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 participate 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. 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 in given point 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 for understanding modern physics that I will still present it here - in its simplest form (the so-called “one-dimensional stationary Schrödinger equation”). The above wave function the probability distribution, denoted by the Greek letter (“psi”), is the solution to the following differential equation(It’s okay if you don’t understand it; the main thing is to take it on faith that this equation indicates that probability behaves like a wave):
where is the distance, is Planck’s constant, and , and are, respectively, the mass, total energy and potential energy of the particle.
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 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.
| Comments: 0 |
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.
Absolutely black body, which 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. At the beginning of the 20th century, scientists penetrated into the hidden depths of matter, into the subatomic building blocks 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.
No. 1 The stationary Schrödinger equation has the form . This equation is written for...
The stationary Schrödinger equation in the general case has the form
, where is the potential energy of the microparticle. For the one-dimensional case. In addition, the particle cannot be inside the potential box, but outside the box, because its walls are infinitely high. Therefore, this Schrödinger equation is written for a particle in a one-dimensional box with infinitely high walls.
Linear harmonic oscillator
ü Particles in a one-dimensional potential box with infinitely high walls
Particles in a three-dimensional potential box with infinitely high walls
Electron in a hydrogen atom
Establish correspondences between quantum mechanical problems and Schrödinger equations for them.
General form stationary equation Schrödinger has the form:
Particle potential energy,
Laplace operator. For simultaneous case
The expression for the potential energy of a harmonic oscillator, that is, a particle performing one-dimensional motion under the action of a quasi-elastic force, has the form U=.
The value of the potential energy of an electron in a potential box with infinitely high walls is U = 0. An electron in a hydrogen-like atom has potential energy For a hydrogen atom Z = 1.
Thus, for an electron in a one-dimensional potential box, the Schrödinger equation has the form:
Using the wave function, which is a solution to the Schrödinger equation, we can determine...
Answer options: (Indicate at least two answer options)
Average values of physical quantities characterizing a particle
The probability that a particle is located in a certain region of space
Particle trajectory
Particle location
The value has the meaning of probability density (probability per unit volume), that is, it determines the probability of a particle being in the corresponding place in space. Then the probability W of detecting a particle in a certain region of space is equal to
Schrödinger equation (specific situations)
No. 1The eigenfunctions of an electron in a one-dimensional potential box with infinitely high walls have the form 
where is the width of the box, a quantum number that has the meaning of a number energy level. If the number of function nodes on the segment and , then equals...
Number of nodes, i.e. the number of points at which the wave function on a segment vanishes is related to the number of the energy level by the relation . Then 
, and by condition this ratio is equal to 1.5. Solving the resulting equation for , we find that
Nuclear reactions.
№1 IN nuclear reaction the letter represents the particle...
From the laws of conservation of mass number and charge number it follows that the charge of the particle is zero, and mass number equals 1. Therefore, the letter denotes a neutron.
ü Neutron
Positron
Electron
The graph shows on a semi-logarithmic scale the dependence of the change in the number of radioactive nuclei of an isotope on time. Constant radioactive decay in is equal to...(round the answer to whole numbers)
The number of radioactive nuclei changes over time according to the law - the initial number of nuclei, - the radioactive decay constant. Taking the logarithm of this expression, we get
ln 
.Hence, 
=0,07
Conservation laws in nuclear reactions.
The reaction cannot proceed due to a violation of the conservation law...
In all fundamental interactions, conservation laws are satisfied: energy, momentum, angular momentum (spin) and all charges (electric, baryon and lepton). These conservation laws not only limit the consequences of various interactions, but also determine all the possibilities of these consequences. To choose the correct answer, you need to check which conservation law prohibits and which allows the given reaction of interconversion of elementary particles. According to the law of conservation of lepton charge in a closed system during any process, the difference between the number of leptons and antileptons is preserved. We agreed to calculate for leptons: . lepton charge and for antileptons: . lepton charge. For all other elementary particles, lepton charges are assumed to be zero. The reaction cannot proceed due to a violation of the law of conservation of lepton charge, because
ü Lepton charge
Baryon charge
Spin angular momentum
The reaction cannot proceed due to a violation of the conservation law...
In all fundamental interactions, conservation laws are satisfied: energy, momentum, angular momentum (spin) and all charges (electric Q, baryon B and lepton L). These conservation laws not only limit the consequences of various interactions, but also determine all the possibilities of these consequences. According to the law of conservation of baryon charge B, for all processes involving baryons and antibaryons, the total baryon charge is conserved. Baryons (n, p nucleons and hyperons) are assigned a baryon charge
B = -1, and for all other particles the baryon charge is B = 0. The reaction cannot proceed due to a violation of the law of baryon charge B, because (+1)+(+1)
Answer options: lepton charge, spin angular momentum, electric charge. Q=0, antiproton (
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 a 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 of a particle in the force field in which it moves, Ψ (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. 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.
Lecture 5. SCHRÖDINGER EQUATION.
Probabilistic meaning of de Broglie waves. Wave function.
De Broglie waves have a specific quantum nature that has no analogy with waves in classical physics. These are not electromagnetic waves, since their propagation in space is not associated with the propagation of any electromagnetic field. The question about the nature of waves can be formulated as a question about the physical meaning of the amplitude of these waves. Instead of amplitude, it is more convenient to choose a wave intensity proportional to the square of the amplitude modulus.
From experiments on electron diffraction it follows that in these experiments an unequal distribution of electron beams reflected in different directions is revealed. From a wave point of view, the presence of maxima in the number of electrons in some directions means that these directions correspond to the highest intensity of de Broglie waves. The intensity of the waves at a given point in space determines the probability density of electrons hitting this point in 1 second.
This served as the basis for a kind of statistical, probabilistic interpretation of de Broglie waves.
The squared magnitude of the de Broglie wave amplitude at a given point is a measure of the probability that a particle is detected at that point.
In order to describe the probability distribution of finding a particle at a given moment in time at some point in space, we introduce a function that is a function of time and coordinates, denoted by the Greek letter ψ and is called wave function or simply psi function.
By definition, the probability that a particle has a coordinate within x, x+dx.
If 
, then is the probability that the particle is in the volume dxdydz.
Therefore, the probability that a particle is located in a volume element dV is proportional to the square of the modulus of the psi function and the volume element dV.
The physical meaning is not the function ψ itself, but the square of its modulus , where ψ* is the function complex conjugate to ψ. The magnitude makes sense probability density, i.e. defines probability of a particle being at a given point in space. In other words, it determines the intensity of de Broglie waves. The wave function is the main characteristic of the state of micro-objects (elementary particles, atoms, molecules).
Nonstationary Schrödinger equation.
Newton's equations in classical mechanics make it possible for macroscopic bodies to solve the main problem of mechanics - given the forces acting on the body (or a system of bodies) and the initial conditions, find the coordinates of the body and its speed for any moment in time, i.e. describe the movement of a body in space and time.
When posing a similar problem in quantum mechanics, it is necessary to take into account restrictions on the possibility of applying the classical concepts of coordinates and momentum to microparticles. Since the state of a microparticle in space at a given moment in time is determined by the wave function, or more precisely, by the probability of finding the particle at point x,y,z at moment t, the fundamental equation of quantum mechanics is an equation with respect to the psi function.
This equation was obtained in 1926 by Schrödinger. Like Newton's equations of motion, Schrödinger's equation is postulated rather than derived. The validity of this equation is proven by the fact that the conclusions obtained with its help are in good agreement with experiments.
The Schrödinger equation has the form
,
here m is the mass of the particle, i is the imaginary unit, is the Laplace operator, the result of the action of which on a certain function
.
U(x,y,z,t) – within the framework of our problems, the potential energy of a particle moving in a force field. From the Schrödinger equation it follows that the type of psi function is determined by the function U, i.e. ultimately, the nature of the forces acting on the particle.
The Schrödinger equation is supplemented by important conditions that are imposed on the psi function. There are three conditions:
1) the function ψ must be finite, continuous and unambiguous;
2) derivatives 
must be continuous
3) the function must be integrable, i.e. integral
must be final. In the simplest cases, the third condition reduces to the normalization condition
This means that the presence of a particle somewhere in space is a reliable event and its probability must be equal to one. The first two conditions are the usual requirements imposed on the desired solution of the differential equation.
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 (U = 0).
Let us compare to it, according to de Broglie’s idea, a plane wave
Let's replace and rewrite
.
Differentiating this expression once with respect to t, and a second time twice with respect to x, we obtain
The energy and momentum of a free particle are related by the relation
Substituting the expressions for E and p 2 into this relationship
The last expression coincides with the Schrödinger equation at U =0.
In the case of particle motion in a force field characterized by potential energy U, energy E and momentum p are related by the relation
The stated reasoning has no evidential value and cannot be considered as a derivation of the Schrödinger equation. Their purpose is to explain how one can come to establish this equation.
| | | next lecture ==> | |
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. The equation
Schrödinger has the form
|
Imaginary unit, y,z,t) -
Potential function of a particle 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
The 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, from which, by imposing boundary conditions, solutions are selected 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. 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 .
§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 can be taken 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, with energy eigenvalue
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 range free particle is continuous.
Thus, a 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 P- 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 result of integration of semi-
A - A eigenfunctions will look like
I graphs of 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. From the figure it follows that, for example, in a quantum state with P= 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 a particle “in a potential well” with infinitely high “walls” cannot have an energy less than
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 P 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: Every new, more general theory, which is a development of the classical one, does not completely reject it, but includes the classical theory, indicating the boundaries of its application, and in certain limiting cases the new theory passes into the old one. Thus, the formulas of kinematics and dynamics of the special theory of relativity transform when 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 Newtonian mechanics.
§ 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 the opposite direction, i.e. she cannot penetrate the barrier. For a microparticle, even with E > U, there is a nonzero probability that the 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 area
General solutions to 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 area 2
the solution depends on the relations E>U or E
(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 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 the momentum of 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 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