From the course of optics it is known that a whole series of optical phenomena can be consistently described from a wave point of view; examples are the well-known phenomena of interference and diffraction of light. On the other hand (let us refer to the Compton effect discussed in the previous paragraph), light just as clearly demonstrates its corpuscular nature. This wave-particle dualism must be considered an experimental fact, and therefore a consistent theory of light must be a particle-wave theory. Of course, in some limiting cases, only wave or only corpuscular descriptions may be sufficient.
It turns out, and at the same time we will again refer to experiment, that particles of matter with non-zero mass (these include, for example, electrons, protons, neutrons, atoms, molecules, etc.) also exhibit wave properties, so between them and photons there is no fundamental difference.
At this point, when moving from macro to micro objects, a certain difficulty arises in understanding the essence of physical phenomena. Indeed, at the level of macrophenomena, the corpuscular and wave descriptions are clearly distinguished. At the level of micro-phenomena, this boundary is largely blurred and the movement of a micro-object becomes both wave and corpuscular. In other words, a situation in which a microobject is to some extent similar to a corpuscle, and to some extent to a wave, becomes more adequate to reality, and this measure depends on the physical conditions of observation of the microobject.
A consistent theory that takes into account this feature of all microparticles is quantum theory. But before moving on to the presentation of its main ideas, it is necessary to establish how one and the same physical object can, in principle, exhibit either corpuscular or wave properties and what comparability exists between these two different methods of description.
In optical phenomena, a criterion for the applicability of the concept of a ray (i.e., a corpuscular picture) has been established and rules for the transition from wave concepts to corpuscular ones have been found. Continuing reasoning in this direction, one can hope! that here lies the transition in the opposite direction: from the corpuscular concepts of classical mechanics to the wave concepts of quantum mechanics.
Corresponding ideas using the optical-mechanical analogy were expressed by the French physicist L. de Broglie in 1924. De Broglie put forward a bold hypothesis that the wave-particle duality is not a feature of optical phenomena alone, but has universal applicability throughout physics of the microworld. In his book "Revolution in Physics" he wrote: "In optics for a century the corpuscular method of consideration was too neglected in comparison with the wave method; was not the opposite error made in the theory of matter? Have we not thought too much about the picture of "particles" and neglected Is it too much of a picture of waves?”
The following considerations also led him to the assumption of wave properties in material particles. At the end of the 20s of the XIX century. V. Hamilton drew attention to the amazing analogy between geometric optics and classical (Newtonian) mechanics. It was shown that the basic laws of these branches of physics, which are so different at first glance, can be represented in a mathematically identical form. As a result, instead of considering the motion of a particle in an external field with potential energy, one can study the propagation of a light beam in an optically inhomogeneous medium with an appropriately selected refractive index. Of course, this equivalence of descriptions also allows for a reverse transition.
The noted analogy was extended by Hamilton only to geometric optics and classical mechanics. But, as already noted, geometric optics is an approximation of more general wave optics and does not describe the purely wave properties of light. In turn, classical mechanics also has a limited range of applicability: as is known, it cannot explain the existence of discrete energy levels in atomic systems.
De Broglie's idea was to extend the analogy between optics and mechanics and compare wave mechanics with wave optics, attempting to apply the latter to intra-atomic phenomena. “An attempt to attribute to the electron, and in general to all particles, like photons, a dual nature, to endow them with wave corpuscular properties interconnected by a quantum of action (Planck’s constant) - such a task seemed extremely necessary and fruitful... It is necessary to create a new mechanics of a wave nature, which will treat old mechanics as wave optics is to geometric optics,” wrote de Broglie in his book “Revolution in Physics.”
For the discovery of the wave properties of matter, L. de Broglie was awarded the Nobel Prize in 1929.
Let us now turn to the formal side of the issue. Let us have a microparticle (for example, an electron) with mass M moving in a vacuum at a constant speed. Using the corpuscular description, we attribute energy to the particle E and momentum in accordance with the formulas (consider the general case of a relativistic particle).
. (1.2.1)
On the other hand, in the wave picture we use the concepts of frequency and wavelength (or wave number). If both descriptions are different aspects of the same physical object, then there must be an unambiguous relationship between them. Following de Broglie, let us transfer to the case of particles of matter the same rules of transition from one picture to another, which are valid when applied to light:
(1.2.2)
Relations (1.2.2) are called de Broglie formula. The wavelength associated with the particle is given by
(1.2.3)
They call her De Broglie wavelength. It is not difficult to understand, by analogy with light, that it is precisely this wavelength that should appear in the criteria for the applicability of wave or corpuscular pictures.
The simplest type of wave in a vacuum with a certain frequency and wave vector is a plane monochromatic wave
He expressed a bold hypothesis about the similarity between light and particles of matter, that if light has corpuscular properties, then material particles, in turn, should have wave properties. The movement of any particle with momentum is associated with a wave process with a wavelength:
This expression is called de Broglie wavelength for a material particle.
The existence of de Broglie waves can only be established on the basis of experiments in which the wave nature of particles is manifested. Since the wave nature of light manifests itself in the phenomena of diffraction and interference, then for particles that, according to de Broglie’s hypothesis, have wave properties, these phenomena should also be detected.
The difficulties in observing the wave properties of particles were due to the fact that these properties do not manifest themselves in macroscopic phenomena.
It was not possible to detect such a short wavelength in any of the experiments. However, if we consider electrons whose mass is very small, the wavelength will become sufficient for its experimental detection. In 1927, de Broglie's hypothesis was confirmed experimentally in the experiments of American physicists Davisson and Germer.
Simple calculations show that the wavelengths associated with the particles must be very small, i.e. significantly shorter than the wavelengths of visible light. Therefore, particle diffraction could be detected not on a conventional diffraction grating for visible light (with a lattice constant), but on crystals in which the atoms are located in a certain order at distances from each other ≈ .
That is why, in their experiments, Davisson and Germer studied the reflection of electrons from a nickel single crystal belonging to the cubic system.
Experience scheme shown in Fig. 20.1. In a vacuum, a narrow beam of monoenergetic electrons, obtained using cathode ray tube 1, was directed to target 2 (the surface of a nickel single crystal, ground perpendicular to the large diagonal of the crystal cell). The reflected electrons were captured by detector 3 connected to a galvanometer. The detector, which could be installed at any angle relative to the incident beam, captured only those electrons that experienced elastic reflection from the crystal.
The strength of the electric current in the galvanometer was used to determine the number of electrons registered by the detector. It turned out that when electron beams are reflected from the surface of a metal, a picture is observed that cannot be predicted on the basis of classical theory. The number of electrons reflected in some directions was greater and in others less than expected. That is, it arose selective reflection in certain directions. Electron scattering occurred especially intensely at an angle at an accelerating voltage.
It turned out to be possible to explain the results of the experiment only on the basis of wave concepts of electrons. Nickel atoms located on a polished surface form a regular reflective diffraction grating. The rows of atoms are perpendicular to the plane of incidence. Row spacing d= 0,091 nm. This value was known from radiographic studies. The energy of electrons is low and they do not penetrate deep into the crystal, so electron waves are scattered on surface nickel atoms. In some directions, the waves scattered from each atom reinforce each other, in others they are damped. Wave amplification will occur in those directions in which the difference in distances from each atom to the observation point is equal to an integer number of wavelengths (Fig. 20.2).
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For infinitely distant point, the condition for amplification of scattered waves will be written in the form 2dsinθ = nλ (Bragg formula, n− orders of diffraction maxima). For and the value of the diffraction angle corresponds to the wavelength
nm. (20.2)
Therefore, the movement of each electron can be described using a wave with a length of 0.167 nm.
De Broglie's formula (20.1) leads to the same result for wavelength. An electron accelerated in an electric field by a potential difference has kinetic energy. Since the modulus of a particle’s momentum is related to its kinetic energy by the relation , expression (20.1) for the wavelength can be written as: . (20.3)
Substituting numerical values of quantities into (20.3), we obtain:
Both results agree well, which confirms the presence of wave properties in electrons.
In 1927, the wave properties of electrons were confirmed in independent experiments by Thomson and Tartakovsky. They obtained diffraction patterns when electrons passed through thin metal films.
IN Thomson's experiments electrons in an electric field were accelerated to high speeds at an accelerating voltage , which corresponded to electron wavelengths from to (according to formula (20.3)). In this case, calculations were carried out using relativistic formulas. A thin beam of fast electrons was directed onto thick gold foil. The use of fast electrons is due to the fact that slower electrons are strongly absorbed by the foil. A photographic plate was placed behind the foil (Fig. 20.3).
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The action of electrons on a photographic plate is similar to the action of fast X-ray photons when they pass through aluminum foil.
Other evidence of electron diffraction in crystals is provided by similar electron diffraction patterns and x-ray diffraction patterns of the same crystal. Using these images, the lattice constant can be determined. Calculations carried out using two different methods lead to the same results. After prolonged bombardment of the foil with electrons, a central spot surrounded by diffraction rings formed on the photographic plate. The origin of diffraction rings is the same as in the case of X-ray diffraction.
The most clear experimental results confirming the wave nature of electrons were obtained in experiments on electron diffraction
Rice. 20.4
on two slits (Fig. 20.4), made for the first time in 1961 by K. Jonson. These experiments are a direct analogy to Young's experiment for visible light.
A flow of electrons, accelerated by a potential difference of 40 kV, after passing through a double slit in the diaphragm, hit the screen (photographic plate). Dark spots form on the photographic plate where electrons hit. With a large number of electrons on a photographic plate, a typical interference pattern is observed in the form of alternating maxima and minima of electron intensity, completely analogous to the interference pattern for visible light. R 12 – probability of electrons hitting different parts of the screen at a distance x from the center. The maximum probability corresponds to the diffraction maximum, zero probability corresponds to the diffraction minimum
It is characteristic that all the described results of experiments on electron diffraction are observed in the case when electrons fly through the experimental setup “one by one.” This can be achieved at a very low intensity of electron flow, when the average time of flight of an electron from the cathode to the photographic plate is less than the average time between the emission of two subsequent electrons from the cathode. In Fig. Figure 20.5 shows photographic plates after being hit by varying numbers of electrons (exposure increases from Fig. 20.5a to Fig. 20.5c).
The successive impact of an ever-increasing number of single electrons on a photographic plate gradually leads to the appearance of a clear diffraction pattern. The described results mean that in this experiment, electrons, while remaining particles, also exhibit wave properties, and these wave properties are inherent in each electron individually , and not just a system of a large number of particles.
In 1929 Stern and Esterman showed that helium atoms () and hydrogen molecules () also undergo diffraction. For heavy chemical elements, the de Broglie wavelength is very short, so diffraction patterns were either not obtained at all or were very blurry. For light helium atoms and hydrogen molecules, the average wavelength at room temperature is about 0.1 nm, that is, the same order as the crystal lattice constant. The beams of these atoms did not penetrate deep into the crystal, so diffraction of molecules was carried out on flat two-dimensional lattices of the crystal surface, similar to the diffraction of slow electrons on the flat surface of a nickel crystal () in the experiments of Davisson and Germer. As a result, clear diffraction patterns were observed. Later, diffraction by crystal lattices of very slow neutrons was discovered.
Bohr published his results in 1913. For the world of physics, they became both a sensation and a mystery. But England, Germany and France are the three cradles of new physics - were soon overwhelmed by another problem. Einstein was finishing his work on a new theory of gravity(one of its consequences was tested in 1919 during an international expedition, whose participants measured the deflection of a ray of light coming from a star as it passed near the Sun during an eclipse). Despite the enormous success of Bohr's theory, which explained the emission spectrum and other properties of the hydrogen atom, attempts to generalize it to the helium atom and atoms of other elements were not very successful. And although more and more information was accumulated about the corpuscular behavior of light during its interaction with matter, the obvious inconsistency of Bohr’s postulates (Bohr's Atom Mystery) remained unexplained.
In the twenties, several areas of research emerged that led to the creation of the so-called quantum theory. Although these directions seemed at first completely unrelated to each other, later (in 1930) they have all been shown to be equivalent and are simply different formulations of the same idea. Let's follow one of them.
In 1923, Louis de Broglie, then a graduate student, proposed that particles (for example, electrons) should have wave properties. “It seems to me,” he wrote, “... that the main idea of quantum theory is the impossibility of representing a separate portion of energy without associating a certain frequency with it.”
Objects of wave nature exhibit particle properties (for example, light, when emitted or absorbed, behaves like a particle). This was shown by Planck and Einstein and used by Bohr in his model of the atom. Why then cannot objects that we usually think of as particles (say, electrons) exhibit the properties of waves? Really, why? This symmetry between wave and particle was to de Broglie what circular orbits were to Plato, harmonious relationships between integers to Pythagoras, regular geometric forms to Kepler, or the solar system centered on a luminary to Copernicus.
What are these wave properties? De Broglie suggested the following. It was known that a photon is emitted and absorbed in the form of discrete portions, the energy of which is related to frequency by the formula:
At the same time, the relationship between the energy and momentum of a relativistic quantum of light (a particle with zero rest mass) has the form:
Together these ratios give:
From here de Broglie derived the relationship between wavelength and momentum:
for a wave type object - photon, which, judging by observations, was emitted and absorbed in the form of certain portions.
De Broglie further suggested that all objects, regardless of what type they are - wave or corpuscular, are associated with a certain wavelength, expressed through their momentum by exactly the same formula. An electron, for example, and any particle in general corresponds to a wave whose wavelength is equal to:
What kind of wave this was, de Broglie did not yet know at that time. However, if we assume that the electron in some sense has a certain wavelength, then we will obtain certain consequences from this assumption.
Let us consider Bohr's quantum conditions for stationary electron orbits. Let us assume that stable orbits are such that their length fits an integer number of wavelengths, i.e., the conditions for the existence of standing waves are met. Standing waves, whether on a string or in an atom, are motionless and retain their shape over time. For a given size of an oscillating system, they have only certain wavelengths.
Suppose, de Broglie said, that the allowed orbits in the hydrogen atom are only those for which the conditions for the existence of standing waves are met. To do this, an integer number of wavelengths must fit along the length of the orbit (Fig. 89), i.e.
nλ = 2πR, n = 1, 2, 3,…. (38.7)
But the wavelength associated with an electron is expressed in terms of its momentum using the formula:
Then expression (38.7) can be written as:
nh/p = 2πR (38.8)
pR = L = nh/2π (38.9)
The result is Bohr's quantization condition. Thus, if a certain wavelength is associated with an electron, then the Bohr quantization condition means that the electron’s orbit is stable when an integer number of standing waves fits along its length. In other words, the quantum condition now becomes not a special property of the atom, but a property of the electron itself ( and in the end, all other particles).
De Broglie's hypothesis. Diffraction of microparticles. Heisenberg uncertainty principle. Setting the state of the microparticle. The wave function, its statistical meaning and the conditions it must satisfy. The principle of superposition of quantum states. General Schrödinger equation. Schrödinger equation for stationary states.
De Broglie's conjecture
In 1924, French physicist Louis de Broglie hypothesized that all material objects in nature have both corpuscular and wave properties. According to de Broglie's hypothesis, wave-particle duality is a universal property of matter, and therefore any particle (electron, proton, neutron, etc.) has wave properties. In this case, the presence of wave properties in a particle fundamentally changes the nature of its motion and the method of describing such motion.
According to de Broglie's hypothesis, the wave properties of a free particle moving by inertia in the absence of external force fields are described by de Broglie plane wave , the frequency and wavelength of which are related to the corpuscular characteristics of the particle - energy and momentum. This relationship looks like:
.
The direction of propagation of the de Broglie wave coincides with the direction of motion of the particle, and it can be shown that the group velocity of the wave and the velocity of the particle are the same.
In the theory of wave processes, the equation of a plane monochromatic wave propagating in the direction of the axis has the form:
It is often written in complex form:
given that the harmonic function is the real part of the complex function, where is the imaginary unit.
The plane wave equation determines the amplitude of the wave, its circular frequency and wave number. The initial phase of the wave in the expressions for is chosen to be equal to zero. Since for a plane de Broglie wave 
, then the de Broglie plane wave equation can be written as:
.
A de Broglie plane wave describes the wave properties of a free particle that has energy and momentum. By comparing the squared amplitudes of de Broglie waves in different regions of space, one can estimate the probability of finding a particle in these regions. The greater the squared amplitude of the de Broglie wave, the greater the probability of detecting a particle in a given region of space, i.e. its intensity.
De Broglie waves, which are often called waves of matter, like waves of any nature, can be reflected, refracted, interfere with each other, and experience diffraction when interacting with inhomogeneities. Then we can talk, for example, about particle diffraction and observe diffraction effects in various experiments with inhomogeneous media. One of the first experiments on electron diffraction on a crystal was carried out in 1927 by American scientists Clinton Davisson and Lester Germer.
Davisson-Germer experiment.
In the Davisson-Germer experiment, electrons accelerated in an electron gun hit a nickel crystal at a certain grazing angle. By adjusting the magnitude of the accelerating potential difference in the electron gun, the kinetic energy and momentum of the emitted electrons and, consequently, their de Broglie wavelength were changed. The number of electrons reflected from the crystal was measured by the detector current in the experiment. The nickel crystal structure was well known from X-ray diffraction data.
A sharp increase in the number of electrons reflected from the crystal was discovered in cases where the Wulf-Bragg condition was satisfied for de Broglie electron waves (this condition was obtained in experiments on X-ray diffraction on a nickel crystal).
The shortcomings of Bohr's theory pointed to the need to revise the foundations of quantum theory and ideas about the nature of microparticles (electrons, protons, etc.). The question arose about how comprehensive the representation of the electron in the form of a small mechanical particle, characterized by certain coordinates and a certain speed, is.
We already know that a kind of dualism is observed in optical phenomena. Along with the phenomena of diffraction and interference (wave phenomena), phenomena that characterize the corpuscular nature of light (photoelectric effect, Compton effect) are also observed.
In 1924, Louis de Broglie hypothesized that dualism is not a feature of only optical phenomena ,but has a universal character. Particles of matter also have wave properties .
“In optics,” wrote Louis de Broglie, “for a century, the corpuscular method of examination was too neglected in comparison with the wave one; hasn’t the opposite mistake 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 of transition from one picture to another that 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. the motion of a particle can be considered as the motion of a wave.
According to quantum mechanics, the free movement of a particle with mass m and momentum (where υ is the particle speed) can be represented as a plane monochromatic wave ( de Broglie wave) with 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).
Dependence of the wave function on the coordinate X is given by the formula
| , | (3.1.2) |
Where - wave number ,A wave vector directed towards the propagation of the wave or along the movement of the particle:
| . | (3.1.3) |
Thus, monochromatic wave wave vector associated with a freely moving microparticle, proportional to its momentum or inversely proportional to the wavelength.
Since the kinetic energy of a relatively slowly moving particle is , the wavelength can also be expressed through 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 in accordance with the principles common to all wave phenomena. Therefore, the basic geometric patterns of particle diffraction are no different from the patterns of diffraction of any waves. The general condition for the diffraction of waves of any nature is the commensurability of the length of the incident wave λ with distance d between scattering centers: .
Louis de Broglie's hypothesis was revolutionary, even for that revolutionary time in science. However, it was soon confirmed by many experiments.