Type of wave equation physical system is determined by its Hamiltonian, which thereby acquires fundamental significance in the entire mathematical apparatus quantum mechanics.

The form of the Hamiltonian of a free particle has already been established 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 relation for eigenvalues energy and momentum are 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.

Hamiltonian of a system of non-interacting particles equal to the sum 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 considered as an operator kinetic energy, and the second - as the potential energy operator. In particular, the Hamiltonian for one particle located in an external field is

where U(x, y, z) - potential energy particles in an external field.

Substituting expressions (17.2)-(17.5) into 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 in the entire space for any positive value 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 look like

(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 infinite set 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 get

(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 visual physical meaning: there is the probability density of finding a particle in a particular place in space; there is the classical speed 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 classic speed v at every 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 movement 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 free movement particles, and sets the desired common law transformation of the wave function of an arbitrary particle state. For a system of particles, the exponent in (1) should contain the sum over the particles.

General Schrödinger equation. Schrödinger equation for stationary states

The statistical interpretation of de Broglie waves (see § 216) and the Heisenberg uncertainty relation (see 5 215) led to the conclusion that the equation of motion in quantum mechanics, which describes the movement of microparticles in various force fields, there must be an equation from which the experimentally observed values ​​would follow wave properties particles. The main equation must be an equation with respect to the wave function Ψ (x, y, z, t), since it is precisely this, or, more precisely, the quantity |Ψ| 2, determines the probability of a particle being at time t in volume dV, i.e. in the area with coordinates x and x+dx, y and y+dy, z and z+dz. Since the required equation must take into account the wave properties of particles, it must be wave equation, similar to the equation describing electromagnetic waves.

The basic equation of nonrelativistic quantum mechanics was formulated in 1926 by E. Schrödinger. The Schrödinger equation, like all the basic equations of physics (for example, Newton's equations in classical mechanics and Maxwell's equations for electromagnetic field), is not derived, but postulated. The correctness of this equation is confirmed by the agreement with experience of the results obtained with its help, which, in turn, gives it the character of a law of nature. The Schrödinger equation has the form

where h=h/(2π), m is the mass of the particle, ∆ is the Laplace operator ( ),

i - imaginary unit, U (x, y, z, t) - potential function particle in the force field in which it moves, Ψ (x, y, z, t ) - sought after wave function particles.

Equation (217.1) is valid for any particle (with a spin equal to 0; see § 225) moving at a low speed (compared to the speed of light), i.e. at a speed υ<<с. Оно дополняется условиями, накладываемыми на волновую функцию: 1) волновая функция должна быть конечной, однозначной и непрерывной (см. § 216); 2) производные

must be continuous; 3) function |Ψ| 2 must be integrable; this condition in the simplest cases reduces to the condition for normalizing probabilities (216.3).

To arrive at the Schrödinger equation, consider a freely moving particle, which, according to de Broglie’s idea, is associated with a plane wave. For simplicity, we consider the one-dimensional case. The equation of a plane wave propagating along the x axis has the form (see § 154)

Or in a complex recording . Therefore, the plane de Broglie wave has the form

(217.2)

(it is taken into account that ω = E/h, k=p/h). In quantum mechanics, the exponent is taken with a minus sign, but since only |Ψ| has a physical meaning. 2 , then this (see (217.2)) is unimportant. Then

,

; (217.3)

Using the relationship between energy E and momentum p (E = p 2 /(2m)) and substituting expressions (217.3), we obtain the differential equation

which coincides with equation (217.1) for the case U = 0 (we considered a free particle).

If a particle moves in a force field characterized by potential energy U, then the total energy E is the sum of the kinetic and potential energies. Carrying out similar reasoning using the relationship between E and p (for this case p 2 /(2m)=E -U), we arrive at a differential equation coinciding with (217.1).

The above reasoning should not be taken as a derivation of the Schrödinger equation. They only explain how one can arrive at this equation. The proof of the correctness of the Schrödinger equation is the agreement with experience of the conclusions to which it leads.

Equation (217.1) is the general Schrödinger equation. It is also called the time-dependent Schrödinger equation. For many physical phenomena occurring in the microworld, equation (217.1) can be simplified by eliminating the dependence of Ψ on time, in other words, find the Schrödinger equation for stationary states - states with fixed energy values. This is possible if the force field in which the particle moves is stationary, i.e. the function U = U(x, y, z ) does not depend explicitly on time and has the meaning of potential energy. In this case, the solution to the Schrödinger equation can be represented as a product of two functions, one of which is a function of only coordinates, the other - only time, and the dependence on time is expressed by the multiplier

,

where E - the total energy of the particle, constant in the case of a stationary field. Substituting (217.4) into (217.1), we get

from where, after dividing by a common factor e – i (E/ h) t and the corresponding transformations, we arrive at the equation defining the function ψ:

(217.5)

Equation (217.5) is called the Schrödinger equation for stationary states.

This equation includes the total energy E of the particle as a parameter. In the theory of differential equations, it is proven that such equations have an infinite number of solutions, from which solutions that have a physical meaning are selected by imposing boundary conditions. For the Schrödinger equation, such conditions are the conditions for the regularity of wave functions: wave functions must be finite, single-valued and continuous along with their first derivatives. Thus, only those solutions that are expressed by regular functions ψ have real physical meaning . But regular solutions do not take place for any values ​​of the parameter E, but only for a certain set of them, characteristic of a given problem. These energy values ​​are called eigenvalues. Solutions that correspond to eigenvalues ​​of energy are called eigenfunctions. Eigenvalues ​​E can form either a continuous or a discrete series. In the first case, they speak of a continuous, or solid, spectrum, in the second - of a discrete spectrum.

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 Hamilton 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, one should not accept these formal transformations 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 Hamilton 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 Φ 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 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).

Heisenberg was led to the conclusion that the equation of motion in quantum mechanics, which describes the movement of microparticles in various force fields, should be an equation from which the experimentally observed wave properties of particles would follow. The governing equation must be an equation for the wave function Ψ (x, y, z, t), since it is precisely this, or, more precisely, the quantity |Ψ| 2, determines the probability of a particle being present at the moment of time t in volume Δ V, i.e. in the area with coordinates X And x + dx, y And y + dу, z And z+ dz.

The basic equation of nonrelativistic quantum mechanics was formulated in 1926 by E. Schrödinger. The Schrödinger equation, like all the basic equations of physics (for example, Newton's equations in classical mechanics and Maxwell's equations for the electromagnetic field), is not derived, but postulated. The correctness of this equation is confirmed by the agreement with experience of the results obtained with its help, which, in turn, gives it the character of a law of nature.

The general Schrödinger equation is:

Where ? =h/(2π), m- particle mass, Δ - Laplace operator , i- imaginary unit, U(x, y, z, t) is the potential function of a particle in the force field in which it moves, Ψ( x, y, z, t) is the desired wave function of the particle.

Equation (1) is valid for any particle (with a spin equal to 0) moving at a low (compared to the speed of light) speed, i.e. υ "With.

It is supplemented by conditions, superimposed on the wave function:

1) the wave function must be finite, unambiguous and continuous;

2) derivatives must be continuous;

3) function |Ψ| 2 must be integrable (this condition in the simplest cases reduces to the condition for normalizing the probabilities).

Equation (1) is called time-dependent Schrödinger equation.

For many physical phenomena occurring in the microworld, equation (1) can be simplified by eliminating the dependence of Ψ on time, i.e. find the Schrödinger equation for stationary states - states with fixed energy values. This is possible if the force field in which the particle moves is stationary, i.e. the function U = U(x, y,z) does not depend explicitly on time and has the meaning of potential energy. In this case, the solution to the Schrödinger equation can be represented in the form

. (2)

Equation (2) called the Schrödinger equation for stationary states.

This equation includes total energy as a parameter E particles. In the theory of differential equations, it is proven that such equations have an infinite number of solutions, from which solutions that have a physical meaning are selected by imposing boundary conditions. For the Schrödinger equation such conditions are conditions for the regularity of wave functions: The new functions must be finite, unambiguous and continuous along with their first derivatives.

Thus, only those solutions that are expressed by regular functions Ψ have real physical meaning. But regular solutions do not take place for any parameter values E, but only for a certain set of them, characteristic of a given task. These energy values ​​are called eigenvalues . Solutions that correspond to energy eigenvalues ​​are called eigenfunctions . Eigenvalues E can form either a continuous or discrete series. In the first case, they speak of a continuous, or solid, spectrum, in the second - of a discrete spectrum.

Particle in a one-dimensional rectangular "potential well"with infinitely high “walls”

Let us carry out a qualitative analysis of solutions to the Schrödinger equation as applied to a particle in a one-dimensional rectangular “potential well” with infinitely high “walls”. Such a “hole” is described by potential energy of the form (for simplicity, we assume that the particle moves along the axis X)

Where l is the width of the “hole”, and the energy is counted from its bottom (Fig. 2).

The Schrödinger equation for stationary states in the case of a one-dimensional problem will be written in the form:

. (1)

According to the conditions of the problem (infinitely high “walls”), the particle does not penetrate beyond the “hole”, therefore the probability of its detection (and, consequently, the wave function) outside the “hole” is zero. At the boundaries of the “pit” (at X= 0 and x = 1) the continuous wave function must also vanish.

Therefore, the boundary conditions in this case have the form:

Ψ (0) = Ψ ( l) = 0. (2)

Within the “pit” (0 ≤ X≤ 0) the Schrödinger equation (1) will be reduced to the equation:

or . (3)

Where k 2 = 2mE / ? 2.(4)

General solution of differential equation (3):

Ψ ( x) = A sin kx + B cos kx.

Since according to (2) Ψ (0) = 0, then B = 0. Then

Ψ ( x) = A sin kx. (5)

Condition Ψ ( l) = A sin kl= 0 (2) is executed only when kl = nπ, Where n- integers, i.e. it is necessary that

k = nπ/l. (6)

From expressions (4) and (6) it follows that:

(n = 1, 2, 3,…), (7)

i.e., the stationary Schrödinger equation, which describes the motion of a particle in a “potential well” with infinitely high “walls,” is satisfied only for the eigenvalues E p, depending on an integer p. Therefore, the energy E p particles in a “potential well” with infinitely high “walls” accept only certain discrete values, i.e. quantized.

Quantized energy values E p are called energy levels and the number p, which determines the energy levels of a particle is called principal quantum number. Thus, a microparticle in a “potential well” with infinitely high “walls” can only be at a certain energy level E p, or, as they say, the particle is in a quantum state p.

Substituting into (5) the value k from (6), we find the eigenfunctions:

.

Constant of integration A we find from the normalization condition, which for this case will be written in the form:

.

As a result of integration we obtain , and the eigenfunctions will have the form:

(n = 1, 2, 3,…). (8)

Graphs of eigenfunctions (8) corresponding to energy levels (7) at n= 1,2,3, shown in Fig. 3, A. In Fig. 3, b shows the probability density of detecting a particle at various distances from the “walls” of the hole, equal to ‌‌‌‌‌‌ Ψ n(x)‌ 2 = Ψ n(x)·Ψ n * (x) For n = 1, 2 and 3. It follows from the figure that, for example, in a quantum state with n= 2, a particle cannot be in the middle of the “hole,” while equally often it can be in its left and right parts. This behavior of the particle indicates that the concepts of particle trajectories in quantum mechanics are untenable.

From expression (7) it follows that the energy interval between two adjacent levels is equal to:

For example, for an electron with well dimensions l= 10 -1 m (free electrons in metal) , Δ E n ≈ 10 -35 · n J ≈ 10 -1 6 n eV, i.e. The energy levels are located so closely that the spectrum can practically be considered continuous. If the dimensions of the well are comparable to atomic ones ( l ≈ 10 -10 m), then for the electron Δ E n ≈ 10 -17 n J ≈ 10 2 n eV, i.e. Obviously discrete energy values ​​(line spectrum) are obtained.

Thus, applying the Schrödinger equation to a particle in a “potential well” with infinitely high “walls” leads to quantized energy values, while classical mechanics does not impose any restrictions on the energy of this particle.

In addition, a quantum mechanical consideration of this problem leads to the conclusion that a particle “in a potential well” with infinitely high “walls” cannot have an energy less than the minimum energy equal to π 2 ? 2 /(2t1 2). The presence of a nonzero minimum energy is not accidental and follows from the uncertainty relation. Coordinate uncertainty Δ X particles in a "pit" wide l equal to Δ X= l.

Then, according to the uncertainty relation, the impulse cannot have an exact, in this case zero, value. Momentum uncertainty Δ r ≈ h/l. This spread of momentum values ​​corresponds to the kinetic energy E min ≈(Δ p) 2 / (2m) = ? 2 / (2ml 2). All other levels ( p> 1) have an energy exceeding this minimum value.

From formulas (9) and (7) it follows that for large quantum numbers ( n"1) Δ E n / E p ≈ 2/n“1, i.e. adjacent levels are located closely: the closer the more p. If n is very large, then we can talk about an almost continuous sequence of levels and the characteristic feature of quantum processes - discreteness - is smoothed out. This result is a special case of Bohr's correspondence principle (1923), according to which the laws of quantum mechanics should transform into the laws of classical physics at large values ​​of quantum numbers.