Laboratory computer workshop. —§23 Wave (field) properties of particles Which particles have wave properties
You can of course call it bullshit,
but I have met such nonsense that in
compared with her, this seems sensible
dictionary.
L. Carroll
What is the planetary model of the atom and what is its disadvantage? What is the essence of Bohr's model of the atom? What is the hypothesis about the wave properties of particles? What predictions does this hypothesis give about the properties of the microworld?
Lesson-lecture
CLASSICAL ATOMIC MODELS AND THEIR DISADVANTAGES... The ideas that atoms are not indivisible particles and contain elementary charges as constituent particles were first expressed in late XIX v. The term "electron" was proposed in 1881 by the English physicist George Stoney. In 1897, the electronic hypothesis received experimental confirmation in the studies of Emil Wiechert and Joseph John Thomson. From that moment on, the creation of various electronic models of atoms and molecules began.
Thomson's first model assumed that the positive charge is uniformly dispersed throughout the atom, and electrons are interspersed into it, like raisins in a bun.
The inconsistency of this model with the experimental data became clear after the 1906 experiment by Ernest Rutherford, who investigated the process of scattering of α-particles by atoms. From the experience, it was concluded that the positive charge is concentrated inside the formation, which is significantly smaller than the size of the atom. This formation was called an atomic nucleus, the dimensions of which were 10 -12 cm, and the dimensions of the atom were 10 -8 cm. In accordance with the classical concepts of electromagnetism, a Coulomb attraction force must act between each electron and the nucleus. The dependence of this force on distance should be the same as in the law of universal gravitation. Therefore, the movement of electrons in an atom should be similar to the movement of planets Solar system... So was born planetary model of the atom Rutherford.
The short lifetime of the atom and the continuous spectrum of radiation, following from the planetary model, showed its inconsistency in describing the motion of electrons in an atom.
Further investigation of the stability of the atom gave a stunning result: calculations showed that in a time of 10 -9 s, the electron must fall onto the nucleus due to the loss of energy due to radiation. In addition, such a model gave continuous, rather than discrete, emission spectra of atoms.
THEORY OF THE BORON ATOM... The next important step in the development of atomic theory was taken by Niels Bohr. The most important hypothesis put forward by Bohr in 1913 was the hypothesis of the discrete structure of the energy levels of an electron in an atom. This position is illustrated in the energy diagrams (Fig. 21). Traditionally, energy diagrams are plotted along the vertical axis.
Rice. 21 Satellite energy in the Earth's gravitational field (a); energy of an electron in an atom (b)
The difference between the motion of a body in a gravitational field (Fig. 21, a) from the motion of an electron in an atom (Fig. 21, b) in accordance with Bohr's hypothesis is that the energy of a body can continuously change, and the energy of an electron at negative values can take the series discrete values shown in the figure by segments blue... These discrete values were called energy levels or, in other words, energy levels.
Of course, the idea of discrete energy levels was taken from Planck's hypothesis. According to Bohr's theory, the change in the energy of an electron could only occur in a jump (from one energy level to another). During these transitions, a quantum of light is emitted (transition down) or absorbed (transition up), the frequency of which is determined from the Planck formula hv = E quantum = ΔE of the atom, i.e., the change in the energy of the atom is proportional to the frequency of the emitted or absorbed light quantum.
Bohr's theory perfectly explained the line character of atomic spectra. However, the theory did not actually give an answer to the question of the reason for the discreteness of the levels.
WAVES OF MATTER... The next step in the development of the theory of the microworld was taken by Louis de Broglie. In 1924, he suggested that the motion of microparticles should be described not as classical mechanical motion, but as some kind of wave motion. It is from the laws of wave motion that recipes for calculating various observable quantities must be obtained. So in science along with waves electromagnetic field waves of matter appeared.
Hypothesis about wave character particle motion was as bold as Planck's hypothesis of the discrete properties of the field. An experiment that directly confirms de Broglie's hypothesis was staged only in 1927. In this experiment, the diffraction of electrons by a crystal was observed, similar to the diffraction of an electromagnetic wave.
Bohr's theory was an important step in understanding the laws of the microworld. It was the first to introduce the provision on discrete values of the energy of an electron in an atom, which corresponded to experiment and subsequently entered the quantum theory.
The hypothesis of waves of matter made it possible to explain the discrete nature of energy levels. It was known from the theory of waves that a wave limited in space always has discrete frequencies. An example is a wave in such musical instrument like a flute. The frequency of sounding in this case is determined by the dimensions of the space that the wave is limited by (the dimensions of the flute). It turns out that this is a common property of waves.
But in accordance with Planck's hypothesis, the frequencies of a quantum of an electromagnetic wave are proportional to the energy of the quantum. Consequently, the energy of the electron must take on discrete values.
De Broglie's idea turned out to be very fruitful, although, as already mentioned, a direct experiment confirming the wave properties of the electron was carried out only in 1927. In 1926, Erwin Schrödinger derived the equation that the electron wave must obey, and, having solved this equation in relation to atom of hydrogen, received all the results that Bohr's theory was capable of giving. In fact, this was the beginning of the modern theory describing processes in the microworld, since the wave equation was easily generalized for a variety of systems - many-electron atoms, molecules, crystals.
The development of the theory led to the understanding that the wave corresponding to a particle determines the probability of finding a particle at a given point in space. This is how the concept of probability entered the physics of the microworld.
According to the new theory, the wave corresponding to the particle completely determines the motion of the particle. But the general properties of waves are such that the wave cannot be localized at any point in space, that is, it makes no sense to talk about the coordinates of the particle at a given moment in time. The consequence of this was the complete exclusion from the physics of the microworld of such concepts as the trajectory of a particle and electron orbits in the atom. The beautiful and visual planetary model of the atom, as it turned out, does not correspond to the real movement of electrons.
All processes in the microcosm are of a probabilistic nature. By calculations, only the probability of a particular process can be determined
In conclusion, let's return to the epigraph. Hypotheses about waves of matter and field quanta seemed to be nonsense to many physicists brought up in the traditions of classical physics. The fact is that these hypotheses are deprived of the usual clarity that we have when making observations in the macrocosm. However, the subsequent development of the science of the microworld led to such ideas that ... (see the epigraph to the paragraph).
- What experimental facts did Thomson's model of the atom contradict?
- What of Bohr's model of the atom remained in modern theory and what was discarded?
- What ideas contributed to de Broglie's hypothesis about the waves of matter?
4.4.1. De Broglie's hypothesis
An important stage in the creation of quantum mechanics was the discovery of the wave properties of microparticles. The idea of wave properties was originally expressed as a hypothesis by the French physicist Louis de Broglie.
For many years, physics has been dominated by the theory that light is an electromagnetic wave. However, after the work of Planck (thermal radiation), Einstein (photoelectric effect) and others, it became obvious that light has corpuscular properties.
To explain some physical phenomena, it is necessary to consider light as a stream of particles-photons. The corpuscular properties of light do not reject, but complement its wave properties.
So, photon is an elementary particle of light with wave properties.
Photon momentum formula
| . | (4.4.3) |
According to de Broglie, the motion of a particle, for example, an electron, is similar to a wave process with a wavelength λ determined by formula (4.4.3). These waves are called de Broglie waves... Consequently, particles (electrons, neutrons, protons, ions, atoms, molecules) can exhibit diffraction properties.
K. Davisson and L. Jermer were the first to observe electron diffraction on a single crystal of nickel.
The question may arise: what happens to individual particles, how are the maxima and minima formed during the diffraction of individual particles?
Experiments on the diffraction of beams of electrons of very low intensity, that is, as it were, of individual particles, have shown that in this case the electron is not "smeared" in different directions, but behaves like a whole particle. However, the probability of deflection of an electron in separate directions as a result of interaction with a diffracted object is different. The electrons are most likely to hit those places that, according to the calculation, correspond to the diffraction maxima, they are less likely to hit the places of minima. Thus, wave properties are inherent not only to the collective of electrons, but also to each electron separately.
4.4.2. Wave function and its physical meaning
Since a wave process is associated with a microparticle, which corresponds to its motion, the state of particles in quantum mechanics is described by a wave function that depends on coordinates and time:.
If the force field acting on the particle is stationary, that is, independent of time, then the ψ-function can be represented as a product of two factors, one of which depends on time and the other on coordinates:
this implies physical meaning wave function:
4.4.3. Uncertainty ratio
One of the important provisions of quantum mechanics are the uncertainty relations proposed by W. Heisenberg.
Let the position and momentum of the particle be measured simultaneously, while the inaccuracies in the definitions of the abscissa and the projection of the momentum on the abscissa axis are equal to Δx and Δp x, respectively.
In classical physics, there are no restrictions that prohibit with any degree of accuracy to simultaneously measure both one and the other quantity, that is, Δx → 0 and Δp x → 0.
In quantum mechanics, the situation is fundamentally different: Δx and Δр x, corresponding to the simultaneous determination of x and р x, are related by the dependence
Formulas (4.4.8), (4.4.9) are called uncertainty relations.
Let us explain them with one model experiment.
When studying the phenomenon of diffraction, attention was drawn to the fact that a decrease in the slit width during diffraction leads to an increase in the width of the central maximum. A similar phenomenon will occur in the case of electron diffraction by the slit in the model experiment. A decrease in the slit width means a decrease in Δ x (Fig. 4.4.1), this leads to a greater "smearing" of the electron beam, that is, to a greater uncertainty in the momentum and particle velocity.
Rice. 4.4.1. Explanation of the uncertainty relation.
The uncertainty relation can be represented as
| , | (4.4.10) |
where ΔE is the uncertainty of the energy of a certain state of the system; Δt is the time interval at which it exists. Relation (4.4.10) means that the shorter the lifetime of any state of the system, the more uncertain its energy value. Energy levels E 1, E 2, etc. have a certain width (Figure 4.4.2)), depending on the time the system is in the state corresponding to this level.
Rice. 4.4.2 Energy levels E 1, E 2, etc. have a certain width.
The "blurring" of the levels leads to the uncertainty of the energy ΔE of the emitted photon and its frequency Δν during the transition of the system from one energy level to another:
,
where m is the mass of the particle; ; E and E n are its total and potential energies (potential energy is determined by the force field in which the particle is located, and for the stationary case does not depend on time)
If the particle moves only along a certain line, for example, along the OX axis (one-dimensional case), then the Schrödinger equation is greatly simplified and takes the form
 | (4.4.13) |
One of the most simple examples the use of the Schrödinger equation is to solve the problem of the motion of a particle in a one-dimensional potential well.
4.4.5. Application of the Schrödinger equation to the hydrogen atom. Quantum numbers
Describing the states of atoms and molecules using the Schrödinger equation is a rather complex problem. It is most simply solved for one electron in the field of the nucleus. Such systems correspond to a hydrogen atom and hydrogen-like ions (singly ionized helium atom, doubly ionized lithium atom, etc.). However, in this case, the solution of the problem is difficult, therefore, we restrict ourselves to only a qualitative presentation of the issue.
First of all, the potential energy should be substituted into the Schrödinger equation (4.4.12), which for two interacting point charges - e (electron) and Ze (nucleus) - located at a distance r in vacuum, is expressed as follows:
This expression is a solution to the Schrödinger equation and completely coincides with the corresponding formula of Bohr's theory (4.2.30)
Figure 4.4.3 shows the levels of possible values of the total energy of the hydrogen atom (E 1, E 2, E 3, etc.) and a graph of the dependence of the potential energy E n on the distance r between the electron and the nucleus. As the principal quantum number n increases, r increases (see 4.2.26), and the total (4.4.15) and potential energies tend to zero. Kinetic energy also tends to zero. The shaded region (E> 0) corresponds to the state of a free electron.
Rice. 4.4.3. The levels of possible values of the total energy of the hydrogen atom are shown.
and a graph of potential energy versus distance r between an electron and a nucleus.
Second quantum number - orbital l, which for a given n can take on the values 0, 1, 2,…., n-1. This number characterizes the orbital angular momentum L i of the electron relative to the nucleus:
The fourth quantum number - spin m s... It can take only two values (± 1/2) and characterizes the possible values of the electron spin projection:
| .(4.4.18) |
The state of an electron in an atom with a given n and l is denoted as follows: 1s, 2s, 2p, 3s, etc. Here the digit indicates the value of the principal quantum number, and the letter indicates the orbital quantum number: the symbols s, p, d, f correspond to the values l = 0, 1, 2.3, etc.
By the beginning of the 20th century, both phenomena that confirm the presence of wave properties in light (interference, polarization, diffraction, etc.) and phenomena that were explained from the standpoint of the corpuscular theory (photoelectric effect, Compton effect, etc.) were known in optics. At the beginning of the 20th century, a number of effects were discovered for particles of matter that are outwardly similar to optical phenomena characteristic of waves. So, in 1921, Ramsauer, while studying the scattering of electrons by argon atoms, found that with a decrease in the electron energy from several tens of electron-volts, the effective cross section for elastic scattering of electrons on argon increases (Figure 4.1).
But at an electron energy of ~ 16 eV, the effective cross section reaches a maximum and decreases with a further decrease in the electron energy. At an electron energy of ~ 1 eV, it becomes close to zero, and then begins to increase again.
Thus, near ~ 1 eV, electrons do not seem to experience collisions with argon atoms and fly through the gas without scattering. The same behavior is characteristic of the cross section for the scattering of electrons by other atoms of inert gases, as well as by molecules (the latter was discovered by Townsend). This effect is analogous to the formation of a Poisson spot when light is diffracted by a small screen.
Another interesting effect is the selective reflection of electrons from the surface of metals; it was studied in 1927 by American physicists Davisson and Germer, and also independently of them English physicist J.P. Thomson.
Parallel beam of monoenergetic electrons from cathode ray tube(Figure 4.2) was directed onto a nickel plate. The reflected electrons were captured by a collector connected to a galvanometer. The collector is installed at any angle relative to the incident beam (but in the same plane with it).
As a result of the Davisson-Jermer experiments, it was shown that the angular distribution of scattered electrons has the same character as the distribution of X-rays scattered by the crystal (Figure 4.3). When studying the X-ray diffraction by crystals, it was found that the distribution of diffraction maxima is described by the formula
where is the constant crystal lattice, is the diffraction order, is the wavelength of the X-ray radiation.
In the case of neutron scattering by a heavy nucleus, a typical diffraction distribution of scattered neutrons also appeared, similar to that observed in optics when light is diffracted by an absorbing disk or ball.
The French scientist Louis de Broglie in 1924 expressed the idea that particles of matter have both corpuscular and wave properties. At the same time, he assumed that a plane monochromatic wave corresponds to a particle moving freely at a constant speed
where and are its frequency and wave vector.
Wave (4.2) propagates in the direction of particle motion (). Such waves are called phase waves, waves of matter or de Broglie waves.
De Broglie's idea was to expand the analogy between optics and mechanics, and compare wave optics with wave mechanics, trying to apply the latter to intra-atomic phenomena. An attempt to ascribe to the electron, and in general to all particles, like photons, a dual nature, to endow them with wave and corpuscular properties interconnected by the quantum of action - such a task seemed extremely necessary and fruitful. “… It is necessary to create a new mechanics of a wave nature, which will relate to the old mechanics as wave optics to geometric optics,” de Broglie wrote in his book “The Revolution in Physics”.
A particle of mass moving with speed has energy
and momentum
and the state of motion of a particle is characterized by a four-dimensional vector of energy-momentum ().
On the other hand, in the wave pattern we use the concept of frequency and wave number (or wavelength), and the corresponding 4-vector for a plane wave is ().
Since both of the above descriptions are different aspects of the same physical object, there must be an unambiguous connection between them; the relativistically invariant relation between the 4-vectors is
Expressions (4.6) are called de Broglie's formulas... The de Broglie wavelength is thus determined by the formula
(here). It is this wavelength that should appear in the formulas for the wave description of the Ramsauer - Townsend effect and the Davisson - Jermer experiments.
For electrons accelerated electric field with potential difference B, de Broglie wavelength nm; at kV = 0.0122 nm. For a hydrogen molecule with energy J (at = 300 K) = 0.1 nm, which coincides in order of magnitude with the wavelength of X-ray radiation.
Taking into account (4.6), formula (4.2) can be written in the form of a plane wave
corresponding particle with momentum and energy.
De Broglie waves are characterized by phase and group velocities. Phase Velocity is determined from the condition of constancy of the wave phase (4.8) and for a relativistic particle is equal to
that is, it is always greater than the speed of light. Group speed de Broglie waves is equal to the particle velocity:
From (4.9) and (4.10), the relationship between the phase and group velocities of de Broglie waves follows:
What is the physical meaning of de Broglie waves and what is their relationship with particles of matter?
Within the framework of the wave description of the motion of a particle, a significant epistemological complexity was presented by the question of its spatial localization. De Broglie waves (4.2), (4.8) fill all space and exist indefinitely. The properties of these waves are always and everywhere the same: their amplitude and frequency are constant, the distances between wave surfaces are constant, etc. On the other hand, microparticles retain their corpuscular properties, that is, they have a certain mass localized in a certain area of space. In order to get out of this situation, the particles began to be represented not by monochromatic de Broglie waves, but by sets of waves with close frequencies (wave numbers) - wave packets:
in this case, the amplitudes differ from zero only for waves with wave vectors enclosed in the interval (). Since the group velocity of the wave packet is equal to the speed of the particle, it was proposed to represent the particle in the form of a wave packet. But this idea is untenable for the following reasons. A particle is a stable formation and does not change as such during its movement. The same properties must be possessed by a wave packet claiming to represent a particle. Therefore, it is necessary to require that in the course of time the wave packet retains its spatial shape or, at least, its width. However, since the phase velocity depends on the momentum of the particle, then (even in a vacuum!) There should be a dispersion of de Broglie waves. As a result, the phase relations between the waves of the packet are violated, and the packet spreads. Therefore, the particle represented by such a packet must be unstable. This conclusion is contrary to experience.
Further, the opposite assumption was put forward: the particles are primary, and the waves represent their formations, that is, they arise like sound in a medium consisting of particles. But such a medium should be sufficiently dense, because it makes sense to talk about waves in a medium of particles only when the average distance between particles is very small in comparison with the wavelength. And in experiments in which the wave properties of microparticles are found, this is not done. But even if this difficulty is overcome, the point of view indicated must still be rejected. Indeed, it means that wave properties are inherent in systems of many particles, and not in individual particles. Meanwhile, the wave properties of particles do not disappear even at low intensities of the incident beams. In the experiments of Biberman, Sushkin and Fabrikant, carried out in 1949, beams of electrons were used so weak that the average time interval between two successive passages of an electron through a diffraction system (crystal) was 30,000 (!) Times longer than the time spent by one electron to pass the entire device. Under these conditions, the interaction between electrons, of course, did not play any role. Nevertheless, with a sufficiently long exposure on a photographic film placed behind the crystal, a diffraction pattern appeared, which did not differ in any way from the pattern obtained with a short exposure with electron beams, the intensity of which was 10 7 times higher. It is only important that in both cases the total number of electrons falling on the photographic plate is the same. This shows that individual particles also have wave properties. The experiment shows that one particle does not give a diffraction pattern, each individual electron causes blackening of the photographic plate in a small area. The entire diffraction pattern can be obtained only by hitting the plate with a large number of particles.
The electron in the considered experiment fully retains its integrity (charge, mass and other characteristics). This is the manifestation of its corpuscular properties. At the same time, the manifestation of wave properties is also evident. The electron never hits that part of the photographic plate where there should be a minimum of the diffraction pattern. It can be found only near the position of the diffraction maxima. In this case, it is impossible to indicate in advance in which specific direction this particular particle will fly.
The idea that both corpuscular and wave properties are manifested in the behavior of micro-objects is enshrined in the term "Particle-wave dualism" and lies at the heart of quantum theory, where he received a natural interpretation.
Born proposed the following now generally accepted interpretation of the results of the described experiments: the probability of an electron hitting a certain point on a photographic plate is proportional to the intensity of the corresponding de Broglie wave, that is, the square of the wave field amplitude at a given place on the screen. Thus, it is suggested probabilistic statistical interpretation the nature of waves associated with microparticles: the pattern of distribution of microparticles in space can be established only for a large number of particles; for one particle, only the probability of hitting a certain area can be determined.
After getting acquainted with the particle-wave dualism of particles, it is clear that the methods used in classical physics are unsuitable for describing the mechanical state of microparticles. In quantum mechanics, new specific means must be used to describe a state. The most important of them is the concept of wave function, or state function (-function).
The state function is a mathematical image of the wave field that should be associated with each particle. Thus, the function of the state of a free particle is the plane monochromatic de Broglie wave (4.2) or (4.8). For a particle exposed to external influences (for example, for an electron in the field of a nucleus), this wave field can have a very complex form, and it changes over time. The wave function depends on the parameters of the microparticle and on the physical conditions in which the particle is located.
Further, we will see that through the wave function the most complete description of the mechanical state of a micro-object, which is possible in the microcosm, is achieved. Knowing the wave function, one can predict which values of all measured quantities can be observed experimentally and with what probability. The state function carries all the information about the motion and quantum properties of particles; therefore, we speak of setting a quantum state with its help.
According to the statistical interpretation of de Broglie waves, the probability of localization of a particle is determined by the intensity of the de Broglie wave, so that the probability of detecting a particle in a small volume in the vicinity of a point at a time instant is
Taking into account the complexity of the function, we have:
For a plane de Broglie wave (4.2)
that is, it is equally probable to find a free particle anywhere in space.
The value
are called density of probability. The probability of finding a particle at a moment in time in a finite volume, according to the probability addition theorem, is equal to
If in (4.16) to carry out the integration in infinite limits, then the total probability of detecting a particle at the moment of time somewhere in space will be obtained. This is the probability of a certain event, therefore
Condition (4.17) is called normalization condition, and -function satisfying it, - normalized.
We emphasize once again that for a particle moving in a force field, the function is more complex kind than the plane de Broglie wave (4.2).
Since the -function is complex, it can be represented as
where is the modulus of the function, and is the phase factor, in which is any real number. From a joint consideration of this expression and (4.13), it is clear that the normalized wave function is determined ambiguously, but only up to a constant factor. The noted ambiguity is fundamental and cannot be eliminated; however, it is insignificant as it does not affect any physical results. Indeed, the multiplication of a function by an exponential changes the phase of the complex function, but not its modulus, which determines the probability of obtaining in an experiment one or another value of a physical quantity.
The wave function of a particle moving in a potential field can be represented as a wave packet. If, when the particle moves along the axis, the length of the wave packet is equal, then the wave numbers necessary for its formation cannot occupy an arbitrarily narrow interval. The minimum interval width must satisfy the ratio or, after multiplying by,
Similar relations hold for wave packets propagating along the axes and:
Relations (4.18), (4.19) are called the Heisenberg uncertainty relations(or uncertainty principle). According to this fundamental position of quantum theory, any physical system cannot be in states in which the coordinates of its center of inertia and momentum simultaneously take on quite definite, exact values.
Relations similar to those written down must be satisfied for any pair of so-called canonically conjugate quantities. The Planck constant contained in the uncertainty relations sets a limit on the accuracy of the simultaneous measurement of such quantities. At the same time, the uncertainty in the measurements is associated not with the imperfection of the experimental technique, but with the objective (wave) properties of the particles of matter.
Others important point in considering the states of microparticles is the effect of the device on a micro-object. Any measurement process leads to a change in the physical parameters of the state of the microsystem; the lower limit of this change is also set by the uncertainty relation.
In view of the smallness in comparison with macroscopic quantities of the same dimension of action, the uncertainty relations are essential mainly for phenomena of atomic and smaller scales and do not manifest themselves in experiments with macroscopic bodies.
The uncertainty relations, first obtained in 1927 by the German physicist W. Heisenberg, were an important stage in the elucidation of the patterns of intra-atomic phenomena and the construction of quantum mechanics.
As follows from the statistical interpretation of the meaning of the wave function, a particle can be detected with some probability at any point in space where the wave function is nonzero. Therefore, the results of measurement experiments, for example, coordinates, are probabilistic in nature. This means that when carrying out a series of identical experiments on the same systems (that is, when simulating the same physical conditions), different results are obtained each time. However, some values will be more likely than others and will appear more frequently. Most often, those coordinate values will be obtained that are close to the value that determines the position of the maximum of the wave function. If the maximum is clearly expressed (the wave function is a narrow wave packet), then the particle is mainly located near this maximum. Nevertheless, some scatter in the coordinate values (uncertainty of the order of the maximum half-width) is inevitable. The same applies to momentum measurement.
In atomic systems, the quantity is equal in order of magnitude to the orbital area along which, in accordance with the Bohr-Sommerfeld theory, a particle moves in the phase plane. This can be verified by expressing the orbital area in terms of the phase integral. In this case, it turns out that the quantum number (see Lecture 3) satisfies the condition
Unlike Bohr's theory, where equality holds (here is the speed of an electron in the first Bohr orbit in a hydrogen atom, is the speed of light in vacuum,), in the case under consideration in stationary states the average momentum is determined by the size of the system in coordinate space, and the ratio is only in order of magnitude... Thus, applying coordinates and momentum to describe microscopic systems, it is necessary to introduce quantum corrections in the interpretation of these concepts. Such a correction is the uncertainty relation.
The uncertainty relation for energy and time has a slightly different meaning:
If the system is in a stationary state, then it follows from the uncertainty relation that the energy of the system, even in this state, can be measured only with an accuracy not exceeding, where is the duration of the measurement process. Relation (4.20) is also true if we understand the uncertainty of the energy value of a non-stationary state of a closed system, and by means the characteristic time during which the average values of physical quantities in this system change significantly.
The uncertainty relation (4.20) leads to important conclusions regarding the excited states of atoms, molecules, nuclei. Such states are unstable, and it follows from the uncertainty relation that the energies of the excited levels cannot be strictly defined, that is, the energy levels have a certain natural width, where is the lifetime of the excited state. Another example is the alpha decay of a radioactive nucleus. The energy spread of the emitted -particles is related to the lifetime of such a nucleus by the ratio.
For the normal state of the atom, and energy has a very definite meaning, that is. For an unstable particle s, and there is no need to talk about the definite meaning of its energy. If the lifetime of an atom in an excited state is taken equal to s, then the width of the energy level is ~ 10 -26 J and the width of the spectral line arising during the transition of an atom to the normal state, ~ 10 8 Hz.
It follows from the uncertainty relations that the division of total energy into kinetic and potential energy loses its meaning in quantum mechanics. Indeed, one of them depends on momenta, and the other on coordinates. The same variables cannot have definite values at the same time. Energy should be defined and measured only as total energy, without dividing into kinetic and potential.
NAME SHELL OF A CHEMICAL ELEMENT ATOM
§ 1. INITIAL CONCEPTS OF QUANTUM MECHANICS
The theory of atomic structure is based on the laws describing the movement of microparticles (electrons, atoms, molecules) and their systems (for example, crystals). The masses and sizes of microparticles are extremely small compared to the masses and sizes of macroscopic bodies. Therefore, the properties and patterns of motion of an individual microparticle are qualitatively different from the properties and patterns of motion of a macroscopic body studied by classical physics. The motion and interactions of microparticles are described by quantum (or wave) mechanics. It is based on the concept of energy quantization, the wave nature of the motion of microparticles and the probabilistic (statistical) method for describing microobjects.
The quantum nature of the radiation and the absorption of energy. Around the beginning of the XX century. studies of a number of phenomena (radiation of incandescent bodies, photoelectric effect, atomic spectra) have led to the conclusion that energy is distributed and transmitted, absorbed and emitted not continuously, but discretely, in separate portions - quanta. The energy of a microparticle system can also take on only certain values, which are multiples of quanta.
The assumption of quantum energy was first formulated by M. Planck (1900) and later substantiated by A. Einstein (1905). Quantum energy? depends on the radiation frequency v:
where h is Planck's constant)
