1. Introduction

Quantum theory was born in 1900, when Max Planck proposed a theoretical conclusion about the relationship between the temperature of a body and the radiation emitted by that body - a conclusion that had long eluded other scientists. Like his predecessors, Planck proposed that radiation is emitted by atomic oscillators, but At the same time, he believed that the energy of oscillators (and, consequently, the radiation emitted by them) exists in the form of small discrete portions, which Einstein called quanta. The energy of each quantum is proportional to the frequency of radiation. Although the formula derived by Planck aroused universal admiration, the assumptions he made remained incomprehensible, since they contradicted classical physics.

In 1905, Einstein used quantum theory to explain some aspects of the photoelectric effect—the emission of electrons by the surface of a metal exposed to ultraviolet light. Along the way, Einstein noted an apparent paradox: light, which for two centuries had been known to travel as continuous waves, could, under certain circumstances, also behave as a stream of particles.

About eight years later, Niels Bohr extended quantum theory to the atom and explained the frequencies of waves emitted by atoms excited in a flame or an electric charge. Ernest Rutherford showed that the mass of the atom is almost entirely concentrated in the central nucleus, which carries a positive electric charge and is surrounded at relatively large distances by electrons carrying a negative charge, as a result of which the atom as a whole is electrically neutral. Bohr suggested that electrons could only be in certain discrete orbits corresponding to different energy levels, and that the “jump” of an electron from one orbit to another, with lower energy, is accompanied by the emission of a photon, the energy of which is equal to the difference in the energies of the two orbits. Frequency, according to Planck's theory, is proportional to the energy of the photon. Thus, Bohr's model of the atom established a connection between the various spectral lines characteristic of the substance emitting radiation and the atomic structure. Despite its initial success, Bohr's model of the atom soon required modifications to resolve discrepancies between theory and experiment. In addition, quantum theory at that stage did not yet provide a systematic procedure for solving many quantum problems.

A significant new feature of quantum theory emerged in 1924, when de Broglie put forward a radical hypothesis about the wave nature of matter: if electromagnetic waves, such as light, sometimes behave like particles (as Einstein showed), then particles, such as the electron, can under certain circumstances behave like waves. In de Broglie's formulation, the frequency corresponding to a particle is related to its energy, as in the case of a photon (a particle of light), but de Broglie's proposed mathematical expression was an equivalent relationship between the wavelength, the mass of the particle, and its speed (momentum). The existence of electron waves was experimentally proven in 1927 by Clinton Davisson and Lester Germer in the United States and John Paget Thomson in England.

Impressed by Einstein's comments on de Broglie's ideas, Schrödinger attempted to apply the wave description of electrons to the construction of a coherent quantum theory, unrelated to Bohr's inadequate model of the atom. In a certain sense, he intended to bring quantum theory closer to classical physics, which had accumulated many examples of mathematical descriptions of waves. The first attempt, made by Schrödinger in 1925, ended in failure.

The speeds of electrons in Schrödinger's theory II were close to the speed of light, which required the inclusion of Einstein's special theory of relativity and the significant increase in electron mass at very high speeds that it predicted.

One of the reasons for Schrödinger's failure was that he did not take into account the presence of a specific property of the electron, now known as spin (the rotation of the electron around its own axis like a top), about which little was known at that time.

Schrödinger made the next attempt in 1926. This time the electron velocities were chosen so small that there was no need to invoke the theory of relativity.

The second attempt resulted in the derivation of the Schrödinger wave equation, which provides a mathematical description of matter in terms of the wave function. Schrödinger called his theory wave mechanics. The solutions of the wave equation were in agreement with experimental observations and had a profound influence on the subsequent development of quantum theory.

Not long before, Werner Heisenberg, Max Born, and Pascual Jordan published another version of quantum theory, called matrix mechanics, which described quantum phenomena using tables of observable quantities. These tables represent mathematical sets ordered in a certain way, called matrices, on which, according to known rules, various mathematical operations can be performed. Matrix mechanics also allowed for agreement with observed experimental data, but unlike wave mechanics, it did not contain any specific reference to spatial coordinates or time. Heisenberg especially insisted on the rejection of any simple visual representations or models in favor of only those properties that could be determined from experiment.

Schrödinger showed that wave mechanics and matrix mechanics are mathematically equivalent. Now known collectively as quantum mechanics, these two theories provided a long-awaited common framework for describing quantum phenomena. Many physicists preferred wave mechanics because its mathematics was more familiar to them and its concepts seemed more “physical”; operations on matrices are more cumbersome.

Function Ψ. Probability normalization.

The discovery of the wave properties of microparticles indicated that classical mechanics cannot provide a correct description of the behavior of such particles. There was a need to create a mechanics of microparticles that would also take into account their wave properties. The new mechanics created by Schrödinger, Heisenberg, Dirac and others was called wave or quantum mechanics.

Plane de Broglie wave

is a very special wave formation corresponding to the free uniform movement of a particle in a certain direction and with a certain momentum. But a particle, even in free space and especially in force fields, can also perform other movements described by more complex wave functions. In these cases, a complete description of the state of the particle in quantum mechanics is given not by a plane de Broglie wave, but by some more complex complex function

The wave function determines the relative probability of detecting a particle in different places in space. At this stage, when only probability relations are discussed, the wave function is fundamentally determined up to an arbitrary constant factor. If at all points in space the wave function is multiplied by the same constant (generally speaking, complex) number, different from zero, then a new wave function is obtained that describes exactly the same state. It makes no sense to say that Ψ is equal to zero at all points in space, because such a “wave function” never allows us to conclude about the relative probability of detecting a particle in different places in space. But the uncertainty in determining Ψ can be significantly narrowed if we move from relative probability to absolute probability. Let us dispose of the indefinite factor in the function Ψ so that the value |Ψ|2dV gives the absolute probability of detecting a particle in the space volume element dV. Then |Ψ|2 = Ψ*Ψ (Ψ* is the complex conjugate function of Ψ) will have the meaning of the probability density that should be expected when trying to detect a particle in space. In this case, Ψ will still be determined up to an arbitrary constant complex factor, the modulus of which, however, is equal to unity. With this definition, the normalization condition must be met:

where the integral is taken over the entire infinite space. It means that the particle will be detected with certainty throughout space. If the integral of |Ψ|2 is taken over a certain volume V1, we calculate the probability of finding a particle in the space of volume V1.

Normalization (2) may be impossible if integral (2) diverges. This will be the case, for example, in the case of a plane de Broglie wave, when the probability of detecting a particle is the same at all points in space. But such cases should be considered as idealizations of a real situation in which the particle does not go to infinity, but is forced to remain in a limited region of space. Then normalization is not difficult.

So, the direct physical meaning is associated not with the function Ψ itself, but with its module Ψ*Ψ. Why in quantum theory do they operate with wave functions Ψ, and not directly with experimentally observed quantities Ψ*Ψ? This is necessary to interpret the wave properties of matter - interference and diffraction. Here the situation is exactly the same as in any wave theory. It (at least in a linear approximation) accepts the validity of the principle of superposition of the wave fields themselves, and not their intensities, and thus achieves inclusion in the theory of the phenomena of wave interference and diffraction. Likewise, in quantum mechanics the principle of superposition of wave functions is accepted as one of the main postulates, which consists in the following.

The form of the wave equation of a physical system is determined by its Hamiltonian, which therefore acquires fundamental significance in the entire mathematical apparatus of quantum mechanics.

The form of the Hamiltonian of a free particle is established by general requirements related to the homogeneity and isotropy of space and Galileo's principle of relativity. In classical mechanics, these requirements lead to a quadratic dependence of the energy of a particle on its momentum: where the constant is called the mass of the particle (see I, § 4). In quantum mechanics, the same requirements lead to the same relationship for the eigenvalues ​​of energy and momentum - simultaneously measurable conserved (for a free particle) quantities.

But in order for the relationship to hold for all eigenvalues ​​of energy and momentum, it must also be valid for their operators:

Substituting (15.2) here, we obtain the Hamiltonian of a freely moving particle in the form

where is the Laplace operator.

The Hamiltonian of a system of non-interacting particles is equal to the sum of the Hamiltonians of each of them:

where the index a numbers the particles; - Laplace operator, in which differentiation is carried out with respect to the coordinates of the particle.

In classical (non-relativistic) mechanics, the interaction of particles is described by an additive term in the Hamilton function - the potential energy of interaction, which is a function of the coordinates of the particles.

By adding the same function to the Hamiltonian of the system, the interaction of particles in quantum mechanics is described:

the first term can be thought of as the kinetic energy operator and the second term as the potential energy operator. In particular, the Hamiltonian for one particle located in an external field is

where U(x, y, z) is the potential energy of a particle in an external field.

Substituting expressions (17.2)-(17.5) into the general equation (8.1) gives the wave equations for the corresponding systems. Let us write down here the wave equation for a particle in an external field

Equation (10.2), which defines stationary states, takes the form

Equations (17.6), (17.7) were established by Schrödinger in 1926 and are called the Schrödinger equations.

For a free particle, equation (17.7) has the form

This equation has solutions that are finite throughout the entire space for any positive value of energy E. For states with certain directions of motion, these solutions are the eigenfunctions of the momentum operator, and . The complete (time-dependent) wave functions of such stationary states have the form

(17,9)

Each such function - a plane wave - describes a state in which the particle has a certain energy E and momentum. The frequency of this wave is equal to and its wave vector the corresponding wavelength is called the de Broglie wavelength of the particle.

The energy spectrum of a freely moving particle thus turns out to be continuous, extending from zero to Each of these eigenvalues ​​(except only the value is degenerate, and the degeneracy is of infinite multiplicity. Indeed, each non-zero value of E corresponds to an infinite set of eigenfunctions (17, 9), differing in vector directions with the same absolute value.

Let us trace how the limit transition to classical mechanics occurs in the Schrödinger equation, considering for simplicity only one particle in an external field. Substituting the limiting expression (6.1) of the wave function into the Schrödinger equation (17.6), we obtain, by differentiation,

This equation has purely real and purely imaginary terms (recall that S and a are real); equating both of them separately to zero, we obtain two equations:

Neglecting the term containing in the first of these equations, we obtain

(17,10)

i.e., as expected, the classical Hamilton-Jacobi equation for the action of an S particle. We see, by the way, that at classical mechanics is valid up to quantities of the first (and not zero) order inclusive.

The second of the resulting equations after multiplication by 2a can be rewritten in the form

This equation has a clear physical meaning: there is the probability density of finding a particle in a particular place in space, and there is the classical velocity v of the particle. Therefore, equation (17.11) is nothing more than a continuity equation, showing that the probability density “moves” according to the laws of classical mechanics with a classical speed v at each point.

Task

Find the law of wave function transformation under the Galilean transform.

Solution. Let us perform a transformation over the wave function of the free motion of a particle (a plane wave). Since any function can be expanded into plane waves, the transformation law will thereby be found for an arbitrary wave function.

Plane waves in the reference systems K and K" (K" moves relative to K with speed V):

Moreover, the momenta and energies of particles in both systems are related to each other by the formulas

(see I, § 8), Substituting these expressions in we get

In this form, this formula no longer contains quantities characterizing the free movement of a particle, and establishes the desired general law for transforming the wave function of an arbitrary state of a particle. For a system of particles, the exponent in (1) should contain the sum over the particles.

According to the folklore so widespread among physicists, it happened like this: in 1926, a theoretical physicist by name spoke at a scientific seminar at the University of Zurich. He talked about strange new ideas in the air, about how microscopic objects often behave more like waves than like particles. Then an elderly teacher asked to speak and said: “Schrödinger, don’t you see that all this is nonsense? Or don’t we all know that waves are just waves to be described by wave equations?” Schrödinger took this as a personal insult and set out to develop a wave equation to describe particles within the framework of quantum mechanics - and coped with this task brilliantly.

An explanation needs to be made here. In our everyday world, energy is transferred in two ways: by matter when moving from place to place (for example, a moving locomotive or the wind) - particles are involved in such energy transfer - or by waves (for example, radio waves that are transmitted by powerful transmitters and caught by the antennas of our televisions). That is, in the macrocosm where you and I live, all energy carriers are strictly divided into two types - corpuscular (consisting of material particles) or wave. Moreover, any wave is described by a special type of equations - wave equations. Without exception, all waves - ocean waves, seismic rock waves, radio waves from distant galaxies - are described by the same type of wave equations. This explanation is necessary in order to make it clear that if we want to represent the phenomena of the subatomic world in terms of probability distribution waves (see Quantum Mechanics), these waves must also be described by the corresponding wave equation.

Schrödinger applied the classical differential equation of the wave function to the concept of probability waves and obtained the famous equation that bears his name. Just as the usual wave function equation describes the propagation of, for example, ripples on the surface of water, the Schrödinger equation describes the propagation of a wave of the probability of finding a particle at a given point in space. The peaks of this wave (points of maximum probability) show where in space the particle is most likely to end up. Although the Schrödinger equation belongs to the field of higher mathematics, it is so important for understanding modern physics that I will nevertheless present it here - in its simplest form (the so-called “one-dimensional stationary Schrödinger equation”). The above probability distribution wave function, denoted by the Greek letter (psi), is the solution to the following differential equation (it’s okay if you don’t understand it; just take it on faith that this equation shows that probability behaves like a wave): :

The picture of quantum events that Schrödinger's equation gives us is that electrons and other elementary particles behave like waves on the surface of the ocean. Over time, the peak of the wave (corresponding to the location where the electron is most likely to be) moves in space in accordance with the equation that describes this wave. That is, what we traditionally considered a particle behaves much like a wave in the quantum world.

When Schrödinger first published his results, a storm broke out in a teacup in the world of theoretical physics. The fact is that almost at the same time, the work of Schrödinger’s contemporary, Werner Heisenberg, appeared (see Heisenberg’s Uncertainty Principle), in which the author put forward the concept of “matrix mechanics”, where the same problems of quantum mechanics were solved in another, more complex mathematical point view matrix form. The commotion was caused by the fact that scientists were simply afraid that two equally convincing approaches to describing the microworld might contradict each other. The worries were in vain. In the same year, Schrödinger himself proved the complete equivalence of the two theories - that is, the matrix equation follows from the wave equation, and vice versa; the results are identical. Today, it is primarily Schrödinger's version (sometimes called "wave mechanics") that is used because his equation is less cumbersome and easier to teach.

However, it is not so easy to imagine and accept that something like an electron behaves like a wave. In everyday life, we encounter either a particle or a wave. The ball is a particle, sound is a wave, and that's it. In the world of quantum mechanics, everything is not so simple. In fact - and experiments soon showed this - in the quantum world, entities differ from the objects we are familiar with and have different properties. Light, which we think of as a wave, sometimes behaves like a particle (called a photon), and particles like electrons and protons can behave like waves (see the Complementarity Principle).

This problem is usually called the dual or dual particle-wave nature of quantum particles, and it is characteristic, apparently, of all objects of the subatomic world (see Bell's Theorem). We must understand that in the microworld our ordinary intuitive ideas about what forms matter can take and how it can behave simply do not apply. The very fact that we use the wave equation to describe the movement of what we are accustomed to thinking of as particles is clear proof of this. As noted in the Introduction, there is no particular contradiction in this. After all, we have no compelling reasons to believe that what we observe in the macrocosm should be accurately reproduced at the level of the microcosm. Yet the dual nature of elementary particles remains one of the most puzzling and troubling aspects of quantum mechanics for many people, and it is no exaggeration to say that all the troubles began with Erwin Schrödinger.

Encyclopedia by James Trefil “The Nature of Science. 200 laws of the universe."

James Trefil is a professor of physics at George Mason University (USA), one of the most famous Western authors of popular science books.

Max Planck, one of the founders of quantum mechanics, came to the ideas of energy quantization, trying to theoretically explain the process of interaction between recently discovered electromagnetic waves and atoms and, thereby, solve the problem of black body radiation. He realized that to explain the observed emission spectrum of atoms, it is necessary to take for granted that atoms emit and absorb energy in portions (which the scientist called quanta) and only at individual wave frequencies.

A completely black body that completely absorbs electromagnetic radiation of any frequency, when heated, emits energy in the form of waves evenly distributed over the entire frequency spectrum.

The word “quantum” comes from the Latin quantum (“how much, how much”) and the English quantum (“quantity, portion, quantum”). “Mechanics” has long been the name given to the science of the movement of matter. Accordingly, the term “quantum mechanics” means the science of the movement of matter in portions (or, in modern scientific language, the science of the movement of quantized matter). The term “quantum” was coined by the German physicist Max Planck to describe the interaction of light with atoms.

One of the facts of the subatomic world is that its objects - such as electrons or photons - are not at all similar to the usual objects of the macroworld. They behave neither like particles nor like waves, but like completely special formations that exhibit both wave and corpuscular properties depending on the circumstances. It is one thing to make a statement, but quite another to connect together the wave and particle aspects of the behavior of quantum particles, describing them with an exact equation. This is exactly what was done in the de Broglie relation.

In everyday life, there are two ways to transfer energy in space - through particles or waves. In everyday life, there are no visible contradictions between the two mechanisms of energy transfer. So, a basketball is a particle, and sound is a wave, and everything is clear. However, in quantum mechanics things are not so simple. Even from the simplest experiments with quantum objects, it very soon becomes clear that in the microworld the principles and laws of the macroworld that we are familiar with do not apply. Light, which we are accustomed to thinking of as a wave, sometimes behaves as if it consists of a stream of particles (photons), and elementary particles, such as an electron or even a massive proton, often exhibit the properties of a wave.

Most of all, Einstein protested against the need to describe the phenomena of the microworld in terms of probabilities and wave functions, and not from the usual position of coordinates and particle velocities. That's what he meant by "rolling the dice." He recognized that describing the movement of electrons in terms of their speeds and coordinates contradicts the uncertainty principle. But, Einstein argued, there must be some other variables or parameters, taking into account which the quantum mechanical picture of the microworld will return to the path of integrity and determinism. That is, he insisted, it only seems to us that God is playing dice with us, because we do not understand everything. Thus, he was the first to formulate the hidden variable hypothesis in the equations of quantum mechanics. It lies in the fact that in fact electrons have fixed coordinates and speed, like Newton’s billiard balls, and the uncertainty principle and the probabilistic approach to their determination within the framework of quantum mechanics are the result of the incompleteness of the theory itself, which is why it does not allow them for certain define.

Yulia Zotova

You will learn: What technologies are called quantum and why. What is the advantage of quantum technologies over classical ones? What a quantum computer can and cannot do. How physicists make a quantum computer. When it will be created.

French physicist Pierre Simon Laplace raised an important question about whether everything in the world is predetermined by the previous state of the world, or whether a cause can cause several consequences. As expected by the philosophical tradition, Laplace himself in his book “Exposition of the World System” did not ask any questions, but said a ready-made answer that yes, everything in the world is predetermined, however, as often happens in philosophy, the picture of the world proposed by Laplace did not convince everyone and thus his answer gave rise to a debate around the issue that continues to this day. Despite the opinion of some philosophers that quantum mechanics resolved this issue in favor of a probabilistic approach, nevertheless, Laplace’s theory of complete predetermination, or as it is otherwise called the theory of Laplace determinism, is still discussed today.

Gordey Lesovik

Some time ago, a group of co-authors and I began to derive the second law of thermodynamics from the point of view of quantum mechanics. For example, in one of his formulations, which states that the entropy of a closed system does not decrease, typically increases, and sometimes remains constant if the system is energetically isolated. Using known results from quantum information theory, we have derived some conditions under which this statement is true. Unexpectedly, it turned out that these conditions do not coincide with the condition of energy isolation of systems.

Physics professor Jim Al-Khalili explores the most precise and one of the most confusing scientific theories - quantum physics. In the early 20th century, scientists plumbed the hidden depths of matter, the subatomic building blocks of the world around us. They discovered phenomena that were different from anything seen before. A world where everything can be in many places at the same time, where reality only truly exists when we observe it. Albert Einstein resisted the mere idea that the essence of nature was based on chance. Quantum physics implies that subatomic particles can interact faster than the speed of light, which contradicts his theory of relativity.

Introduction

It is known that the course of quantum mechanics is one of the most difficult to understand. This is due not so much to the new and “unusual” mathematical apparatus, but primarily to the difficulty of understanding the revolutionary, from the standpoint of classical physics, ideas underlying quantum mechanics and the complexity of interpreting the results.

In most textbooks on quantum mechanics, the presentation of the material is based, as a rule, on the analysis of solutions to the stationary Schrödinger equations. However, the stationary approach does not allow one to directly compare the results of solving a quantum mechanical problem with similar classical results. In addition, many processes studied in the course of quantum mechanics (such as the passage of a particle through a potential barrier, the decay of a quasi-stationary state, etc.) are in principle non-stationary in nature and, therefore, can be understood in full only on the basis of solutions to the non-stationary equation Schrödinger. Since the number of analytically solvable problems is small, the use of a computer in the process of studying quantum mechanics is especially relevant.

The Schrödinger equation and the physical meaning of its solutions

Schrödinger wave equation

One of the basic equations of quantum mechanics is the Schrödinger equation, which determines the change in states of quantum systems over time. It is written in the form

where H is the Hamiltonian operator of the system, coinciding with the energy operator if it does not depend on time. The type of operator is determined by the properties of the system. For the nonrelativistic motion of a mass particle in a potential field U(r), the operator is real and is represented by the sum of the operators of the kinetic and potential energy of the particle

If a particle moves in an electromagnetic field, then the Hamiltonian operator will be complex.

Although equation (1.1) is a first-order equation in time, due to the presence of an imaginary unit, it also has periodic solutions. Therefore, the Schrödinger equation (1.1) is often called the Schrödinger wave equation, and its solution is called the time-dependent wave function. Equation (1.1) with a known form of the operator H allows one to determine the value of the wave function at any subsequent time, if this value is known at the initial time. Thus, the Schrödinger wave equation expresses the principle of causality in quantum mechanics.

The Schrödinger wave equation can be obtained based on the following formal considerations. In classical mechanics it is known that if energy is given as a function of coordinates and momentum

then the transition to the classical Hamilton-Jacobi equation for the action function S

can be obtained from (1.3) by the formal transformation

In the same way, equation (1.1) is obtained from (1.3) by passing from (1.3) to the operator equation by formal transformation

if (1.3) does not contain products of coordinates and momenta, or contains products of them that, after passing to operators (1.4), commute with each other. Equating after this transformation the results of the action on the function of the operators of the right and left sides of the resulting operator equality, we arrive at the wave equation (1.1). However, these formal transformations should not be taken as a derivation of the Schrödinger equation. The Schrödinger equation is a generalization of experimental data. It is not derived in quantum mechanics, just as Maxwell’s equations are not derived in electrodynamics, the principle of least action (or Newton’s equations) in classical mechanics.

It is easy to verify that equation (1.1) is satisfied for the wave function

describing the free movement of a particle with a certain momentum value. In the general case, the validity of equation (1.1) is proven by agreement with experience of all conclusions obtained using this equation.

Let us show that equation (1.1) implies the important equality

indicating that the normalization of the wave function persists over time. Let us multiply (1.1) on the left by the function *, a the equation complex conjugate to (1.1) by the function and subtract the second from the first resulting equation; then we find

Integrating this relation over all values ​​of the variables and taking into account the self-adjointness of the operator, we obtain (1.5).

If we substitute into relation (1.6) the explicit expression of the Hamiltonian operator (1.2) for the motion of a particle in a potential field, then we arrive at the differential equation (continuity equation)

where is the probability density, and the vector

can be called the probability current density vector.

The complex wave function can always be represented as

where and are real functions of time and coordinates. Thus, the probability density

and the probability current density

From (1.9) it follows that j = 0 for all functions for which the function Ф does not depend on the coordinates. In particular, j= 0 for all real functions.

Solutions of the Schrödinger equation (1.1) in the general case are represented by complex functions. Using complex functions is quite convenient, although not necessary. Instead of one complex function, the state of the system can be described by two real functions and, satisfying two related equations. For example, if the operator H is real, then by substituting the function into (1.1) and separating the real and imaginary parts, we obtain a system of two equations

in this case, the probability density and probability current density will take the form

Wave functions in impulse representation.

The Fourier transform of the wave function characterizes the distribution of momentum in a quantum state. It is required to derive an integral equation for the potential with the Fourier transform as the kernel.

Solution. There are two mutually inverse relationships between the functions and.

If relation (2.1) is used as a definition and an operation is applied to it, then taking into account the definition of a 3-dimensional -function,

as a result, as is easy to see, we get the inverse relation (2.2). Similar considerations are used below in deriving relation (2.8).

then for the Fourier transform of the potential we have

Assuming that the wave function satisfies the Schrödinger equation

Substituting expressions (2.1) and (2.3) here instead of and, respectively, we obtain

In the double integral, we move from integration over a variable to integration over a variable, and then we again denote this new variable by. The integral over vanishes for any value only in the case when the integrand itself is equal to zero, but then

This is the desired integral equation with the Fourier transform of the potential as the kernel. Of course, the integral equation (2.6) can be obtained only under the condition that the Fourier transform of the potential (2.4) exists; for this, for example, the potential must decrease over large distances at least as, where.

It should be noted that from the normalization condition

equality follows

This can be shown by substituting expression (2.1) for the function into (2.7):

If we first perform integration over here, we can easily obtain relation (2.8).

No. 1 The stationary Schrödinger equation has the form . This equation is written for...

The stationary Schrödinger equation in the general case has the form

, where is the potential energy of the microparticle. For the one-dimensional case. In addition, the particle cannot be inside the potential box, but outside the box, because its walls are infinitely high. Therefore, this Schrödinger equation is written for a particle in a one-dimensional box with infinitely high walls.

Linear harmonic oscillator

ü Particles in a one-dimensional potential box with infinitely high walls

Particles in a three-dimensional potential box with infinitely high walls

Electron in a hydrogen atom

Establish correspondences between quantum mechanical problems and Schrödinger equations for them.

The general form of the stationary Schrödinger equation is:

Particle potential energy,

Laplace operator. For simultaneous case

The expression for the potential energy of a harmonic oscillator, that is, a particle performing one-dimensional motion under the action of a quasi-elastic force, has the form U=.

The value of the potential energy of an electron in a potential box with infinitely high walls is U = 0. An electron in a hydrogen-like atom has potential energy For a hydrogen atom Z = 1.

Thus, for an electron in a one-dimensional potential box, the Schrödinger equation has the form:

Using the wave function, which is a solution to the Schrödinger equation, we can determine...

Answer options: (Indicate at least two answer options)

Average values ​​of physical quantities characterizing a particle

The probability that a particle is located in a certain region of space

Particle trajectory

Particle location

The value has the meaning of probability density (probability per unit volume), that is, it determines the probability of a particle being in the corresponding place in space. Then the probability W of detecting a particle in a certain region of space is equal to

Schrödinger equation (specific situations)

No. 1The eigenfunctions of an electron in a one-dimensional potential box with infinitely high walls have the form where is the width of the box, a quantum number meaning the energy level number. If the number of function nodes on the segment and , then equals...

Number of nodes, i.e. the number of points at which the wave function on a segment vanishes is related to the number of the energy level by the relation . Then , and by condition this ratio is equal to 1.5. Solving the resulting equation for , we find that

Nuclear reactions.

№1 In a nuclear reaction, the letter represents a particle...

From the laws of conservation of mass number and charge number it follows that the charge of the particle is zero and the mass number is 1. Therefore, the letter denotes a neutron.

ü Neutron

Positron

Electron

The graph on a semi-logarithmic scale shows the dependence of the change in the number of radioactive nuclei of an isotope on time. The radioactive decay constant is equal to ... (round the answer to whole numbers)

The number of radioactive nuclei changes over time according to the law - the initial number of nuclei, - the radioactive decay constant. Taking the logarithm of this expression, we get

ln .Hence, =0,07

Conservation laws in nuclear reactions.

The reaction cannot proceed due to a violation of the conservation law...

In all fundamental interactions, conservation laws are satisfied: energy, momentum, angular momentum (spin) and all charges (electric, baryon and lepton). These conservation laws not only limit the consequences of various interactions, but also determine all the possibilities of these consequences. To choose the correct answer, you need to check which conservation law prohibits and which allows the given reaction of interconversion of elementary particles. According to the law of conservation of lepton charge in a closed system during any process, the difference between the number of leptons and antileptons is preserved. We agreed to calculate for leptons: . lepton charge and for antileptons: . lepton charge. For all other elementary particles, lepton charges are assumed to be zero. The reaction cannot proceed due to a violation of the law of conservation of lepton charge, because

ü Lepton charge

Baryon charge

Spin angular momentum

Electric charge

The reaction cannot proceed due to a violation of the conservation law...

In all fundamental interactions, conservation laws are satisfied: energy, momentum, angular momentum (spin) and all charges (electric Q, baryon B and lepton L). These conservation laws not only limit the consequences of various interactions, but also determine all the possibilities of these consequences. According to the law of conservation of baryon charge B, for all processes involving baryons and antibaryons, the total baryon charge is conserved. Baryons (n, p nucleons and hyperons) are assigned a baryon charge

B = -1, and for all other particles the baryon charge is B = 0. The reaction cannot proceed due to a violation of the law of baryon charge B, because (+1)+(+1)

Answer options: lepton charge, spin angular momentum, electric charge. Q=0, antiproton (