In developing de Broglie's idea about the wave properties of matter, E. Schrödinger received his famous equation. Schrödinger compared the movement of microparticles complex function coordinates and time, which he called the wave function and designated Greek letter"psi" (). We will call it the psi function.

The psi function characterizes the state of the microparticle. The form of the function is obtained from the solution of the Schrödinger equation, which looks like this:

Here is the particle mass, i - imaginary unit, - Laplace operator, the result of which acts on a certain function is the sum of the second partial derivatives with respect to the coordinates:

The letter U in equation (21.1) denotes the function of coordinates and time, the gradient of which, taken with the opposite sign, determines the force acting on the particle. In the case when the function U does not depend explicitly on time, it has the meaning of the potential energy of the particle.

From equation (21.1) it follows that the form of the psi function is determined by the function U, i.e., ultimately, by the nature of the forces acting on the particle.

The Schrödinger equation is the fundamental equation of non-relativistic quantum mechanics. It cannot be derived from other relations. It should be considered as an initial basic assumption, the validity of which is proven by the fact that all the consequences flowing from it are in the most accurate agreement with experimental facts.

Schrödinger established his equation based on an optical-mechanical analogy. This analogy lies in the similarity of the equations that describe the path of light rays with the equations that determine the trajectories of particles in analytical mechanics. In optics, the path of rays satisfies Fermat’s principle (see § 115 of the 2nd volume); in mechanics, the type of trajectory satisfies the so-called principle of least action.

If the force field in which the particle moves is stationary, then the function V does not explicitly depend on time and, as already noted, has the meaning of potential energy. In this case, the solution to the Schrödinger equation splits into two factors, one of which depends only on the coordinates, the other - only on time:

Here E is the total energy of the particle, which in the case stationary field remains constant. To verify the validity of expression (21.3), let us substitute it into equation (21.1). As a result, we obtain the relation

Reduced by common multiplier we arrive at a differential equation defining the function

Equation (21.4) is called the Schrödinger equation for stationary states. In what follows we will deal only with this equation and for brevity we will simply call it the Schrödinger equation. Equation (21.4) is often written in the form

Let us explain how one can arrive at the Schrödinger equation. For simplicity, we restrict ourselves to the one-dimensional case. Let's consider a freely moving particle.

According to de Broglie's idea, it needs to be associated with a plane wave

(V quantum mechanics It is customary to take the exponent with a minus sign). Replacing in accordance with (18.1) and (18.2) through E and , we arrive at the expression

Differentiating this expression once with respect to t, and a second time twice with respect to x, we obtain

In non-relativistic classical mechanics, the energy E and the momentum of a free particle are related by the relation

Substituting expressions (21.7) for E and into this relation and then reducing by , we obtain the equation

which coincides with equation (21.1), if in the latter we put

In the case of a particle moving in a force field characterized by potential energy U, energy E and momentum are related by the relation

Extending expressions (21.7) for E to this case, we obtain

Multiplying this ratio by and moving the term to the left, we arrive at the equation

coinciding with equation (21.1).

The stated reasoning has no evidentiary force and cannot be considered as a derivation of the Schrödinger equation. Their purpose is to explain how this equation could be arrived at.

In quantum mechanics, the concept plays an important role. An operator is a rule by which one function (let's denote it) is associated with another function (let's denote it). Symbolically this is written as follows:

Here is a symbolic designation of the operator (with the same success one could take any other letter with a “hat” above it, for example, etc.). In formula (21.2), the role of Q is played by the function F, and the role of f is the right-hand side of the formula.

Heisenberg was 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 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 electromagnetic field), is not derived, but postulated. The correctness of this equation is confirmed by agreement with the experience obtained from it using the results, 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 the particle in the force field in which it moves, Ψ( x, y, z, t) - required ox new feature particles.

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 explicitly depend on time and has the meaning of potential energy. IN in this case the solution to the Schrödinger equation can be represented as

. (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, of which, by imposing boundary conditions select solutions that have physical meaning. 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 both continuous and 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's carry out qualitative analysis solutions of 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 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. stationary equation Schrödinger, 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. it is quantized.

Quantized energy values E p are called energy levels and the number p, defining energy levels particles are 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, 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 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 characteristic feature 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 must large values quantum numbers go over to the laws of classical physics.

§ 217. General Schrödinger equation. Schrödinger equation for stationary states

The statistical interpretation of da Broglie waves (see § 216) and the Heisenberg uncertainty relation (see § 215) 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 observables on experimental wave properties of particles. The governing equation must be an equation with respect to the wave function (x, y, z, t ), since it is precisely this, or, more precisely, the quantity, that determines the probability of a particle being at the moment of timet in volumedV , i.e. in the area with coordinatesx And x + dx . y And y + dy . zuz + dz . Since the required equation must take into account the wave properties of particles, it must be a wave equation, similar to the equation describing electromagnetic waves.

Basic equationnonrelativistic quantum mechanics 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 Schrödinger equation has the form

(217.1)

Where,T - particle mass, - Laplace operator ,

- imaginary unit,V (x, y, z , t ) - potential function of a particle in the force field in which it moves,(x, y, z, t ) - the desired wave function of the particle.

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., with the speed It is supplemented by the conditions imposed on the wave function: 1) the wave function must be finite, unambiguous and continuous (see § 216); 2) derivatives must be continuous; 3) the function should 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 plane wave. For simplicity, we consider the one-dimensional case. Equation of a plane wave propagating along an axis X, has the form (see § 154), or in complex notation Therefore flat

the de Broglie wave has the form

(217.2)

(it is taken into account that In quantum mechanics, the exponent is taken with a minus sign,

but since it only has a physical meaning, this (see (217.2)) is unimportant. Then

where

Using the relationship between energyE and impulse and substituting expressions

(217.3), we obtain the differential equation

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

If a particle moves in a force field characterized by potential energyU , That

total energyE consists of typical actual and potential energies. Carrying out similar

reasoning and using the relationship betweenE Andr (for this case we'll come

° to 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 Schroednäger equation. For many physical phenomena occurring in the microworld, equation (217.1) can be simplified by eliminating the dependence 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 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 of time, and the dependence on time is expressed by the multiplier

So

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

whence, after dividing by common factors and corresponding transformations

we arrive at the equation defining the function

(217.5)

Equation (217.5) is 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 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 have a 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 task. These energy values ​​are called proper values. The solutions, which correspond to the energy eigenvalues, are called eigenfunctions. Eigenvalues E can form both permanent

discontinuous and discrete series. In the first case they speak of a continuous, or solid, spectrum, in the second - of a discrete spectrum.

§ 218. The principle of causality ■ quantum mechanics

From the uncertainty relationship, a conclusion is often drawn that the principle of causality is not applicable to phenomena occurring in the microcosm. This is based on the following considerations. In classical mechanics, according to the principle of causality - principle classical determinism, based on the known state of the system at a certain moment in time (completely determined by the values ​​of the coordinates and momenta of all particles of the system) and the forces applied to it, one can absolutely accurately determine its state at any subsequent moment. Hence, classical physics is based on the following understanding of causality: state mechanical system V starting moment time with the known law of interaction of particles is the cause, and its state at a subsequent moment is the effect.

On the other hand, microobjects cannot simultaneously have both a certain coordinate and a certain corresponding projection of momentum (set by the uncertainty relation (215.1)), therefore it is concluded that at the initial moment of time the state of the system is not precisely determined. If the state of the system is not determined at the initial moment of time, then subsequent states cannot be predicted, i.e. the principle of causality is violated.

However, no violation of the principle of causality in relation to micro-objects is observed, since in quantum mechanics the concept of the state of a micro-object takes on a completely different meaning than in classical mechanics. In quantum mechanics, the state of a microobject is completely determined by the wave function (x, y,z, t), the square of the modulus of which(x, y,z, t)\ 2 specifies the probability density of finding a particle at a point with coordinates x, y,z.

In turn, the wave function(x, y,z, t) satisfies the Schrödinger equation (217.1), containing the first derivative of the function with respect to time. This also means that specifying a function (for time t 0) determines its value at subsequent moments. Therefore, in quantum mechanics the initial state

There is a cause, and the state at a subsequent moment is a consequence. This is the form of the principle of causality in quantum mechanics, i.e., specifying a function predetermines its values ​​for any subsequent moments. Thus, the state of a system of microparticles, defined in quantum mechanics, unambiguously follows from the previous state, as required by the principle of causality.

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. In this case, any wave is described special type equations - wave equations. All waves, without exception, are waves of the ocean, seismic waves rocks, 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 region higher mathematics, it is so important to understand modern physics, that I will still present it here - in the 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 the Schrödinger 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, the world theoretical physics a storm broke out in a glass of water. 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 system. mathematical point view matrix form. The commotion was caused by the fact that scientists were simply afraid whether the two in equally convincing approaches to describing the microworld. 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 collide with 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 up with the ideas of energy quantization, trying to theoretically explain the 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.

Absolutely black body, completely absorbing 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 terms scientific language 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 education, exhibiting 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 very soon it becomes clear that in the microworld the principles and laws of the macroworld that we are accustomed to 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 can and cannot quantum computer. How physicists make a quantum computer. When it will be created.

French physicist Pierre Simon Laplace put 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 has resolved this question in favor of the probabilistic approach, however, 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 entropy closed system does not decrease, typically increases, and sometimes remains constant if the system is energetically isolated. Using known results quantum theory information, 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 randomness was at the core of nature. Quantum physics implies that subatomic particles can interact faster speed light, and this contradicts his theory of relativity.

From the statistical interpretation of de Broglie waves (see § and the Heisenberg uncertainty relations (see § 215) it followed that the equation of motion in quantum mechanics, describing the movement of microparticles in various force fields, should be an equation from which the observations would follow - experimentally determined wave properties of particles.

The main equation must be an equation with respect to the wave function, since it is precisely it, or, more precisely, the value |Ф|2, that determines the probability of a particle being present at the moment of time t in volume dV, in the area with coordinates and X+ dx, y+dy,

z and 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. Basic equation nonrelativistic quantum mechanics 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. Equation

Schrödinger has the form

Imaginary unit, y,z,t) -

Potential function particles in the force field in which it moves; z,t) - desired wave function

The equation is valid for any particle (with a spin equal to 0; see § 225) moving at a low (compared to the speed of light) speed, i.e. v With. It is supplemented by conditions imposed on the wave function: 1) the wave function must be finite, unambiguous and continuous (see § 216);

2) derivatives -, -, --, must-

dh doo

we need to be continuous; 3) the function |Ф|2 must be integrable; this condition in the simplest cases reduces to

Normalization condition (216.3).

To arrive at the Schrödinger equation, let us consider a freely moving particle, which, according to de Broglie, is associated with. For simplicity, let us consider the one-dimensional case. Equation of a plane wave propagating along an axis X, has the form (see § 154) t) = A cos - or complex notation t)- Therefore, the plane de Broglie wave has the form

(217.2)

(it is taken into account that - = -). In quantum

The exponent is taken with the sign “-”, since only |Ф|2 has a physical meaning, this is unimportant. Then

Using the relationship between energy E and impulse = --) and substituting

expression (217.3), we obtain the differential equation

which coincides with the equation for the case U- O (we considered a free particle).

If a particle moves in a force field characterized by potential energy U, then the total energy E consists of kinetic and potential energies. Carrying out similar reasoning and using the relationship between ("for

Cases = 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 general Schrödinger equation. It is also called time-dependent Schrödinger equation. For many physical phenomena occurring in the microworld, equation (217.1) can be simplified by eliminating the time dependence, 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=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

It is multiplied by e" = e, so

(217.4)

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

Where, after dividing by the common factor e of the corresponding transformations

Formation, we arrive at the equation defining the function

Equation equation

Schrödinger's theory for stationary states. This equation includes the 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, of which through imposing boundary conditions select solutions that have a physical

For the Schrödinger equation such conditions are 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 have a 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 own. Solutions that correspond to the energy eigenvalues ​​are called own functions. Eigenvalues E can form both a continuous and discrete series. In the first case we talk about continuous, or continuous spectrum in the second - about discrete spectrum.

§ 218. The principle of causality in quantum mechanics

From the uncertainty relationship it is often concluded that

the principle of causality to phenomena occurring in the microcosm. This is based on the following considerations. In classical mechanics, according to the principle of causality - the principle of classical determinism, By the known state of the system at a certain moment in time (completely determined by the values ​​of the coordinates and momenta of all particles of the system) and the forces applied to it, one can absolutely accurately determine its state at any subsequent moment. Consequently, classical physics is based on the following understanding of causality: the state of a mechanical system at the initial moment of time with a known law of particle interaction is the cause, and its state at the subsequent moment is the effect.

On the other hand, microobjects cannot simultaneously have both a certain coordinate and a certain corresponding projection of momentum [are given by the uncertainty relation; therefore, it is concluded that at the initial moment of time the state of the system is not precisely determined. If the state of the system is not certain at the initial moment of time, then subsequent states cannot be predicted, i.e. the principle of causality is violated.

However, no violation of the principle of causality in relation to microobjects is observed, since in quantum mechanics the concept of the state of a microobject takes on a completely different meaning than in classical mechanics. In quantum mechanics, the state of a microobject is completely determined by the wave function whose modulus is squared

2 specifies the probability density of finding a particle at a point with coordinates x, y, z.

In turn, the wave function satisfies the equation

Schrödinger containing the first derivative of the function Ф with respect to time. This also means that specifying a function (for a moment in time determines its value at subsequent moments. Consequently, in quantum mechanics, the initial state is the cause, and the state Ф at the subsequent moment is the effect. This is the form of the principle causality in quantum mechanics, i.e., the specification of a function predetermines its values ​​for any subsequent moments. Thus, the state of a system of microparticles defined in quantum mechanics unambiguously follows from the previous state, as required by the principle of causality. .

§219. Movement of a free particle

Freeparticle - a particle moving in the absence of external fields. Since the free one (let it move along the axis X) forces do not act, then the potential energy of the particle U(x) = const and it can be accepted equal to zero. Then the total energy of the particle coincides with its kinetic energy. In this case, the Schrödinger equation (217.5) for stationary states will take the form

(219.1)

By direct substitution we can verify that a particular solution to equation (219.1) is the function - Where A = const and To= const, s eigenvalue energy

The function = = represents only the coordinate part of the wave function. Therefore, the time-dependent wave function, according to (217.4),

(219.3) is a plane monochromatic de Broglie wave [see (217.2)].

From expression (219.2) it follows that the dependence of energy on momentum

turns out to be usual for nonrelativistic particles. Therefore, the energy of a free particle can take any values(since the wave number To can take any positive values), i.e. energy spectrum free particle is continuous.

So free quantum particle is described by a plane monochromatic de Broglie wave. This corresponds to the time-independent probability density of detecting a particle at a given point in space

that is, all positions of a free particle in space are equally probable.

§ 220. 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 using

Rice. 299

(220.4)

relative to the particle V a one-dimensional rectangular “potential well” with infinitely high “walls”. Such a “well” is described by potential energy of the form (for simplicity, we assume that the particle moves along the axis X)

where is the width of the “pit”, A energy is counted from its bottom (Fig. 299).

The Schrödinger equation (217.5) for stationary states in the case of a one-dimensional problem will be written in the form

According to the conditions of the problem (infinitely high “walls”), the particle does not penetrate beyond the “hole”, so 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 = the continuous wave function must also vanish. Consequently, the boundary conditions in this case have the form

Within the “pit” (0 X the Schrödinger equation (220.1) will be reduced to the equation

The general solution of the differential equation (220.3):

Since according to (220.2) = 0, then IN= 0.

(220.5)

Condition (220.2) = 0 is executed only for where n- integers, i.e. it is necessary that

From expressions (220.4) and (220.6) it follows,

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 eigenvalues ​​depending on the integer p. Therefore, the particle energy in

a “potential well” with infinitely high “walls” takes only certain discrete values, those. quantized.

Quantized energy values ​​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 or, as they say, the particle is in a quantum

Substituting into (220.5) the value To from (220.6), we find the eigenfunctions:

Constant of integration A we find from the normalization condition (216.3), which for this case will be written in the form

IN as a result of integrating semi-

A - A eigenfunctions will look like

I Rafiki eigenfunctions(220.8), corresponding to levels

energy (220.7) at n=1.2, 3 are shown in Fig. 300, A. In Fig. 300, b shows the probability density of detecting a particle at various distances from the “walls” of the hole, equal to =

For n= 1, 2 and 3. It follows from the figure that, for example, in a quantum state with n= 2, the particle cannot be in the middle of the “well,” while equally often it can be in its left and right parts. This behavior of the particle indicates that the ideas about particle trajectories in quantum mechanics are untenable. From expression (220.7) it follows that the energy interval between two

The neighboring levels are equal to

For example, for an electron with well dimensions - 10"1 m (free electrical

Thrones in metal) 10 J

That is, the energy levels are located so closely that the spectrum can practically be considered continuous. If the dimensions of the well are commensurate with atomic m), then for an electron J eV, i.e. Obviously discrete energy values ​​are obtained (line spectrum).

Thus, the application of 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.

Besides,

Consideration of this problem leads to the conclusion that the particle is “in a potential well” with infinitely high “ walls"cannot have less energy

Minimum, equal to [see. (220.7)].

The presence of a nonzero minimum energy is not accidental and follows from the uncertainty relation. Coordinate uncertainty Oh particles in a "pit" wide Ah= Then, according to the uncertainty relation, the impulse cannot have an exact, in this case zero, value. Impulse uncertainty

Such a spread of values

impulse corresponds to kinetic energy

All other levels (n > 1) have energy exceeding this minimum value.

From formulas (220.9) and (220.7) it follows that for large quantum numbers

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 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.

More general interpretation of the principle of correspondence: any new, more general theory, which is a development of the classical one, does not reject it completely, but includes the classical theory, indicating the boundaries of its application, and in certain limiting cases new theory goes into the old one. Thus, the formulas of kinematics and dynamics special theory relativity goes over at v c to the formulas of Newtonian mechanics. For example, although the da Broglie hypothesis attributes wave properties to all bodies, in those cases when we are dealing with macroscopic bodies, their wave properties can be neglected, i.e. apply classical mechanics Newton.

§ 221. The passage of a particle through a potential barrier.

Tunnel effect

the simplest potential barrier of a rectangular shape (Fig. for one-dimensional (along the axis of motion of the particle. For a potential barrier of a rectangular shape with a height and width / we can write

Under the given conditions of the problem, a classical particle, having energy E, or will pass unhindered over the barrier (if E > U), or will be reflected from it (if E< U) will move in reverse side, i.e. she cannot penetrate the barrier. For a microparticle, even with E > U, available excellent from zero the probability that a particle will be reflected from the barrier and will move in the opposite direction. At E there is also a non-zero probability that the particle will end up in the region x> those. will penetrate the barrier. Similar seemingly paradoxical conclusions follow directly from the solution of the Schrödinger equation, describing

412

describing the movement of a microparticle under the conditions of this problem.

Equation (217.5) for stationary states for each of the highlighted Figs. 301, A region has

(for regions

(for region

General solutions these differential equations:

Solution (221.3) also contains waves (after multiplication by a time factor) propagating in both directions. However, in the area 3 there is only a wave that has passed through the barrier and propagates from left to right. Therefore, the coefficient of formula (221.3) should be taken equal to zero.

In the area 2 the decision depends on relations E>U or E Of physical interest is the case when the total energy of the particle is less than the height of the potential barrier, since at E the laws of classical physics clearly do not allow a particle to penetrate the barrier. In this case, according to q= - imaginary number, where

(for region

(for area 2);

Meaning q and 0, we obtain solutions to the Schrödinger equation for three regions in the following form:

(for region 3).

IN in particular for the region 1 the complete wave function, according to (217.4), will have the form

In this expression, the first term represents a plane wave of type (219.3), propagating in the positive direction of the axis X(corresponds to a particle moving towards the barrier), and the second is a wave propagating in the opposite direction, i.e. reflected from the barrier (corresponds to a particle moving from the barrier to the left).

(for region 3).

In the area 2 the function no longer corresponds to plane waves propagating in both directions, since the exponents of the exponents are not imaginary, but real. It can be shown that for the special case of a high and wide barrier, when 1,

The qualitative nature of the functions is illustrated in Fig. 301, from which it follows that the wave-

The function is not equal to zero even inside the barrier, but in the region 3, if the barrier is not very wide, it will again have the form of de Broglie waves with the same impulse, i.e., with the same frequency, but with a smaller amplitude. Consequently, we found that a particle has a nonzero probability of passing through a potential barrier of finite width.

Thus, quantum mechanics leads to a fundamentally new specific quantum phenomenon, called tunnel effect, as a result of which a microobject can “pass” through a potential barrier. via A joint solution of the equations for a rectangular potential barrier gives (assuming that the transparency coefficient is small compared to unity)

where is a constant factor that can be equal to one; U- potential barrier height; E - particle energy; - width of the barrier.

From expression (221.7) it follows that D strongly depends on mass T particles, width/barrier and from (U - The wider the barrier, the less likely a particle will pass through it.

For a potential barrier of arbitrary shape (Fig. 302), satisfying the conditions of the so-called semiclassical approximation(a fairly smooth shape of the curve), we have

Where U= U(x).

From the classical point of view, the passage of a particle through a potential barrier at E impossible, since the particle, being in the barrier region, would have to have negative kinetic energy. The tunnel effect is a specific quantum effect.

The passage of a particle through a region into which, according to the laws of classical mechanics, it cannot penetrate, can be explained by the uncertainty relation. Momentum uncertainty Ar on the segment Ah = is Ar > -. Associated with this scatter in the values ​​of momentum is the kinetic

302

Czech energy may be

sufficient for complete

the particle energy turned out to be greater than the potential.

The foundations of the theory of tunnel transitions are laid in the works of L. I. Mandelshtam

Tunneling through a potential barrier underlies many phenomena in solid state physics (for example, phenomena in the contact layer at the boundary of two semiconductors), atomic and nuclear physics (for example, decay, the occurrence of thermonuclear reactions).

§ 222. Linear harmonic oscillator

In quantum mechanics

Linear harmonic oscillator- a system undergoing one-dimensional motion under the action of a quasi-elastic force is a model used in many problems of classical and quantum theory (see § 142). Spring, physical and mathematical pendulums are examples of classical harmonic oscillators.

Potential energy of a harmonic oscillator [see. (141.5)] is equal to

Where is the natural frequency of the oscillator; T - particle mass.

Dependence (222.1) has the form of a parabola (Fig. 303), i.e. The “potential well” in this case is parabolic.

The amplitude of small oscillations of a classical oscillator is determined by its total energy E(see Fig. 17).

Dinger taking into account expression (222.1) for potential energy. Then the stationary states of the quantum oscillator are determined by the Schrödinger equation of the form

= 0, (222.2)

Where E - total energy of the oscillator. In the theory of differential equations

It is proved that equation (222.2) can be solved only for the eigenvalues ​​of energy

(222.3)

Formula (222.3) shows that the energy of a quantum oscillator can

have only discrete values, i.e. quantized. The energy is limited from below by a nonzero minimum value of energy, as for a rectangular “well” with infinitely high “walls” (see § 220). = Su-

existence of minimum energy - it is called energy of zero-point vibrations - is typical for quantum systems and is a direct consequence of the uncertainty relation.

The presence of zero-point oscillations means that the particle cannot be at the bottom of the “potential well” (regardless of the shape of the well). In fact, “falling to the bottom of the hole” is associated with the vanishing of the particle’s momentum, and at the same time its uncertainty. Then the uncertainty of the coordinate becomes arbitrarily large, which, in turn, contradicts the presence of the particle in

"potential hole".

The conclusion about the presence of energy of zero-point oscillations of a quantum oscillator contradicts the conclusions of the classical theory, according to which the lowest energy that an oscillator can have is equal to zero (corresponds to a particle at rest in the equilibrium position). For example, according to the conclusions of classical physics at T= 0, the energy of the vibrational motion of the atoms of the crystal should vanish. Consequently, light scattering due to atomic vibrations should also disappear. However, experiment shows that the intensity of light scattering with decreasing temperature is not equal to zero, but tends to a certain limiting value, indicating that when T 0 vibrations of atoms in a crystal do not stop. This confirms the presence of zero oscillations.

From formula (222.3) it also follows that the energy levels of a linear harmonic oscillator are located at equal distances from each other (see Fig. 303), namely, the distance between adjacent energy levels is equal to and the minimum energy value is =

A rigorous solution to the problem of a quantum oscillator leads to another significant difference from the classical