—§23 Wave (field) properties of particles. Download the book "General and Inorganic Chemistry" (5.36Mb) Assumption of the wave nature of particle motion
The shortcomings of Bohr's theory indicated the need to revise the foundations of quantum theory and ideas about the nature of microparticles (electrons, protons, etc.). The question arose of how exhaustive the representation of an electron is in the form of a small mechanical particle, characterized by certain coordinates and a certain speed.
We already know that a kind of dualism is observed in optical phenomena. Along with the phenomena of diffraction, interference (wave phenomena), phenomena are also observed that characterize the corpuscular nature of light (the photoelectric effect, the Compton effect).
In 1924, Louis de Broglie hypothesized that dualism is not a feature of only optical phenomena ,but is universal. Particles of matter also have wave properties .
“In optics,” wrote Louis de Broglie, “for a century, the corpuscular method of consideration was too neglected in comparison with the wave; Has the reverse error been made in the theory of matter? Assuming that particles of matter, along with corpuscular properties, also have wave properties, de Broglie transferred to the case of particles of matter the same rules for the transition from one picture to another, which are valid in the case of light.
If a photon has energy and momentum, then a particle (for example, an electron) moving at a certain speed has wave properties, i.e. particle motion can be considered as wave motion.
According to quantum mechanics, the free motion of a particle with mass m and momentum (where υ is the particle velocity) can be represented as a plane monochromatic wave ( de Broglie wave) with a wavelength
| (3.1.1) |
propagating in the same direction (for example, in the direction of the axis X) in which the particle moves (Fig. 3.1).
The dependence of the wave function on the coordinate X is given by the formula
| , | (3.1.2) |
where - wave number ,a wave vector directed in the direction of wave propagation or along the motion of the particle:
| . | (3.1.3) |
In this way, wave vector of a monochromatic wave associated with a freely moving microparticle, proportional to its momentum or inversely proportional to its wavelength.
Since the kinetic energy of a relatively slowly moving particle , then the wavelength can also be expressed in terms of energy:
| . | (3.1.4) |
When a particle interacts with some object - with a crystal, molecule, etc. – its energy changes: the potential energy of this interaction is added to it, which leads to a change in the motion of the particle. Accordingly, the nature of the propagation of the wave associated with the particle changes, and this occurs according to the principles common to all wave phenomena. Therefore, the basic geometric regularities of particle diffraction do not differ in any way from the regularities of diffraction of any waves. The general condition for the diffraction of waves of any nature is the commensurability of the incident wave length λ with distance d between scattering centers: .
The hypothesis of Louis de Broglie was revolutionary, even for that revolutionary time in science. However, it was soon confirmed by many experiments.
By the beginning of the 20th century, both phenomena were known in optics that confirmed the presence of wave properties in light (interference, polarization, diffraction, etc.), and phenomena that were explained from the standpoint of corpuscular theory (photoelectric effect, Compton effect, etc.). At the beginning of the 20th century, a number of effects were discovered for particles of matter, outwardly similar to optical phenomena characteristic of waves. So, in 1921, Ramsauer, while studying the scattering of electrons on argon atoms, found that as the electron energy decreases 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 also characteristic of the cross section for electron scattering 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 during light diffraction on a small screen.
Another interesting effect is the selective reflection of electrons from the surface of metals; it was studied in 1927 by the American physicists Davisson and Germer, and independently of them English physicist J. P. Thomson.
A parallel beam of monoenergetic electrons from a 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 a crystal (Figure 4.3). When studying the diffraction of X-rays on crystals, it was found that the distribution of diffraction maxima is described by the formula
where is the lattice constant, is the diffraction order, is the X-ray wavelength.
In the case of neutron scattering by a heavy nucleus, a typical diffraction distribution of scattered neutrons also arose, 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 the particles of matter have both corpuscular and wave properties. At the same time, he suggested that a particle moving freely at a constant speed corresponds to a plane monochromatic wave
where and are its frequency and wave vector.
Wave (4.2) propagates in the direction of motion of the particle (). Such waves are called phase waves, matter waves or de Broglie waves.
De Broglie's idea was to expand the analogy between optics and mechanics, and to compare wave optics with wave mechanics, trying to apply the latter to intra-atomic phenomena. An attempt to attribute to an electron, and in general to all particles, like photons, a dual nature, to endow them with wave and corpuscular properties interconnected by a 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 “Revolution in Physics”.
A particle of mass moving at a speed has energy
and momentum
and the state of particle motion is characterized by a four-dimensional energy-momentum vector ().
On the other hand, in the wave pattern we use the concept of frequency and wave number (or wavelength), and the 4-vector corresponding to a plane wave is ().
Since both of the above descriptions are different aspects of the same physical object, there must be an unambiguous relationship between them; the relativistically invariant relation between 4-vectors is
Expressions (4.6) are called de Broglie 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-rays.
Taking into account (4.6), formula (4.2) can be written as a plane wave
the corresponding particle with momentum and energy.
De Broglie waves are characterized by phase and group velocities. Phase speed is determined from the condition of constancy of the phase of the wave (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 speed of the particle:
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 connection 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 the entire space and exist for an unlimited time. The properties of these waves are always and everywhere the same: their amplitude and frequency are constant, the distances between the wave surfaces are unchanged, etc. On the other hand, microparticles retain their corpuscular properties, that is, they have a certain mass localized in a certain region of space. In order to get out of this situation, particles began to be represented not by monochromatic de Broglie waves, but by sets of waves with similar frequencies (wave numbers) - wave packets:
in this case, the amplitudes are nonzero only for waves with wave vectors contained in the interval (). Since the group velocity of the wave packet is equal to the velocity 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 motion. The wave packet that claims to represent a particle must have the same properties. Therefore, it is necessary to require that, over 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 must be a dispersion of de Broglie waves. As a result, the phase relationships 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: particles are primary, and waves represent their formations, that is, they arise, like sound in a medium consisting of particles. But such a medium must be dense enough, because it makes sense to talk about waves in a medium of particles only when the average distance between particles is very small compared to the wavelength. And in experiments in which the wave properties of microparticles are found, this is not performed. But even if this difficulty is overcome, the indicated point of view 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 the particles do not disappear even at low intensities of the incident beams. In the experiments of Biberman, Sushkin and Fabrikant, carried out in 1949, such weak electron beams were used 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 through the entire device. Under such conditions, the interaction between the electrons, of course, played no role. Nevertheless, with a sufficiently long exposure, a diffraction pattern appeared on a photographic film placed behind the crystal, which did not differ in any way from the pattern obtained with a short exposure to electron beams, the intensity of which was 10 7 times greater. It is only important that in both cases the total number of electrons falling on the photographic plate be 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 only be obtained by hitting the plate with a large number of particles.
The electron in the considered experiment completely retains its integrity (charge, mass and other characteristics). This shows its corpuscular properties. At the same time, the manifestation of wave properties is also evident. The electron never hits that section of the photographic plate where there should be a minimum of the diffraction pattern. It can only appear near the position of the diffraction maxima. In this case, it is impossible to specify in advance in which specific direction a given 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 underlies 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, to the square of the wave field amplitude at a given location on the screen. Thus, it is proposed probabilistic-statistical interpretation the nature of the waves associated with microparticles: the regularity of the 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 wave-particle duality 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 the state. The most important of these is the concept of wave function, or state function (-functions).
The state function is a mathematical image of the wave field that should be associated with each particle. Thus, the state function of a free particle is a plane monochromatic de Broglie wave (4.2) or (4.8). For a particle subjected to external action (for example, for an electron in the field of a nucleus), this wave field can have a very complex form, and it changes with 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 the most complete description of the mechanical state of a micro-object is achieved through the wave function, which is only possible in the micro-world. Knowing the wave function, it is possible to 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, one speaks of setting a quantum state with its help.
According to the statistical interpretation of de Broglie waves, the probability of particle localization 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 is
Taking into account the complexity of the function, we have:
For a plane de Broglie wave (4.2)
that is, it is equally likely to find a free particle anywhere in space.
the value
called probability density. Probability of finding a particle at a time in a finite volume, according to the probability addition theorem, is equal to
If in (4.16) integration is performed in infinite limits, then the total probability of detecting a particle at a moment of time somewhere in space will be obtained. This is the probability of a certain event, so
Condition (4.17) is called normalization condition, and - a function that satisfies it, - normalized.
We emphasize once again that for a particle moving in a force field, the role is played by a function of a more complex form than the de Broglie plane 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 the joint consideration of this expression and (4.13) it is clear that the normalized wave function is defined ambiguously, but only up to a constant factor. The noted ambiguity is fundamental and cannot be eliminated; however, it is insignificant, since it does not affect any physical results. Indeed, multiplying a function by an exponent changes the phase of the complex function, but not its modulus, which determines the probability of obtaining one or another value of a physical quantity in an experiment.
The wave function of a particle moving in a potential field can be represented by a wave packet. If, when a particle moves along an 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 relation or, after multiplying by,
Similar relationships hold for wave packets propagating along the axes and:
Relations (4.18), (4.19) are called 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 hold for any pair of so-called canonically conjugate quantities. Planck's 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 measurements is connected not with the imperfection of the experimental technique, but with the objective (wave) properties of matter particles.
Other important point in considering the states of microparticles is the impact of the device on the 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, the effects of the uncertainty relations are significant mainly for atomic and smaller scale phenomena and do not appear in experiments with macroscopic bodies.
Uncertainty relations, first obtained in 1927 by the German physicist W. Heisenberg, were an important step in elucidating the patterns of intra-atomic phenomena and building 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 experiments on measurement, for example, coordinates, are of a probabilistic nature. This means that when conducting a series of identical experiments on identical systems (that is, when reproducing 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 values of the coordinate 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 values of the coordinate (an uncertainty of the order of the half-width of the maximum) is unavoidable. The same applies to the measurement of momentum.
In atomic systems, the magnitude is equal in order of magnitude to the area of the orbit along which, in accordance with the Bohr-Sommerfeld theory, a particle moves in the phase plane. This can be verified by expressing the area of the orbit in terms of the phase integral. In this case, it turns out that the quantum number (see lecture 3) satisfies the condition
In contrast to the Bohr theory, where equality takes place (here is the electron velocity in the first Bohr orbit in the hydrogen atom, is the speed of light in vacuum), in the considered case in stationary states, the average momentum is determined by the dimensions of the system in the coordinate space, and the ratio is only in order of magnitude. Thus, using coordinates and momentum to describe microscopic systems, it is necessary to introduce quantum corrections into 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 only be measured with an accuracy not exceeding, where is the duration of the measurement process. Relation (4.20) is also valid if we understand the uncertainty of the value of the energy of the non-stationary state of a closed system, and by - 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, and 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 some 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 relation.
For the normal state of the atom, and energy has a well-defined value, that is. For an unstable particle s, and there is no need to talk about a certain value of its energy. If the lifetime of an atom in an excited state is taken equal to c, then the width of the energy level is ~10 -26 J and the width of the spectral line that occurs during the transition of an atom to the normal state, ~10 8 Hz.
It follows from the uncertainty relations that the division of the total energy into kinetic and potential loses its meaning in quantum mechanics. Indeed, one of them depends on the momenta, and the other - on the coordinates. The same variables cannot have certain values at the same time. Energy should be defined and measured only as total energy, without division into kinetic and potential.
Light has both wave and particle properties. Wave properties appear during the propagation of light (interference, diffraction). Corpuscular properties are manifested in the interaction of light with matter (photoelectric effect, emission and absorption of light by atoms).
The properties of a photon as a particle (energy E and momentum p) are related to its wave properties (frequency ν and wavelength λ) by the relations
; , (19)
where h=6.63×10 -34 J is Planck's constant.
Trying to overcome the difficulties of the Bohr model of the atom, the French physicist Louis de Broglie in 1924 put forward the hypothesis that the combination of wave and corpuscular properties is inherent not only in light, but also in any material body. That is, particles of matter (for example, electrons) have wave properties. suggested, According to de Broglie, each body of mass m, moving with a speed υ, corresponds to a wave process with a wavelength
The most pronounced wave properties are manifested in micro-objects (elementary particles). Due to the small mass, the de Broglie wavelength turns out to be comparable with the interatomic distance in crystals. Under these conditions, the interaction of a particle beam with a crystal lattice gives rise to diffraction phenomena. Electrons with energy 150 eV corresponds to the wavelength λ»10 -10 m. Interatomic distances in crystals are of the same order. If a beam of such electrons is directed to a crystal, then they will scatter according to the laws of diffraction. A diffraction pattern (electron diffraction pattern) recorded on photographic film contains information about the structure of a three-dimensional crystal lattice.
Figure 6 Illustration of the wave properties of matter
To illustrate the wave properties of particles, a thought experiment is often used - the passage of an electron beam (or other particles) through a slot of width Δx. From the point of view of the wave theory, after diffraction by the slit, the beam will broaden with an angular divergence θ»λ/Δх. From the corpuscular point of view, the broadening of the beam after passing through the slit is explained by the appearance of a certain transverse momentum in the particles. The spread in the values of this transverse momentum ("uncertainty") is
(21)
Ratio (22)
is called the uncertainty relation. This ratio in corpuscular language reflects the presence of wave properties in particles.
An experiment on the passage of an electron beam through two closely spaced slits can serve as an even clearer illustration of the wave properties of particles. This experiment is analogous to Young's optical interference experiment.
4. 10 Quantum model of the atom Experimental facts (electron diffraction, the Compton effect, the photoelectric effect, and many others) and theoretical models, such as the Bohr model of the atom, clearly show that the laws of classical physics become inapplicable for describing the behavior of atoms and molecules and their interaction with light. During the decade between 1920 and 1930 a number of prominent physicists of the twentieth century. (de Broglie, Heisenberg, Born, Schrödinger, Bohr, Pauli, etc.) was engaged in the construction of a theory that could adequately describe the phenomena of the microworld. As a result, quantum mechanics was born, which became the basis of all modern theories of the structure of matter, one might say, the basis (together with the theory of relativity) of physics of the twentieth century.
The laws of quantum mechanics are applicable in the microcosm, at the same time we are macroscopic objects and live in the macrocosm governed by completely different, classical laws. Therefore, it is not surprising that many of the provisions of quantum mechanics cannot be verified by us directly and are perceived as strange, impossible, unusual. Nevertheless, quantum mechanics is probably the most experimentally confirmed theory, since the consequences of calculations performed according to the laws of this theory are used in almost everything that surrounds us and have become part of human civilization (suffice it to mention those semiconductor elements, work which currently allow the reader to see the text on the monitor screen, the coverage of which, by the way, is also calculated using quantum mechanics).
Unfortunately, the mathematical apparatus used by quantum mechanics is rather complicated, and the ideas of quantum mechanics can only be stated verbally and therefore not convincingly enough. With this remark in mind, we will try to give at least some idea of these ideas.
The basic concept of quantum mechanics is the concept of the quantum state of some micro-object, or micro-system (it can be a separate particle, atom, molecule, set of atoms, etc.).
Quantum model of the atom differs from the planetary one in the first place in that the electron in it does not have a precisely defined coordinate and speed, so it makes no sense to talk about the trajectory of its movement. It is possible to determine (and draw) only the boundaries of the region of its predominant movement (orbitals).
The state of some micro-object or micro-system (it can be a separate particle, atom, molecule, set of atoms, etc.) can be characterized by setting quantum numbers: values of energy, momentum, moment of momentum, projection of this moment of momentum onto some axle, charge, etc.
SCHROEDINGER EQUATION for the motion of an electron in the Coulomb field of the nucleus of the hydrogen atom is used to analyze the quantum model of the atom. As a result of solving this equation, a wave function is obtained, which depends not only on the coordinate and time t, but also on 4 parameters that have a discrete set of values and are called quantum numbers. They have names: principal, azimuthal, magnetic and magnetic spin.
Principal quantum number n can take integer values 1, 2, ... . It determines the energy of an electron in an atom
Where E i is the ionization energy of the hydrogen atom (13.6 eV).
AZIMUTHAL (ORBITAL) quantum number l
determines the modulus of the angular momentum of an electron during its orbital motion 
(24)
where s is the spin quantum number, which has only one value for each particle. For example, for an electron s = (similarly, for a proton and a neutron). For a photon, s = 1.
Degenerate states of an electron with the same energy are called.
MULTIPLE DEGENERATION is equal to the number of states with the same energy.
BRIEF recording the state of an electron in an atom: NUMBER, equal to the main quantum number, and the letter that determines the azimuthal quantum number:
Table 1 Brief record of the state of an electron in an atom
De Broglie's hypothesis. De Broglie waves.
As mentioned earlier, light (and radiation in general) has a dual nature: in some phenomena (interference, diffraction, etc.) light manifests itself as waves, in other phenomena with no less convincingness - as particles. This prompted de Broglie (in 1923) to express the idea that material particles must also have wave properties, i.e. extend a similar wave-particle duality to particles with non-zero rest mass.
If a wave is associated with such a particle, it can be expected that it propagates in the direction of velocity υ particles. De Broglie did not express anything definite about the nature of this wave. We will not yet clarify their nature, although we immediately emphasize that these waves are not electromagnetic. They have, as we shall see below, a specific nature for which there is no analogue in classical physics.
So, de Broglie hypothesized that the relation for momentum p=ћω/c, related to photons, has a universal character, i.e., particles can be associated with a wave whose length
This formula is called de Broglie formulas, and λ is de Broglie wavelength particles with momentum R.
De Broglie also suggested that the beam of particles incident on the double slit should interfere behind them.
The second relation, independent of formula (3.13.1), is the relationship between the energy E particles and the frequency ω of the de Broglie wave:
Basically the energy E is always defined up to the addition of an arbitrary constant (unlike Δ E), therefore, the frequency ω is a fundamentally unobservable quantity (in contrast to the de Broglie wavelength).
With frequency ω and wavenumber k two speeds are connected - phase υ f and group u:
(3.13.3)
Multiplying the numerator and denominator of both expressions by ћ taking into account (3.13.1) and (3.13.2), we obtain, restricting ourselves to considering only the nonrelativistic case, i.e. assuming E = p 2 /2m(kinetic energy):
(3.13.4)
From this it can be seen that the group velocity is equal to the velocity of the particle, i.e., it is in principle an observable quantity, in contrast to υ f - due to ambiguity E.
From the first formula (3.13.4) it follows that the phase velocity of the de Broglie waves
(3.13.5)
i.e., it depends on the frequency ω, which means that de Broglie waves have dispersion even in a vacuum. Further, it will be shown that, in accordance with the modern physical interpretation, the phase velocity of de Broglie waves has a purely symbolic meaning, since this interpretation classifies them as fundamentally unobservable quantities. However, what has been said can be seen immediately, since E in (3.13.5) is defined, as already mentioned, up to the addition of an arbitrary constant.
Establishing the fact that, according to (3.13.4), the group velocity of de Broglie waves is equal to the velocity of the particle, played in its time important role in the development of the fundamental foundations of quantum physics, and primarily in the physical interpretation of de Broglie waves. First, an attempt was made to consider particles as wave packets of a very small extent and thus solve the paradox of the duality of particle properties. However, such an interpretation turned out to be erroneous, since all the harmonic waves that make up the packet propagate with different phase velocities. In the presence of a large dispersion, which is characteristic of de Broglie waves even in vacuum, the wave packet "spreads out". For particles with a mass of the order of the mass of an electron, the packet spreads almost instantly, while the particle is a stable formation.
Thus, the representation of a particle in the form of a wave packet turned out to be untenable. The problem of the duality of particle properties required a different approach to its solution.
Let's return to de Broglie's hypothesis. Let us find out in what phenomena the wave properties of particles can manifest themselves, if they, these properties, really exist. We know that regardless of the physical nature of the waves, these are interference and diffraction. The directly observable quantity in them is the wavelength. In all cases, the de Broglie wavelength is determined by formula (3.13.1). Let us use it to make some estimates.
First of all, let us make sure that the de Broglie hypothesis does not contradict the concepts of macroscopic physics. Let us take as a macroscopic object, for example, a grain of dust, assuming that its mass m= 1mg and rate V= 1 µm/s. Its corresponding de Broglie wavelength
(3.13.6)
That is, even for such a small macroscopic object as a grain of dust, the de Broglie wavelength turns out to be immeasurably smaller than the dimensions of the object itself. Under such conditions, no wave properties, of course, can manifest themselves in the conditions of dimensions accessible to measurement.
The situation is different, for example, for an electron with kinetic energy K and momentum . Its de Broglie wavelength
(3.13.7)
where K must be measured in electron volts (eV). At K\u003d 150 eV, the de Broglie wavelength of an electron is, according to (3.13.7), λ \u003d 0.1 nm. The lattice constant has the same order of magnitude. Therefore, just as in the case of X-rays, the crystal structure can be a suitable lattice for obtaining de Broglie wave diffraction of electrons. However, de Broglie's hypothesis seemed so unrealistic that it was not subjected to experimental verification for quite some time.
Experimentally, de Broglie's hypothesis was confirmed in the experiments of Davisson and Germer (1927). The idea behind their experiments was as follows. If the electron beam has wave properties, then we can expect, even without knowing the mechanism of reflection of these waves, that their reflection from the crystal will have the same interference character as that of x-rays.
In one series of experiments by Davisson and Germer, to detect diffraction maxima (if any), the accelerating voltage of electrons and simultaneously the position of the detector were measured D(counter of reflected electrons). In the experiment, a single crystal of nickel (cubic system), ground as shown in Fig. 3.13, was used. If it is rotated around the vertical axis in Fig.3.13.1
The position corresponding to the figure, then in this position
the ground surface is covered with regular rows of atoms perpendicular to the plane of incidence (plane of the pattern), the distance between which d= 0.215nm. The detector was moved in the plane of incidence by changing the angle θ. At angle θ = 50 0 and accelerating voltage V= 54B, a particularly distinct maximum of reflected Fig.3.13.2 was observed.
electrons, the polar diagram of which is shown in Fig.3.13.2. This maximum can be interpreted as a first-order interference maximum from a flat diffraction grating with the above period in accordance with the formula
What can be seen from Fig.3.13.3. In this figure, each thick dot is a projection of a chain of atoms located on a straight line perpendicular to the plane of the figure. Period d can be measured independently, for example by x-ray diffraction. Fig.3.13.3.
The de Broglie wavelength calculated by formula (3.13.7) for V= 54B is equal to 0.167nm. The corresponding wavelength, found from formula (3.13.8), is 0.165 nm. The agreement is so good that the result obtained should be recognized as a convincing confirmation of the de Broglie hypothesis.
Other experiments confirming de Broglie's hypothesis were those of Thomson and Tartakovsky . In these experiments, an electron beam was passed through a polycrystalline foil (according to the Debye method in the study of X-ray diffraction). As in the case of X-rays, a system of diffraction rings was observed on a photographic plate located behind the foil. The resemblance of both paintings is striking. The suspicion that the system of these rings is generated not by electrons, but by secondary X-ray radiation resulting from the incidence of electrons on the foil, is easily dissipated if a magnetic field is created in the path of scattered electrons (bring a permanent magnet). It does not affect x-rays. This kind of test showed that the interference pattern was immediately distorted. This clearly indicates that we are dealing with electrons.
G. Thomson carried out experiments with fast electrons (tens of keV), P.S. Tarkovsky - with relatively slow electrons (up to 1.7 keV).
For successful observation of the diffraction of waves by crystals, it is necessary that the wavelength of these waves be comparable with the distances between the nodes of the crystal lattice. Therefore, to observe the diffraction of heavy particles, it is necessary to use particles with sufficiently low velocities. Corresponding experiments on the diffraction of neutrons and molecules upon reflection from crystals were carried out and also fully confirmed de Broglie's hypothesis when applied to heavy particles as well.
Thanks to this, it was experimentally proved that wave properties are a universal property of all particles. They are not caused by any features of the internal structure of a particular particle, but reflect their general law of motion.
The experiments described above were carried out using particle beams. Therefore, a natural question arises: do the observed wave properties express the properties of a beam of particles or individual particles?
To answer this question, in 1949 V. Fabrikant, L. Biberman and N. Sushkin carried out experiments in which such weak electron beams were used that each electron passed through the crystal one by one, and each scattered electron was recorded by a photographic plate. At the same time, it turned out that individual electrons hit different points of the photographic plate in a completely random way at first glance (Fig. 3.13.4 a). Meanwhile, with a sufficiently long exposure, a diffraction pattern appeared on the photographic plate (Fig. 3.13.4 b), which is absolutely identical to the diffraction pattern from a conventional electron beam. So it was proved that individual particles also have wave properties.
Thus, we are dealing with micro-objects that simultaneously have both corpuscular and wave-
properties. This allows us to say further
about electrons, but the conclusions we will come to have Fig.3.13.4.
general meaning and apply equally to any particles.
Paradoxical behavior of microparticles.
The experiments considered in the previous paragraph force us to state that we face one of the most mysterious paradoxes: what does the statement "an electron is both a particle and a wave" mean?»?
Let's try to understand this issue with the help of a thought experiment similar to Young's experiment on the study of the interference of light (photons) from two slits. After the passage of an electron beam through two slits, a system of maxima and minima is formed on the screen, the position of which can be calculated using the formulas of wave optics, if each electron is associated with a de Broglie wave.
In the phenomenon of interference from two slits, the very essence of quantum theory is hidden, so we will pay special attention to this issue.
If we are dealing with photons, then the paradox (particle - wave) can be eliminated by assuming that the photon, due to its specificity, splits into two parts (at slits), which then interfere.
What about electrons? After all, they never split - this is established quite reliably. An electron can pass either through slot 1 or through slot 2 (Fig. 3.13.5). Therefore, their distribution on the screen E should be the sum of distributions 1 and 2 (Fig. 3.13.5 a) - it is shown by a dotted curve. Fig.13.13.5.
Although the logic in this reasoning is impeccable, such a distribution is not carried out. Instead, we observe a completely different distribution (Figure 3.13.5 b).
Is this not the collapse of pure logic and common sense? After all, everything looks as if 100 + 100 = 0 (at point P). Indeed, when either slit 1 or slit 2 is open, then, say, 100 electrons per second arrive at point P, and if both slits are open, then not a single one!..
Moreover, if we first open slot 1, and then gradually open slot 2, increasing its width, then, according to common sense, the number of electrons arriving at point P every second should increase from 100 to 200. In reality, from 100 to zero.
If a similar procedure is repeated, registering particles, for example, at point O (see Fig. 3.13.5 b), then a no less paradoxical result arises. As slit 2 opens (with slit 1 open), the number of particles at point O grows not to 200 per second, as one would expect, but to 400!
How opening slit 2 can affect the electrons that seem to pass through slit 1? That is, the situation is such that each electron, passing through some gap, "feels" the neighboring gap, correcting its behavior. Or, like a wave, it passes through both slots at once (!?). After all, otherwise the interference pattern cannot arise. An attempt to determine through which slit this or that electron passes leads to the destruction of the interference pattern, but this is a completely different question.
What is the conclusion? The only way to "explain" these paradoxical results is to create a mathematical formalism that is compatible with the results obtained and always correctly predicts the observed phenomena. Moreover, of course, this formalism must be internally consistent.
And such a formalism was created. He assigns to each particle some complex psi-function Ψ( r, t). Formally, it has the properties of classical waves, so it is often called wave function. The behavior of a free uniformly moving particle in a certain direction is described by a plane de Broglie wave
But more details about this function, its physical meaning and the equation that governs its behavior in space and time, will be discussed in the next lecture.
Returning to the behavior of electrons when passing through two slits, we must recognize: the fact that in principle it is impossible to answer the question through which slit an electron passes(without destroying the interference pattern), incompatible with the idea of a trajectory. Thus, electrons, generally speaking, cannot be assigned trajectories.
However, under certain conditions, namely, when the de Broglie wavelength of a microparticle becomes very small and can be much smaller, for example, the distance between the slits or atomic dimensions, the concept of a trajectory again becomes meaningful. Let us consider this question in more detail and formulate more correctly the conditions under which one can use the classical theory.
Uncertainty principle
In classical physics, an exhaustive description of the state of a particle is determined by dynamic parameters, such as coordinates, momentum, angular momentum, energy, etc. However, the real behavior of microparticles shows that there is a fundamental limit to the accuracy with which such variables can be specified and measured.
A deep analysis of the reasons for the existence of this limit, which is called uncertainty principle, conducted by W. Heisenberg (1927). Quantitative ratios expressing this principle in specific cases are called uncertainty relations.
The peculiarity of the properties of microparticles is manifested in the fact that not for all variables certain values are obtained during measurements. There are pairs of quantities that cannot be determined exactly at the same time.
The most important are two uncertainty relations.
The first of them limits the accuracy of the simultaneous measurement of the coordinates and the corresponding projections of the particle's momentum. For projection, for example, on the axis X it looks like this:
The second relation establishes the energy measurement uncertainty, Δ E, for a given time interval Δ t:
Let us explain the meaning of these two relations. The first of these states that if the position of the particle, for example, along the axis X known with uncertainty Δ x, then at the same moment the projection of the particle momentum onto the same axis can be measured only with the uncertainty Δ p= ћ/Δ x. Note that these restrictions do not apply to the simultaneous measurement of the particle coordinate along one axis and the momentum projection along the other: the quantities x and p y , y and p x, etc. can all have exact values at the same time.
According to the second relation (3.13.11) for measuring energy with an error Δ E time is needed, not less than Δ t=ћ /Δ E. An example is the "blurring" of the energy levels of hydrogen-like systems (except for the ground state). This is due to the fact that the lifetime in all excited states of these systems is on the order of 10 -8 s. The smearing of the levels leads to a broadening of the spectral lines (natural broadening), which is actually observed. The same applies to any unstable system. If its lifetime before decay is of the order of τ, then, due to the finiteness of this time, the energy of the system has an irremovable uncertainty no less than Δ E≈ ћ/τ.
Let us point out more pairs of quantities that cannot be exactly determined at the same time. These are any two projections of the angular momentum of the particle. So there is no state in which all three and even any two of the three projections of angular momentum have certain values.
Let us discuss in more detail the meaning and possibilities of the relation Δ x·Δ p x ≥ ћ . First of all, let's pay attention to the fact that it determines the fundamental limit of uncertainties Δ x and Δ p x , with which the state of the particle can be characterized classically, i.e. coordinate x and momentum projection p x . The more precisely x, the less accurate it is possible to establish p x , and vice versa.
We emphasize that the true meaning of relation (3.13.10) reflects the fact that in nature there are objectively no particle states with precisely defined values of both variables, x and p X. At the same time, we are forced, since the measurements are carried out with the help of macroscopic instruments, to attribute to the particles classical variables that are not characteristic of them. The costs of such an approach express the uncertainty relations.
After the need to describe the behavior of particles by wave functions became clear, the uncertainty relations arise in a natural way - as a mathematical consequence of the theory.
Considering the uncertainty relation (3.13.10) to be universal, let us estimate how it would affect the motion of a macroscopic body. Take a very small ball of mass m= 1mg. Let us determine, for example, using a microscope, its position with an error Δ x≈ 10 -5 cm (it is due to the resolution of the microscope). Then the uncertainty of the ball speed Δυ = Δ p/m≈ (ћ /Δ x)/m~ 10 -19 cm/s. Such a value is inaccessible to any measurement, and therefore the deviation from the classical description is completely insignificant. In other words, even for such a small (but macroscopic) ball, the concept of a trajectory is applicable with a high degree of accuracy.
An electron in an atom behaves differently. A rough estimate shows that the uncertainty of the speed of an electron moving along the Bohr orbit of a hydrogen atom is comparable to the speed itself: Δυ ≈ υ. In this situation, the idea of the motion of an electron in a classical orbit loses all meaning. And generally speaking, when microparticles move in very small regions of space, the concept of a trajectory turns out to be untenable.
At the same time, under certain conditions, the motion of even microparticles can be considered classically, that is, as motion along a trajectory. This happens, for example, when charged particles move in electromagnetic fields(v cathode ray tubes, accelerators, etc.). These motions can be considered classically, since for them the limitations due to the uncertainty relation are negligible compared to the quantities themselves (coordinates and momentum).
The gap experience. The uncertainty relation (3.13.10) manifests itself in any attempt to accurately measure the position or momentum of a microparticle. And each time we come to a "disappointing" result: the refinement of the position of the particle leads to an increase in the uncertainty of the momentum, and vice versa. To illustrate this situation, consider the following example.
Let's try to determine the coordinate x freely moving with momentum p particles, placing on its path perpendicular to the direction of motion a screen with a slot of width b(fig.3.13.6). Before the particle passes through the slit, its momentum projection p x has the exact value: p x = 0. This means that Δ p x = 0, but
Coordinate x particles is completely indeterminate according to (3.13.10): we cannot say Fig.3.13.6.
whether the particle will pass through the slit.
If the particle passes through the slit, then in the plane of the slit the coordinate x will be registered with uncertainty Δ x ≈ b. In this case, due to diffraction, the particle will most likely move within the angle 2θ, where θ is the angle corresponding to the first diffraction minimum. It is determined by the condition under which the difference in the path of the waves from both edges of the slot will be equal to λ (this is proved in wave optics):
As a result of diffraction, there is an uncertainty in the value p x - projections of the momentum, the spread of which
Given that b≈ Δ X and p= 2π ћ /λ., we obtain from the two previous expressions:
which agrees in order of magnitude with (3.13.10).
Thus, an attempt to determine the coordinate x particles, indeed, led to the appearance of uncertainty Δ p in the momentum of the particle.
An analysis of many situations related to measurements shows that measurements in the quantum domain are fundamentally different from classical measurements. Unlike the latter, there is a natural limit to the accuracy of measurements in quantum physics. It is in the very nature of quantum objects and cannot be overcome by any improvement in instruments and measurement methods. Relation (3.13.10) establishes one of these limits. The interaction between a microparticle and a macroscopic measuring device cannot be made arbitrarily small. Measuring, for example, the coordinates of a particle, inevitably leads to a fundamentally unremovable and uncontrollable distortion of the state of the microparticle, and hence to an uncertainty in the value of the momentum.
Some Conclusions.
The uncertainty relation (3.13.10) is one of the fundamental provisions of quantum theory. This relation alone is sufficient to obtain a number of important results, in particular:
1. A state in which the particle would be at rest is impossible.
2. When considering the motion of a quantum object, it is necessary in many cases to abandon the very concept of a classical trajectory.
3. The division of total energy often loses its meaning E particle (as a quantum object) to the potential U and kinetic K. Indeed, the first, i.e. U, depends on the coordinates, and the second depends on the momentum. The same dynamic variables cannot have a definite value at the same time.
Home > WorkshopWave properties of microparticles.
The development of ideas about the corpuscular-wave properties of matter received in the hypothesis of the wave nature of the movement of microparticles. Louis de Broglie, from the idea of symmetry in nature for particles of matter and light, attributed to any microparticle some internal periodic process (1924). Combining the formulas E \u003d hν and E \u003d mc 2, he obtained a ratio showing that any particle has its own wavelength: λ B \u003d h / mv \u003d h / p, where p is the momentum of the wave-particle. For example, for an electron having an energy of 10 eV, the de Broglie wavelength is 0.388 nm. Later it was shown that the state of a microparticle in quantum mechanics can be described by a certain complex wave function of the coordinates Ψ(q), and the square of the modulus of this function |Ψ| 2 defines the probability distribution of coordinate values. This function was first introduced into quantum mechanics by Schrodinger in 1926. Thus, the de Broglie wave does not carry energy, but only reflects the “phase distribution” of some probabilistic periodic process in space. Consequently, the description of the state of microcosm objects is probabilistic, unlike macrocosm objects, which are described by the laws of classical mechanics. To prove de Broglie's idea about the wave nature of microparticles, the German physicist Elsasser suggested using crystals to observe electron diffraction (1925). In the USA, K. Davisson and L. Germer discovered the phenomenon of diffraction during the passage of an electron beam through a nickel crystal plate (1927). Independently of them, the diffraction of electrons when passing through a metal foil was discovered by J.P. Thomson in England and P.S. Tartakovsky in the USSR. So the idea of de Broglie about the wave properties of matter found experimental confirmation. Subsequently, diffractive, and therefore wave, properties were discovered in atomic and molecular beams. Corpuscular-wave properties are possessed not only by photons and electrons, but also by all microparticles. The discovery of wave properties in microparticles showed that such forms of matter as field (continuous) and matter (discrete), which, from the point of view of classical physics, were considered qualitatively different, under certain conditions, they can exhibit properties inherent in both forms. This speaks of the unity of these forms of matter. A complete description of their properties is possible only on the basis of opposite, but complementary ideas.Electron diffraction.
A diffraction grating is used to obtain the spectrum of light waves and determine their length. It is a collection of a large number of narrow slits separated by opaque gaps, for example, a glass plate with scratches (strokes) applied to it. As with two slits (see lab. work 2), when a plane monochromatic wave passes through such a grating, each slit will become a source of secondary coherent waves, as a result of which an interference pattern will appear as a result of their addition. The condition for the occurrence of interference maxima on a screen located at a distance L from the diffraction grating is determined by the path difference between the waves from neighboring slots. If at the observation point the path difference is equal to an integer number of waves, then they will be amplified and the maximum of the interference pattern will be observed. The distance between the maxima for light of a certain wavelength λ is determined by the formula: h 0 = λL/d. The value d is called the grating period and is equal to the sum of the widths of the transparent and opaque gaps. To observe electron diffraction, metal crystals are used as a natural diffraction grating. The period d of such a natural diffraction grating corresponds to the characteristic distance between the atoms of the crystal. The installation scheme for observing electron diffraction is shown in Figure 1. Passing through the potential difference U between the cathode and anode, the electrons acquire kinetic energy Ekin. = Ue, where e is the electron charge. From the formula of kinetic energy E kin. = (m e v 2)/2 you can find the speed of the electron: . Knowing the electron mass m e, one can determine its momentum and, accordingly, the de Broglie wavelength.
According to the same scheme, an electron microscope was created in the 30s, giving a magnification of 10 6 times. Instead of light waves, it uses the wave properties of a beam of electrons accelerated to high energies in a deep vacuum. Significantly smaller objects were studied than with a light microscope, and in terms of resolution, the improvement was thousands of times. Under favorable conditions, it is possible to photograph even individual large atoms, the most closely located details of an object with a size of about 10 -10 m. Without it, it was hardly possible to control defects in microcircuits, to obtain pure substances to develop microelectronics, molecular biology etc.
Laboratory work No. 7. The order of the work.
Open a working window.
A). By moving the slider on the right side of the working window, set an arbitrary value of the accelerating voltage U ( until you move the slider, the buttons will be inactive!!!) and write down this value. Click the button Start. Observe on the screen of the working window how the interference pattern appears during the diffraction of electrons on a metal foil. Note that the hitting of electrons at different points on the screen is random, but the probability of electrons hitting certain areas of the screen is zero, and other than zero. That is why the interference pattern appears. Wait until the concentric circles of the interference pattern clearly appear on the screen and press the button Test. Attention! Until the interference pattern becomes clear enough, the Test button will be inactive. It will become active after the mouse cursor, when hovering over this button, changes its view from an arrow to a hand!!! The screen will show graphic image probability distribution of electrons along the x axis, corresponding to the interference pattern. Drag the measuring ruler to the plot area. Use the right mouse button to zoom in on the graph and determine the distance between the two extreme interference maxima with an accuracy of tenths of a millimeter. Write down this value. By dividing this value by 4 you get the distance h 0 between the maxima of the interference pattern. Write it down. Use the right mouse button to return the image to its original state. Using the formulas in the theoretical part, determine the de Broglie wavelength. Substitute this value in the test window and click the button Check Right!!! B). Using the formulas in the theoretical part, find the electron velocity from the accelerating voltage and write it down. Substitute this value in the test window and click the button Check. If the calculations are correct, an inscription will appear Right!!! Calculate the momentum of an electron and use de Broglie's formula to find the wavelength. Compare the value obtained with that found from the interference pattern. V). Change the voltage and pressing the button Test repeat points A and B. Show your test results to your teacher. Based on the results of the measurements, make a table:| Electron speed v | Electron momentum p | ||||
Laboratory work No. 7. Report form.
The heading states:
NAME OF THE LABORATORY WORK
Exercise. Electron diffraction.
A). Found distance h 0 . Wavelength calculation λ.
B). Calculations of electron speed, momentum and wavelength.
V). Repeat items A and B.Table with results:
| h 0 (distance between maxima) | Electron speed v | Electron momentum p | |||
G). Analysis of results. Answers on questions.
D). Determination of the de Broglie wavelength for a car. Answers on questions. Conclusions.
1. What is the essence of Louis de Broglie's hypothesis?
2. What experiments confirmed this hypothesis?
3. What is the specificity of the description of the state of the objects of the microcosm, in contrast to the description of the objects of the macrocosm?
4. Why did the discovery of wave properties of microparticles, along with the manifestation of corpuscular properties of electromagnetic waves (light), make it possible to talk about the corpuscular-wave dualism of matter? Explain the essence of these representations.
5. How does the de Broglie wavelength depend on the mass and speed of the microparticle?
6. Why don't macro objects show wave properties?
Lab #8 DESCRIPTION
Diffraction of photons. Uncertainty relation.
Working window
The view of the working window is shown in Fig. 1.1. The working window shows the photon diffraction model. The test buttons are located in the lower right part of the window. The calculated parameters are entered into the window under the test buttons. In the upper position of the switch, this is the uncertainty of the photon momentum, and in the lower position, the product of the momentum uncertainty and the x-coordinate uncertainty. In the windows below, the number of correct answers and the number of attempts are recorded. By moving the sliders, you can change the photon wavelength and the size of the slit.
Figure 1.1.
To measure the distance from the maximum of the diffraction pattern to the minimum, the slider located to the right of the model window is used. The measurements are carried out for several values of the gap sizes. The test system records the number of correctly given answers and the total number of attempts.
Laboratory work number 8. Theory
Uncertainty relation.
PURPOSE OF THE WORK: Using the example of photon diffraction, to give students an idea of the uncertainty relation. Using the model of photon diffraction by a slit, it is clear to demonstrate that the more accurately the x coordinate of a photon is determined, the less accurately the value of its momentum projection p x is determined.
Uncertainty relation
In 1927, W. Heisenberg discovered the so-called uncertainty relations, according to which the uncertainties of coordinates and momenta are interconnected by the relation:
, where 
, h Planck's constant. The peculiarity of the description of the microcosm is that the product of the uncertainty (accuracy of determination) of the position Δx and the uncertainty (accuracy of determination) of the momentum Δp x must always be equal to or greater than a constant equal to –. It follows from this that a decrease in one of these quantities should lead to an increase in the other. It is well known that any measurement is associated with certain errors, and by improving measuring instruments, it is possible to reduce errors, i.e., increase the accuracy of measurement. But Heisenberg showed that there are conjugate (additional) characteristics of a microparticle, the exact simultaneous measurement of which is fundamentally impossible. Those. uncertainty is a property of the state itself, it is not related to the accuracy of the device. For other conjugate quantities - energy E and time t the ratio looks like: 
. This means that for the characteristic evolution time of the system Δ t, the error in determining its energy cannot be less than 
. From this relation follows the possibility of the emergence of the so-called virtual particles from nothing for a period of time less than 
and having energy Δ E. In this case, the law of conservation of energy will not be violated. Therefore, according to modern concepts, vacuum is not a void in which there are no fields and particles, but a physical entity in which virtual particles constantly appear and disappear. One of the basic principles of quantum mechanics is uncertainty principle discovered by Heisenberg. Obtaining information about some quantities that describe a micro-object inevitably leads to a decrease in information about other quantities that are additional to the first ones. Instruments that record quantities related by uncertainty relations are of different types, they are complementary to each other. Measurement in quantum mechanics means any process of interaction between classical and quantum objects that occurs apart from and independently of any observer. If in classical physics the measurement did not perturb the object itself, then in quantum mechanics each measurement destroys the object, destroying its wave function. For a new measurement, the object must be prepared again. In this regard, N. Bohr put forward Pcomplementarity principle, the essence of which is that for a complete description of the objects of the microworld, it is necessary to use two opposite, but complementary representations. Photon diffraction as an illustration of the uncertainty relation
From the point of view of quantum theory, light can be considered as a stream of light quanta - photons. When a monochromatic plane wave of light is diffracted by a narrow slit, each photon passing through the slit hits a certain point on the screen (Fig. 1.). It is impossible to predict exactly where the photon will hit. However, in aggregate, falling into different points of the screen, photons give a diffraction pattern. When a photon passes through a slit, we can say that its x coordinate was determined with an error Δx, which is equal to the size of the slit. If the front of a plane monochromatic wave is parallel to the plane of the screen with a slit, then each photon has a momentum directed along the z axis perpendicular to the screen. Knowing the wavelength, this momentum can be accurately determined: p = h/λ.
However, after passing through the slit, the direction of the pulse changes, as a result of which a diffraction pattern is observed. The momentum modulus remains constant, since the wavelength does not change during the diffraction of light. Deviation from the original direction occurs due to the appearance of the component Δp x along the x axis (Fig. 1.). It is impossible to determine the value of this component for each competitive photon, but its maximum value in absolute value determines the width of the 2S diffraction pattern. The maximum value of Δp x is a measure of the uncertainty of the photon momentum, which arises when determining its coordinates with an error of Δx. As can be seen from the figure, the maximum value of Δp x is: Δp x = psinθ, . If L>> s , then we can write: sinθ =s/ L and Δp x = p(s/ L).
Laboratory work No. 8. The order of the work.
Familiarize yourself with the theoretical part of the work.
Open a working window.A). By moving the sliders on the right side of the working window, set arbitrary values of the wavelength λ and the slit size Δx. Write down these values. Click the button Test. Using the right mouse button, zoom in on the diffraction pattern. Using the slider to the right of the diffraction pattern image, determine the maximum distance s that photons are deflected along the x-axis, and write it down. Use the right mouse button to return the image to its original state. Using the formulas in the theoretical part, determine Δp x . Substitute this value in the test window and click the button Check. If the calculations are correct, an inscription will appear Right!!!B). Using the found values, find the product Δp x Δx. Substitute this value in the test window and click the button Check. If the calculations are correct, an inscription will appear Right!!!.V). Change the slot size and by pressing the button Test repeat points A and B. Show your test results to your teacher. Make a table according to the results of measurements:| Δx (slit width) | Photon momentum p | Δp x (calculated) | ||||
Laboratory work No. 8. Report form.
General requirements for registration.
The work is carried out on sheets of A4 paper, or on double notebook sheets.
The heading states:
Surname and initials of the student, group number
NAME OF THE LABORATORY WORK
Each task of laboratory work is made out as its section and should have a heading. In the report for each task, answers to all questions should be given and, if indicated, conclusions are drawn and the necessary drawings are given. results test tasks must be shown to the teacher. In tasks that include measurements and calculations, the measurement data and the data of the calculations performed should be given.
Exercise. Uncertainty relation.
A). Wavelength λ and slit size Δx. Measured maximum distance s. Calculations of the photon momentum and Δp x .
B). Calculations of the product Δp x Δx.
V). Repeat items A and B.Table with results:
| Δx (slit width) | Photon momentum p | Δp x (calculated) | ||||
G). Analysis of results. Conclusions. Answers on questions.
D). Answers on questions.
Control questions to check the assimilation of the topic of laboratory work:
1. Explain why it follows from the uncertainty relation that it is impossible to simultaneously accurately determine the conjugate quantities?
2. The energy spectra of radiation are associated with the transition of electrons from higher energy levels to lower ones. This transition takes place over a certain period of time. Is it possible to absolutely accurately determine the energy of radiation?
3. State the essence of the uncertainty principle.
4. What is the role of the device in the microworld?
5. From the uncertainty relation, explain why, in photon diffraction, a decrease in the size of the slit leads to an increase in the width of the diffraction pattern?
6. State the essence of Bohr's complementarity principle.
7. What is vacuum according to modern concepts?
Lab #9 DESCRIPTION
Thermal motion (1)
Working window
The view of the working window is shown in Fig. 6.1. The left part of the working window shows a model of the thermal motion of particles in a volume, which is divided into two parts by a partition. With the mouse, the partition can be moved to the left (by pressing the left mouse button on its upper part) or removed (by clicking on its lower part).
R 
Figure 6.1.
In the right part of the working window are given: temperature (in the right and left parts of the simulated volume), instantaneous particle velocities, and the number of collisions of particles with walls during the observation process. button Start the movement of particles is started, while the initial velocities and the location of the particles are set randomly. In the box next to the button Start the number of particles is set. Button Stop stops the movement. By pressing the button Continue the movement is resumed, and the windows for recording the number of collisions with the walls are cleared. With button Heat it is possible to increase the temperature in the right part of the simulated volume. Button Off turns off the heating. The switch to the right of the control buttons can set several different operating modes.
To open the working window, click on its image.
Laboratory work number 9. Theory
