf... or: zu Grunde die ganze Geschichte... or: s"wird was gescheites d"raus werden... [nonsense... to hell with the whole thing... (German) ] Prince Andrei did not hear and wanted to pass, but Chernyshev introduced Prince Andrei to Pful, noting that Prince Andrei came from Turkey, where the war was so happily over. Pful almost looked not so much at Prince Andrei as through him, and said laughing: “Da muss ein schoner taktischcr Krieg gewesen sein.” [“It must have been a correctly tactical war.” (German)] - And, laughing contemptuously, he walked into the room from which voices were heard. Apparently, Pfuel, who was always ready for ironic irritation, was now especially excited by the fact that they dared to inspect his camp without him and judge him. Prince Andrei, from this one short meeting with Pfuel, thanks to his Austerlitz memories, compiled a clear description of this man. Pfuel was one of those hopelessly, invariably, self-confident people to the point of martyrdom, which only Germans can be, and precisely because only Germans are self-confident on the basis of an abstract idea - science, that is, an imaginary knowledge of perfect truth. The Frenchman is self-confident because he considers himself personally, both in mind and body, to be irresistibly charming to both men and women. An Englishman is self-confident on the grounds that he is a citizen of the most comfortable state in the world, and therefore, as an Englishman, he always knows what he needs to do, and knows that everything he does as an Englishman is undoubtedly good. The Italian is self-confident because he is excited and easily forgets himself and others. The Russian is self-confident precisely because he knows nothing and does not want to know, because he does not believe that it is possible to completely know anything. The German is the worst self-confident of all, and the firmest of all, and the most disgusting of all, because he imagines that he knows the truth, a science that he himself invented, but which for him is the absolute truth. This, obviously, was Pfuhl. He had a science - the theory of physical movement, which he derived from the history of the wars of Frederick the Great, and everything that he encountered in modern history wars of Frederick the Great, and everything that he encountered in modern times military history, seemed to him nonsense, barbarism, an ugly clash, in which so many mistakes were made on both sides that these wars could not be called wars: they did not fit the theory and could not serve as the subject of science. In 1806, Pfuel was one of the drafters of the plan for the war that ended with Jena and Auerstätt; but in the outcome of this war he did not see the slightest proof of the incorrectness of his theory. On the contrary, the deviations made from his theory, according to his concepts, were the only reason for the entire failure, and he, with his characteristic joyful irony, said: “Ich sagte ja, daji die ganze Geschichte zum Teufel gehen wird.” [After all, I said that the whole thing would go to hell (German)] Pfuel was one of those theorists who love their theory so much that they forget the purpose of theory - its application to practice; In his love for theory, he hated all practice and did not want to know it. He even rejoiced at failure, because failure, which resulted from a deviation in practice from theory, only proved to him the validity of his theory. He said a few words with Prince Andrei and Chernyshev about the real war with the expression of a man who knows in advance that everything will be bad and that he is not even dissatisfied with it. The unkempt tufts of hair sticking out at the back of his head and the hastily slicked temples especially eloquently confirmed this. He walked into another room, and from there the bassy and grumbling sounds of his voice were immediately heard. Before Prince Andrei had time to follow Pfuel with his eyes, Count Bennigsen hurriedly entered the room and, nodding his head to Bolkonsky, without stopping, walked into the office, giving some orders to his adjutant. The Emperor was following him, and Bennigsen hurried forward to prepare something and have time to meet the Emperor. Chernyshev and Prince Andrei went out onto the porch. The Emperor got off his horse with a tired look. Marquis Paulucci said something to the sovereign. The Emperor, bowing his head to the left, listened with a dissatisfied look to Paulucci, who spoke with particular fervor. The Emperor moved forward, apparently wanting to end the conversation, but the flushed, excited Italian, forgetting decency, followed him, continuing to say: “Quant a celui qui a conseille ce camp, le camp de Drissa, [As for the one who advised the Drissa camp,” said Paulucci, while the sovereign, entering the steps and noticing Prince Andrei, peered into an unfamiliar face .