Quantum optics. Thermal radiation. quantum optics Particles or waves

QUANTUM OPTICS

QUANTUM OPTICS

A branch of statistical optics that studies the microstructure of light fields and optical fields. phenomena in which a quantum is visible. nature of light. The idea of ​​quantum. the structure of radiation is introduced in German. physicist M. Planck in 1900.

Statistical interference structure fields were first observed by S.I. Vavilov (1934), and he also proposed the term “microstructure of light.”

Light is a complex physical. an object whose state is determined by an infinite number of parameters. This also applies to monochromatic radiation, which is classical. description is fully characterized by amplitude, frequency, phase and polarization. The problem of completely determining the light field cannot be solved due to insurmountable technical difficulties. difficulties associated with an infinite number of measurements of field parameters. Additional Quantum contributes significantly to the complexity of solving this problem. character of measurements, since they are associated with the registration of photons by photodetectors.

Advances in laser physics and improvements in the technology for recording weak light fluxes determined the development and tasks of laser vision. Dolaser light sources according to their statistics. St. you are of the same type as noise generators that have a Gaussian . The state of their fields is almost completely determined by the shape of the radiation spectrum and its intensity. With the advent of quantum. generators and quantum. amplifiers K. o. received at its disposal a wide range of sources with very diverse, including non-Gaussian, statistical data. har-kami.

The simplest characteristic of the field is its cf. intensity. A more complete characterization of the spatiotemporal distribution of field intensity, determined from experiments on recording photons over time with one detector. Quantum studies provide even more complete information about the state of the field. its decomposition quantities that can be partially determined from experiments on the joint registration of photons in a field of several. receivers, or in the study of multiphoton processes in the plant.

Center. concepts in quantum theory that determine the state of the field and the picture of its fluctuations, phenomena. so-called correlation functions or field correlators. They are defined as quantum mechanical. averages of field operators (see QUANTUM FIELD THEORY). The degree of complexity of correlators determines the rank, and the higher it is, the more subtle the statistical data. The holy fields are characterized by it. In particular, these functions determine the pattern of joint registration of photons over time by an arbitrary number of detectors. Correlation functions play an important role in nonlinear optics. The higher the degree of optical nonlinearity. process, the higher the rank correlators are needed to describe it. Of particular importance in K. o. has the concept of quantum coherence. There are partial and full fields. A fully coherent wave in its effect on systems is as similar as possible to the classical one. monochromatic wave. This means that quantum. fluctuations of the coherent field are minimal. The radiation of lasers with a narrow spectral band is close in its characteristics to completely coherent.

Correlative research. functions of higher orders allows you to study physics. in emitting systems (for example, in lasers). Methods of K. o. make it possible to determine the details of intermol. responsible for changes in photocount statistics when light is scattered in a medium.

Physical encyclopedic dictionary. - M.: Soviet Encyclopedia. . 1983 .

QUANTUM OPTICS

The branch of optics that studies statistics. properties of light fields and the quantum manifestation of these properties in the processes of interaction of light with matter. The idea of ​​the quantum structure of radiation was introduced by M. Planck in 1900. A light field, like any physical field. the field, due to its quantum nature, is a statistical object, i.e. its state is determined in a probabilistic sense. Since the 60s intensive study of statistics began. distribution.) Further, the quantum process of spontaneous production of photons is an irreducible source of significant fluctuations of the fields studied by cosmos; finally, the registration of light by photodetectors itself - photocounts - is a discrete quantum one. noise of radiation generators, in the medium, etc., nonlinear optics; on the one hand, in nonlinear optics. processes, a statistical change occurs. properties of the light field, on the other hand, field statistics influence the course of nonlinear processes. correlation functions, or field correlators. They are defined as quantum mechanical. averages from field operators (see also Quantum field theory). The simplest characteristics of a field are its and cf. intensity. These characteristics are found from experiments, for example, light intensity - by measuring the rate of electron photoemission in a photomultiplier. Theoretically, these quantities are described (without taking into account field polarization) by a field correlator in which - Hermitian conjugate components of the electric operator. fields
at a space-time point x=(r,t). Operator expressed through - destruction operator (see Secondary quantization)photon " k"th fashion field Uk(r):

Accordingly, it is expressed through the birth operator Sign< . . . >denotes quantum averaging over field states, and if it is considered with matter, then also over states of matter. information about the state of the field is contained in the correlator G 1,1 (x 1 , x 2). In general, a detailed determination of the field state requires knowledge of the correlation. functions of higher orders (ranks). The standard form of correlators, due to its connection with the registration of photon absorption, is accepted as normally ordered:

that's all P creation operators are to the left of all annihilation operators. The order of the correlator is equal to the sum n+m It is practically possible to study low-order correlators. Most often this is a correlator G 2,2 (X 1 ,X 2 ;X 2 ,X 1), which characterizes fluctuations in the intensity of radiation, it is found from experiments on the joint counting of photons by two detectors. The correlator is defined similarly Gn,n(x 1 ,. . .x p;x p,. ..x 1) from registration of photon counts P receivers or from data n-photon absorption. G n,m s P№T possible only in nonlinear optics. experiments. In stationary measurements, the condition for the invariance of the correlator Gn,m in time requires the fulfillment of the law of conservation of energy:

where w 6 are the harmonic frequencies of the operators, respectively. In particular, G 2,l are found from the spatial pattern of interference of three-wave interaction in the process of destruction of one and creation of two quanta (see. Interaction of light waves). Of the nonstationary correlators, the one of particular interest is G 0,1 (x), determining the strength of the quantum field. Magnitude | G 0,1 (x)| 2 gives the field intensity value only in spec. cases, in particular for coherent fields. p(n,T) - probability of implementation exactly P photocounts in a time interval T. This characteristic contains hidden information about correlators of arbitrarily high orders. Revealing hidden information, in particular determining the distribution of radiation intensity by a source, is the subject of the so-called. inverse problem of photon counting in cosmos. Photon counting is an experiment that has a fundamentally quantum nature, which is clearly manifested when the intensity I the registered field does not fluctuate. Even in this case, it is caused by a temporally random sequence of photocounts with Poisson distribution

where b is the sensitivity characteristic of the photodetector, the so-called. its effectiveness. Meaning g(x 1 ,X 2) tends to 1 as the space-time points are spaced apart X 1 and X 2, which corresponds to the statistical independence of photocounts in them. When combining points x 1 =x 2 =x difference g (x, X)from one ( g- 1) characterizes the level of fluctuation of radiation intensity and manifests itself in the difference in the number of coincidences of photocounts obtained during their simultaneous and independent registration by two detectors. Fluctuations in the intensity of a single-mode field are characterized by the magnitude

where it is convenient to carry out averaging over states | n> (see State vector)With density matrix

in a cut R p - probability of realization of the field mode in the state with P photons. For thermal radiation the probability R p given Bose- Einstein statistics:

where cf. number of photons in mode This is a highly fluctuating field, for which g= 2. It is characterized positively. correlation g- 1>0 in the simultaneous registration of two photons. Such cases of intensity fluctuations, when g> 1, called in to. grouping of photons. g-1=0 represent the fields located in the so-called. coherent states, uk-rykh This specially allocated in K. o. a class of fields with non-fluctuating intensity is generated, for example, by classically moving electric charges. Coherent fields max. are simply described in the so-called. R(a)-Glauber representation (see Quantum coherence). In this view

Where

Expression (**) can be considered as corresponding to the classic. expression for g, in Krom R(a) is considered to be the distribution function of complex amplitudes a classic. fields and for which always P(a)>0. The latter leads to the condition g>1, i.e. to the possibility in the classical Grouping fields only. This is explained by the fact that fluctuations in the intensity of classical fields simultaneously cause the same change in photocounts in both photodetectors.

R(a) == d 2 (a - a 0) = d d -

two-dimensional d-function in the complex plane a. Thermal classic fields are characterized by positive f-tion (which describes the grouping in them). For quantum fields R(a) is a real function, but in the finite domain of the argument a it can be negative. meaning, then it represents the so-called. quasi-probabilities. Statistics of photo counts for fields with a precisely specified number N>1 photons in fashion P n = d nN(d nN - Kronecker symbol) is essentially non-classical. For this condition g = 1 - 1/N, which corresponds to negative. correlations: g- 1 <0. Такие случаи наз. в К. о. антигруппировкой фотонов, к-рую можно объяснить тем, что фотона одним из детекторов уменьшает вероятность фотоотсчёта в другом. Эффект антигруппировки наблюдается и в свете, резонансно рассеянном одним атомом. В этом случае регистрируемые кванты спонтанно рождаются в среднем через определ. интервалы времени и вероятность одноврем. рождения двух квантов равна нулю, что и даёт нулевую вероятность их одноврем. регистрации. многофотонные процессы. К. о. находит всё более широкую область применения. Так, напр., в связи с проектированием оптич. системы для регистрации гравитац. волн и постановкой т. н. невозмущающих оптич. экспериментов, в к-рых уровень флуктуации, в т. ч. квантовых, сводится к минимуму, внимание исследователей привлекают такие состояния поля, наз. "сжатыми", в к-рых флуктуации интересующей величины (подобной интенсивности или фазе идеально стабилизированного лазера) могут быть в принципе сведены до нуля.Lit.: Glauber R., Optical coherence and photon statistics, in the book: Quantum optics and quantum radiophysics, trans. from English and French, M., 1966; Klauder J., Sudarshan E., Fundamentals of Quantum Optics, trans. from English, M.. 1970; Perina Ya., Coherence of light, trans. from English, M., 1974; Spectroscopy of optical mixing and photons, ed. G, Cummins, E. Pike, trans. from English, M., 1978; Klyshko D.N., Fotony i, M., 1980; Crosignani B., Di Porto P., Bertolotti M., Statistical properties of scattered light, trans. from English, M., 1980. S.G. Przhibelsky.

Physical encyclopedia. In 5 volumes. - M.: Soviet Encyclopedia. Editor-in-chief A. M. Prokhorov. 1988 .

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QUANTUM OPTICS, a branch of optics in which the principles of quantum mechanics (wave-particle duality, state vectors, Heisenberg and Schrödinger ideas, etc.) are used to study the properties of light and its interaction with matter.

The origin of the quantum theory of light dates back to 1900, when M. Plath, to explain the spectral distribution of electromagnetic energy emitted by a thermal source, postulated the absorption and emission of it in discrete portions. The idea of ​​discreteness formed the basis for the derivation of the formula that bears his name and served as the impetus for the creation of quantum mechanics. However, it remained unclear whether the source of discreteness was the matter or the light itself. In 1905, A. Einstein published the theory of the photoelectric effect, in which he showed that it can be explained if light is considered as a stream of particles (light quanta), later called photons. Photons have energy E = hv (h is Planck's constant, v is the frequency of light) and propagate at the speed of light. Later, N. Bohr showed that atoms can emit light in discrete portions. Thus, light is considered both as an electromagnetic wave and as a stream of photons. A quantized light field is a statistical object and its state is determined in a probabilistic sense.

The creation in 1960 of a laser - a fundamentally new source of radiation compared to thermal one - stimulated research into the statistical characteristics of its radiation. These studies involve measuring the distribution of laser photons and field coherence. Non-laser light sources are essentially sources of random light fields with Gaussian field statistics. While studying the statistics of laser radiation, R. Glauber introduced the concept of a coherent state, which corresponds well to the radiation of a laser operating in a regime above the lasing threshold. In 1977, the American physicist J. Kimble first observed the so-called antibunching of photons (see below), which could be explained using quantum theory.

Since the end of the 20th century, quantum optics has been intensively developing. It is closely related to nonlinear and atomic optics, quantum information theory. One of the most convenient ways to determine the state of the light field is to measure correlation functions. The simplest of them is the field correlation function, which characterizes the connection of fields at different spatiotemporal points. It fully characterizes the field of a thermal radiation source, but does not allow one to distinguish sources with other statistical properties from thermal ones. In this regard, an important role is played by the correlation function of the number of photons (intensities) of the second order g (2) (τ), which contains information on the distribution of delay times τ of photon emission. It is used to measure the effects of bunching and antibunching of photons. Light from the source enters the beam splitter plate (Fig. 1), after which it is fed to two photodetectors. Registration of a photon is accompanied by the appearance of a pulse at the detector output. Pulses from the detectors enter a device that measures the delay time between them. The experiment is repeated many times. In this way, the distribution of delay times, which is associated with the function g (2) (τ), is measured. Figure 2 shows the dependence g (2) (τ) for three typical light sources - thermal, laser and resonant fluorescence. As τ → ∞, the values ​​of the functions for the thermal source and resonant fluorescence approach unity. For laser radiation g (2) (τ) = 1 and photon statistics are Poisson. For a thermal source g(2)(0) = 2 and it is more likely to detect two photons arriving immediately after each other (photon grouping effect). In the case of resonant fluorescence, the probability of an atom emitting two photons at once is zero (photon antibunching). The value g (2) (0) = 0 is due to the fact that there is a delay time between two successive acts of photon emission by one atom. This effect is explained by the complete quantum theory, which describes both the medium and the electromagnetic field from a quantum point of view.

Closely related to the antibunching effect is sub-Poisson photon statistics, for which the distribution function is narrower than the Poisson distribution. Therefore, the level of fluctuations in photon beams with sub-Poisson statistics is less than the level of fluctuations of coherent radiation. In the limiting case, such nonclassical fields have a strictly defined number of photons (the so-called Fock state of the field). In quantum theory, the number of photons is a discrete variable.

Nonlinear optics methods can be used to create non-classical light fields in which, compared to coherent fields, the level of quantum fluctuations of some continuous variables, for example, quadrature components or Stokes parameters characterizing the state of field polarization, is reduced. Such fields are called compressed. The formation of compressed fields can be interpreted in classical language. Let us express the electric field strength E through the quadrature components a and b: E(t) = a(t)cosωt + b(t)sinωt, where a(t) and b(t) are random functions, ω = 2πν is the circular frequency, t - time. By applying such a field to a degenerate optical parametric amplifier (OPPA) with a pump frequency of 2ω, one quadrature component (for example, a) can be amplified due to its phase sensitivity, and the other quadrature (b) can be suppressed. As a result, fluctuations in quadrature a increase, and in quadrature b decrease. The transformation of the noise level in the VOPU is shown in Figure 3. In Figure 3, b, the area of ​​fluctuations is compressed compared to the input state (Figure 3, a). Quantum fluctuations of the vacuum and coherent states behave in a similar way with parametric amplification. Of course, in this case the quantum-mechanical uncertainty relation is not violated (there is, as it were, a redistribution of fluctuations between quadratures). In parametric processes, as a rule, radiation is formed with super-Poisson photon statistics, for which the level of fluctuations exceeds that for coherent light.

To record compressed fields, balanced homodyne detectors are used, which can record only one quadrature. Thus, the level of fluctuations during photodetection of compressed light can be below the level of the standard quantum limit (shot noise) corresponding to the detection of coherent light. In squeezed light, fluctuations can be suppressed by up to 90% relative to the coherent state. Nonlinear optics methods also produce polarization-squeezed light in which fluctuations in at least one of the Stokes parameters are suppressed. Compressed light is of interest for precision optical-physical experiments, in particular for recording gravitational waves.

From a quantum point of view, the considered parametric process is the process of decay of a pump photon with a frequency of 2ω into two photons with a frequency of ω. In other words, photons in compressed light are created in pairs (biphotons), and their distribution function is radically different from Poisson (there are only an even number of photons). This is another unusual property of compressed light in the language of discrete variables.

If pump photons in a parametric process decay into two photons that differ in frequencies and/or polarizations, then such photons are correlated (connected) with each other. Let us denote the frequencies of the generated photons as ω 1 and ω 2, and let the photons have vertical (V) and horizontal (H) polarizations, respectively. The state of the field in this case is written in quantum language as |ψ) = |V) 1 |H) 2. It turns out that at a certain orientation of a nonlinear optical crystal in which a parametric process is observed, photons of the same frequency propagating in the same direction can be produced with orthogonal polarizations. As a result, the state of the field takes the form:

(*)

(The appearance of the coefficient in front of the bracket is due to the normalization condition.)

The state of photons described by the relation (*) is called entangled; this means that if a photon of frequency ω 1 is polarized vertically, then a photon of frequency ω 2 is horizontally polarized, and vice versa. An important property of the entangled state (*) is that measuring the polarization state of one photon projects the state of a photon of another frequency into an orthogonal state. States of type (*) are also called Einstein-Podolsky-Rosen pairs and entangled Bell states. The quantum states of atomic systems, as well as the states of atoms and photons, can be in an entangled state. Experiments have been carried out using photons in entangled states to test Bell's inequality, quantum teleportation and quantum dense coding.

Based on parametric optical interactions, as well as the effect of cross-interactions, quantum non-destructive measurements of the quadrature components and the number of photons were carried out, respectively. The use of quantum optics methods in processing optical images makes it possible to improve their recording, storage and reading (see Quantum image processing).

Quantum fluctuations of the electromagnetic field in a vacuum state can manifest themselves in a unique way: they lead to the appearance of an attractive force between conducting uncharged plates (see Casimir effect).

Quantum optics also includes the theory of fluctuations of laser radiation. Its consistent development is based on quantum theory, which gives correct results for photon statistics and laser radiation linewidth.

Quantum optics also studies the interaction of atoms with a light field, the effect of light on two- and three-level atoms. At the same time, a number of interesting and unexpected effects associated with atomic coherence were discovered: quantum beats (see Interference of states), Hanle effect, photon echo, etc.

Quantum optics also studies the cooling of atoms when interacting with light and the production of a Bose-Einstein condensate, as well as the mechanical effect of light on atoms for the purpose of capturing and controlling them.

Lit.: Klyshko D.N. Non-classical light // Advances in physical sciences. 1996. T. 166. Issue. 6; Bargatin I.V., Grishanin B.A., Zadkov V.N. Entangled quantum states of atomic systems // Ibid. 2001. T. 171. Issue. 6; Physics of quantum information / Edited by D. Bouwmeister et al. M., 2002; Scully M. O., Zubairi M. S. Quantum optics. M., 2003; Shleikh V. P. Quantum optics in phase space. M., 2005.

Definition 1

Quantum optics is a branch of optics whose main task is the study of phenomena in which the quantum properties of light can manifest themselves.

Such phenomena may be:

• photoelectric effect;
• thermal radiation;
• Raman effect;
• Compton effect;
• stimulated emission, etc.

Fundamentals of Quantum Optics

Unlike classical optics, quantum optics represents a more general theory. The main problem it addresses is to describe the interaction of light with matter, while taking into account the quantum nature of objects. Quantum optics also deals with the description of the process of light propagation under special (specific conditions).

A more accurate solution to such problems requires a description of both matter (including the propagation medium) and light exclusively from the position of the existence of quanta. At the same time, scientists often simplify the task when describing it when one of the components of the system (for example, a substance) is described in the format of a classical object.

Often in calculations, for example, only the state of the active medium is quantized, while the resonator is considered classical. However, if its length is an order of magnitude higher than the wavelength, it can no longer be considered classical. The behavior of an excited atom placed in such a resonator will be more complex.

The tasks of quantum optics are aimed at studying the corpuscular properties of light (that is, its photons and corpuscular particles). According to M. Planck's hypothesis about the properties of light, proposed in 1901, it is absorbed and emitted only in separate portions (photons, quanta). A quantum represents a material particle with a certain mass $m_ф$, energy $E$ and momentum $p_ф$. Then the formula is written:

Where $h$ represents Planck's constant.

$v=\frac(c)(\lambda)$

Where $\lambda$ is the frequency of light

$c$ will be the speed of light in vacuum.

The main optical phenomena explained by quantum theory include light pressure and the photoelectric effect.

Photoelectric effect and light pressure in quantum optics

Definition 2

The photoelectric effect is a phenomenon of interactions between photons of light and matter, in which the radiation energy will be transferred to the electrons of the substance. There are such types of photoelectric effect as internal, external and valve.

The external photoelectric effect is characterized by the release of electrons from the metal at the moment of its irradiation with light (at a certain frequency). The quantum theory of the photoelectric effect states that each act of absorption of a photon by an electron occurs independently of the others.

An increase in radiation intensity is accompanied by an increase in the number of incident and absorbed photons. When energy is absorbed by a substance of frequency $ν$, each of the electrons turns out to be capable of absorbing only one photon, while taking away energy from it.

Einstein, applying the law of conservation of energy, proposed his equation for the external photoelectric effect (an expression of the law of conservation of energy):

$hv=A_(out)+\frac(mv^2)(2)$

$A_(out)$ is the work function of an electron leaving the metal.

The kinetic energy of the emitted electron is obtained by the formula:

$E_k=\frac(mv^2)(2)$

From Einstein’s equation it turns out that if $E_k=0$, then it is possible to obtain the very minimum frequency (red limit of the photoelectric effect) at which it will be possible:

$v_0 = \frac (A_(out)) h$

The pressure of light is explained by the fact that, as particles, photons have a certain momentum, which they transfer to the body through the process of absorption and reflection:

Such a phenomenon as light pressure is also explained by the wave theory, according to which (if we refer to de Broglie’s hypothesis), any particle also has wave properties. The relationship between the momentum $P$ and the wavelength $\lambda$ is shown by the equation:

$P=\frac(h)(\lambda)$

Compton effect

Note 1

The Compton effect is characterized by incoherent scattering of photons by free electrons. The very concept of incoherence means the non-interference of photons before and after scattering. The effect changes the frequency of photons, and after scattering the electrons receive part of the energy.

The Compton effect provides experimental evidence of the manifestation of the corpuscular properties of light as a stream of particles (photons). The phenomena of the Compton effect and the photoelectric effect are important proof of quantum concepts of light. At the same time, phenomena such as diffraction, interference, and polarization of light confirm the wave nature of light.

The Compton effect represents one of the proofs of wave-particle duality of microparticles. The law of conservation of energy is written as follows:

$m_ec^2+\frac(hc)(\lambda)=\frac(hc)(\lambda)+\frac(m_ec^2)(scrt(1-\frac(v^2)(c^2)) )$

The inverse Compton effect represents an increase in the frequency of light when scattered by relativistic electrons with higher than photon energy. In this interaction, energy is transferred to the photon from the electron. The energy of scattered photons is determined by the expression:

$e_1=\frac(4)(3)e_0\frac(K)(m_ec^2)$

Where $e_1$ and $e_0$ are the energies of the scattered and incident photons, respectively, and $k$ is the kinetic energy of the electron.

Introduction

1. The emergence of the doctrine of quanta

Photoelectric effect and its laws

1 Laws of the photoelectric effect

3. Kirchhoff's law

4. Stefan-Boltzmann laws and Wien displacements

Formulas of Rayleigh - Jeans and Planck

Einstein's equation for the photoelectric effect

Photon, its energy and momentum

Application of the photoelectric effect in technology

Light pressure. Experiments by P.N. Lebedev

The chemical action of light and its applications

Wave-particle duality

Conclusion

Bibliography

Introduction

Optics is a branch of physics that studies the nature of optical radiation (light), its propagation and phenomena observed during the interaction of light and matter. According to tradition, optics is usually divided into geometric, physical and physiological. We will look at quantum optics.

Quantum optics is the branch of optics that deals with the study of phenomena in which the quantum properties of light are manifested. Such phenomena include: thermal radiation, photoelectric effect, Compton effect, Raman effect, photochemical processes, stimulated emission (and, accordingly, laser physics), etc. Quantum optics is a more general theory than classical optics. The main problem addressed by quantum optics is the description of the interaction of light with matter, taking into account the quantum nature of objects, as well as the description of the propagation of light under specific conditions. In order to accurately solve these problems, it is necessary to describe both matter (the propagation medium, including vacuum) and light exclusively from quantum positions, but simplifications are often resorted to: one of the components of the system (light or matter) is described as a classical object. For example, often in calculations related to laser media, only the state of the active medium is quantized, and the resonator is considered classical, but if the length of the resonator is on the order of the wavelength, then it can no longer be considered classical, and the behavior of an atom in an excited state placed in such a resonator will be much more complex.

1. The emergence of the doctrine of quanta

Theoretical studies of J. Maxwell showed that light is electromagnetic waves of a certain range. Maxwell's theory received experimental confirmation in the experiments of G. Hertz. From Maxwell's theory it followed that light falling on any body exerts pressure on it. This pressure was discovered by P. N. Lebedev. Lebedev's experiments confirmed the electromagnetic theory of light. According to the works of Maxwell, the refractive index of a substance is determined by the formula n=εμ −−√, i.e. associated with the electrical and magnetic properties of this substance ( ε And μ - respectively, the relative dielectric and magnetic permeability of the substance). But Maxwell’s theory could not explain such a phenomenon as dispersion (the dependence of the refractive index on the wavelength of light). This was done by H. Lorentz, who created the electronic theory of the interaction of light with matter. Lorentz suggested that electrons under the influence of the electric field of an electromagnetic wave perform forced oscillations with a frequency v, which is equal to the frequency of the electromagnetic wave, and the dielectric constant of the substance depends on the frequency of changes in the electromagnetic field, therefore, n=f(v). However, when studying the emission spectrum of an absolutely black body, i.e. a body that absorbs all radiation of any frequency incident on it, physics could not, within the framework of electromagnetic theory, explain the distribution of energy over wavelengths. The discrepancy between the theoretical (dashed) and experimental (solid) curves of the distribution of radiation power density in the spectrum of an absolutely black body (Fig. 19.1), i.e. the difference between theory and experiment was so significant that it was called the “ultraviolet catastrophe.” Electromagnetic theory also could not explain the appearance of line spectra of gases and the laws of the photoelectric effect.

Rice. 1.1

A new theory of light was put forward by M. Planck in 1900. According to M. Planck’s hypothesis, the electrons of atoms emit light not continuously, but in separate portions - quanta. Quantum energy Wproportional to oscillation frequency ν :

W=hν,

Where h- proportionality coefficient, called Planck’s constant:

h=6,62⋅10−34 J With

Since radiation is emitted in portions, the energy of an atom or molecule (oscillator) can only take on a certain discrete series of values ​​that are multiples of an integer number of electron portions ω , i.e. be equal hν,2hν,3hνetc. There are no oscillations whose energy is intermediate between two consecutive integers that are multiples of hν. This means that at the atomic-molecular level, vibrations do not occur with any amplitude values. The permissible amplitude values ​​are determined by the oscillation frequency.

Using this assumption and statistical methods, M. Planck was able to obtain a formula for the energy distribution in the radiation spectrum that corresponds to experimental data (see Fig. 1.1).

Quantum ideas about light, introduced into science by Planck, were further developed by A. Einstein. He came to the conclusion that light is not only emitted, but also propagates in space and is absorbed by matter in the form of quanta.

The quantum theory of light has helped explain a number of phenomena observed when light interacts with matter.

2. Photoelectric effect and its laws

The photoelectric effect occurs when a substance interacts with absorbed electromagnetic radiation.

There are external and internal photoeffects.

External photoeffectis the phenomenon of electrons being ejected from a substance under the influence of light incident on it.

Internal photoeffectis the phenomenon of an increase in the concentration of charge carriers in a substance, and therefore an increase in the electrical conductivity of a substance under the influence of light. A special case of the internal photoelectric effect is the gate photoeffect - the phenomenon of the appearance under the influence of light of an electromotive force in the contact of two different semiconductors or a semiconductor and a metal.

The external photoelectric effect was discovered in 1887 by G. Hertz, and studied in detail in 1888-1890. A. G. Stoletov. In experiments with electromagnetic waves, G. Hertz noticed that a spark jumping between the zinc balls of the spark gap occurs at a smaller potential difference if one of them is illuminated with ultraviolet rays. When studying this phenomenon, Stoletov used a flat capacitor, one of the plates of which (zinc) was solid, and the second was made in the form of a metal mesh (Fig. 1.2). The solid plate was connected to the negative pole of the current source, and the mesh plate was connected to the positive pole. The inner surface of the negatively charged capacitor plate was illuminated by light from an electric arc, the spectral composition of which includes ultraviolet rays. While the capacitor was not illuminated, there was no current in the circuit. When illuminating the zinc plate TOultraviolet ray galvanometer Grecorded the presence of current in the circuit. In the event that the grid became the cathode A,there was no current in the circuit. Consequently, the zinc plate, when exposed to light, emitted negatively charged particles. At the time the photoelectric effect was discovered, nothing was known about electrons, discovered by J. Thomson only 10 years later, in 1897. After the discovery of the electron by F. Lenard, it was proven that negatively charged particles emitted under the influence of light are electrons called photoelectrons.

Rice. 1.2

Stoletov conducted experiments with cathodes made of different metals in a setup, the diagram of which is shown in Figure 1.3.

Rice. 1.3

Two electrodes were soldered into a glass container from which air had been pumped out. Inside the cylinder, through a quartz “window”, transparent to ultraviolet radiation, light enters the cathode K. The voltage supplied to the electrodes can be changed using a potentiometer and measured with a voltmeter V.Under the influence of light, the cathode emitted electrons that closed the circuit between the electrodes, and the ammeter recorded the presence of current in the circuit. By measuring the current and voltage, you can plot the dependence of the photocurrent strength on the voltage between the electrodes I=I(U) (Fig. 1.4). From the graph it follows that:

In the absence of voltage between the electrodes, the photocurrent is non-zero, which can be explained by the presence of kinetic energy in the photoelectrons upon emission.

At a certain voltage between the electrodes UHThe strength of the photocurrent ceases to depend on voltage, i.e. reaches saturation IH.

Rice. 1.4

Saturation photocurrent strength IH=qmaxt, Where qmaxis the maximum charge carried by photoelectrons. It is equal qmax=net, Where n- the number of photoelectrons emitted from the surface of the illuminated metal in 1 s, e- electron charge. Consequently, with saturation photocurrent, all electrons that leave the metal surface in 1 s arrive at the anode during the same time. Therefore, by the strength of the saturation photocurrent, one can judge the number of photoelectrons emitted from the cathode per unit time.

If the cathode is connected to the positive pole of the current source, and the anode to the negative pole, then in the electrostatic field between the electrodes the photoelectrons will be inhibited, and the photocurrent strength will decrease as the value of this negative voltage increases. At a certain value of negative voltage U3 (called retardation voltage), the photocurrent stops.

According to the kinetic energy theorem, the work of the retarding electric field is equal to the change in the kinetic energy of the photoelectrons:

A3=−eU3;Δ Wk=mυ2max2,

eU3=mυ2max2.

This expression was obtained under the condition that the speed υ ≪c, Where With- speed of light.

Therefore, knowing U3, the maximum kinetic energy of photoelectrons can be found.

In Figure 1.5, ADependency graphs are shown If(U)for different light fluxes incident on the photocathode at a constant light frequency. Figure 1.5, b shows the dependence graphs If(U)for a constant luminous flux and different frequencies of light incident on the cathode.

Rice. 1.5

Analysis of the graphs in Figure 1.5, a shows that the strength of the saturation photocurrent increases with increasing intensity of the incident light. If, based on these data, we construct a graph of the dependence of the saturation current on the light intensity, we will obtain a straight line that passes through the origin of coordinates (Fig. 1.5, c). Therefore, the saturation photon strength is proportional to the intensity of light incident on the cathode

If∼ I.

As follows from the graphs in Figure 1.5, bdecreasing the frequency of incident light , the magnitude of the retardation voltage increases with increasing frequency of the incident light. At U3 decreases, and at a certain frequency ν 0 delay voltage U30=0. At ν <ν 0 photoelectric effect is not observed. Minimum frequency ν 0(maximum wavelength λ 0) incident light, at which the photoelectric effect is still possible, is called red border of the photoelectric effect.Based on the data in graph 1.5, byou can build a dependence graph U3(ν ) (Fig. 1.5, G).

Based on these experimental data, the laws of the photoelectric effect were formulated.

1 Laws of the photoelectric effect

1. The number of photoelectrons ejected in 1 s. from the surface of the cathode, proportional to the intensity of light incident on this substance.

2. The kinetic energy of photoelectrons does not depend on the intensity of the incident light, but depends linearly on its frequency.

3. The red limit of the photoelectric effect depends only on the type of cathode substance.

4. The photoelectric effect is practically inertia-free, since from the moment the metal is irradiated with light until the electrons are emitted, a time passes of ≈10−9 s.

3. Kirchhoff's law

Kirchhoff, relying on the second law of thermodynamics and analyzing the conditions of equilibrium radiation in an isolated system of bodies, established a quantitative relationship between the spectral density of energetic luminosity and the spectral absorption capacity of bodies. The ratio of the spectral density of energetic luminosity to the spectral absorptivity does not depend on the nature of the body; it is a universal function of frequency (wavelength) and temperature for all bodies (Kirchhoff’s law):

For black body , therefore it follows from Kirchhoff’s law that R,Tfor a black body is equal to r,T. Thus, the universal Kirchhoff function r,Tthere is nothing more than spectral density of the energy luminosity of a black body.Therefore, according to Kirchhoff's law, for all bodies the ratio of the spectral density of energetic luminosity to the spectral absorptivity is equal to the spectral density of energetic luminosity of a black body at the same temperature and frequency.

Using Kirchhoff's law, the expression for the energetic luminosity of a body (3.2) can be written as

For gray body

(3.2)

Energetically, the luminosity of a black body (depends only on temperature).

Kirchhoff's law describes only thermal radiation, being so characteristic of it that it can serve as a reliable criterion for determining the nature of radiation. Radiation that does not obey Kirchhoff's law is not thermal.

4. Stefan-Boltzmann laws and Wien displacements

From Kirchhoff's law (see (4.1)) it follows that the spectral density of the energy luminosity of a black body is a universal function, therefore finding its explicit dependence on frequency and temperature is an important task in the theory of thermal radiation. The Austrian physicist I. Stefan (1835-1893), analyzing experimental data (1879), and L. Boltzmann, using the thermodynamic method (1884), solved this problem only partially, establishing the dependence of the energy luminosity Reon temperature. According to the Stefan-Boltzmann law,

those. the energetic luminosity of a black body is proportional to the fourth power of its thermodynamic temperature;  - Stefan-Boltzmann constant: its experimental value is 5.6710 -8W/(m 2 K 4). Stefan-Boltzmann law, defining the dependence Reon temperature does not provide an answer regarding the spectral composition of black body radiation. From the experimental curves of the function r,Tfrom wavelength  at different temperatures (Fig. 287) it follows that the energy distribution in the black body spectrum is uneven. All curves have a clearly defined maximum, which shifts toward shorter wavelengths as the temperature rises. Area enclosed by the curve r,Tfrom  and x-axis, proportional to energetic luminosity Reblack body and, therefore, according to the Stefan-Boltzmann law, the fourth power of temperature.

The German physicist W. Wien (1864-1928), relying on the laws of thermo- and electrodynamics, established the dependence of the wavelength  max , corresponding to the maximum of the function r,T, on temperature T.According to Wien's displacement law,

(199.2)

i.e. wavelength  max , corresponding to the maximum value of the spectral density of energy luminosity r,Tblack body, is inversely proportional to its thermodynamic temperature, b-constant Guilt; its experimental value is 2.910 -3mK. Expression (199.2) is therefore called displacement lawThe fault is that it shows a shift in the position of the maximum of the function r,Tas the temperature increases into the region of short wavelengths. Wien's law explains why, as the temperature of heated bodies decreases, long-wave radiation increasingly dominates in their spectrum (for example, the transition of white heat to red heat when a metal cools).

5. Formulas of Rayleigh - Jeans and Planck

From the consideration of the Stefan-Boltzmann and Wien laws it follows that the thermodynamic approach to solving the problem of finding the universal Kirchhoff function r,Tdid not give the desired results. The following rigorous attempt to theoretically deduce the relationship r,Tbelongs to the English scientists D. Rayleigh and D. Jeans (1877-1946), who applied the methods of statistical physics to thermal radiation, using the classical law of uniform distribution of energy over degrees of freedom.

Rayleigh formula - Jeans for the spectral density of the energy luminosity of a black body has the form

(200.1)

where   = kT- average energy of the oscillator with natural frequency  . For an oscillator oscillating, the average values ​​of kinetic and potential energies are the same, therefore the average energy of each vibrational degree of freedom   = kT.

As experience has shown, expression (200.1) is consistent with experimental data onlyin the region of fairly low frequencies and high temperatures. In the region of high frequencies, the Rayleigh-Jeans formula sharply diverges from experiment, as well as from Wien’s displacement law (Fig. 288). In addition, it turned out that the attempt to obtain the Stefan-Boltzmann law (see (199.1)) from the Rayleigh-Jeans formula leads to absurdity. Indeed, the energetic luminosity of a black body calculated using (200.1) (see (198.3))

while according to the Stefan-Boltzmann law Reproportional to the fourth power of temperature. This result was called the "ultraviolet catastrophe." Thus, within the framework of classical physics it was not possible to explain the laws of energy distribution in the spectrum of a black body.

In the region of high frequencies, good agreement with experiment is given by Wien’s formula (Wien’s law of radiation), obtained by him from general theoretical considerations:

Where r,T- spectral density of energy luminosity of a black body, WITHAnd A -constant values. In modern notation using Planck's constant, which was not yet known at that time, Wien's radiation law can be written as

The correct expression for the spectral density of the energy luminosity of a black body, consistent with experimental data, was found in 1900 by the German physicist M. Planck. To do this, he had to abandon the established position of classical physics, according to which the energy of any system can change continuously,i.e., it can take any arbitrarily close values. According to the quantum hypothesis put forward by Planck, atomic oscillators emit energy not continuously, but in certain portions - quanta, and the energy of the quantum is proportional to the oscillation frequency (see (170.3)):

(200.2)

Where h= 6,62510-34Js is Planck's constant. Since radiation is emitted in portions, the energy of the oscillator  can only accept certain discrete values,multiples of an integer number of elementary portions of energy  0:

In this case, the average energy    oscillator cannot be taken equal kT.In the approximation that the distribution of oscillators over possible discrete states obeys the Boltzmann distribution, the average oscillator energy

and the spectral density of the energy luminosity of a black body

Thus, Planck derived the formula for the universal Kirchhoff function

(200.3)

which brilliantly agrees with experimental data on the energy distribution in the spectra of black body radiation over the entire range of frequencies and temperatures.The theoretical derivation of this formula was presented by M. Planck on December 14, 1900 at a meeting of the German Physical Society. This day became the date of birth of quantum physics.

In the region of low frequencies, i.e. at h<<kT(quantum energy is very small compared to the energy of thermal motion kT), Planck's formula (200.3) coincides with the Rayleigh-Jeans formula (200.1). To prove this, let us expand the exponential function into a series, limiting ourselves to the first two terms for the case under consideration:

Substituting the last expression into Planck’s formula (200.3), we find that

i.e., we obtained the Rayleigh-Jeans formula (200.1).

From Planck's formula one can obtain the Stefan-Boltzmann law. According to (198.3) and (200.3),

Let us introduce a dimensionless variable x=h/(kt); d x=hd  /(k T); d=kTd x/h.Formula for Reconverted to the form

(200.4)

Where because Thus, indeed, Planck’s formula allows us to obtain the Stefan-Boltzmann law (cf. formulas (199.1) and (200.4)). Additionally, substitution of numeric values k, sAnd hgives the Stefan-Boltzmann constant a value that is in good agreement with experimental data. We obtain Wien's displacement law using formulas (197.1) and (200.3):

Where

Meaning  max , at which the function reaches its maximum, we will find it by equating this derivative to zero. Then, by entering x=hc/(kTmax ), we get the equation

Solving this transcendental equation by the method of successive approximations gives x=4.965. Hence, hc/(kTmax )=4.965, from where

i.e., we obtained Wien's displacement law (see (199.2)).

From Planck's formula, knowing the universal constants h,kAnd With,you can calculate the Stefan-Boltzmann constants  and Wine b.On the other hand, knowing the experimental values  And b,values ​​can be calculated hAnd k(this is exactly how the numerical value of Planck’s constant was first found).

Thus, Planck’s formula not only agrees well with experimental data, but also contains particular laws of thermal radiation, and also allows one to calculate constants in the laws of thermal radiation. Therefore, Planck's formula is a complete solution to the basic problem of thermal radiation posed by Kirchhoff. Its solution became possible only thanks to Planck's revolutionary quantum hypothesis.

6. Einstein's equation for the photoelectric effect

Let's try to explain the experimental laws of the photoelectric effect using Maxwell's electromagnetic theory. An electromagnetic wave causes electrons to undergo electromagnetic oscillations. At a constant amplitude of the electric field strength vector, the amount of energy received by the electron in this process is proportional to the frequency of the wave and the “swinging” time. In this case, the electron must receive energy equal to the work function at any wave frequency, but this contradicts the third experimental law of the photoelectric effect. As the frequency of the electromagnetic wave increases, more energy is transferred to the electrons per unit time, and photoelectrons should be emitted in greater numbers, and this contradicts the first experimental law. Thus, it was impossible to explain these facts within the framework of Maxwell’s electromagnetic theory.

In 1905, to explain the phenomenon of the photoelectric effect, A. Einstein used quantum concepts of light, introduced in 1900 by Planck, and applied them to the absorption of light by matter. Monochromatic light radiation incident on a metal consists of photons. A photon is an elementary particle with energy W0=hν.Electrons in the surface layer of the metal absorb the energy of these photons, with one electron completely absorbing the energy of one or more photons.

If the photon energy W0 equals or exceeds the work function, then the electron is ejected from the metal. In this case, part of the photon energy is spent on performing the work function AV, and the rest goes into the kinetic energy of the photoelectron:

W0=AB+mυ2max2,

hν=AB+mυ2max2 - Einstein's equation for the photoelectric effect.

It represents the law of conservation of energy as applied to the photoelectric effect. This equation is written for the single-photon photoelectric effect, when we are talking about the ejection of an electron not associated with an atom (molecule).

Based on quantum concepts of light, the laws of the photoelectric effect can be explained.

It is known that the light intensity I=WSt, Where W- energy of incident light, S- surface area on which light falls, t- time. According to quantum theory, this energy is carried by photons. Hence, W=Nf hν, Where

Characteristics of thermal radiation:

The glow of bodies, i.e. the emission of electromagnetic waves by bodies, can be achieved through various mechanisms.

Thermal radiation is the emission of electromagnetic waves due to the thermal movement of molecules and atoms. During thermal motion, atoms collide with each other, transfer energy, go into an excited state, and when transitioning to the ground state, they emit an electromagnetic wave.

Thermal radiation is observed at all temperatures other than 0 degrees. Kelvin, at low temperatures long infrared waves are emitted, and at high temperatures visible waves and UV waves are emitted. All other types of radiation are called luminescence.

Let's place the body in a shell with an ideal reflective surface and pump out the air from the shell. (Fig. 1). Radiations leaving the body are reflected from the walls of the shell and are again absorbed by the body, i.e. there is a constant exchange of energy between the body and the radiation. In an equilibrium state, the amount of energy emitted by a body with a unit volume is in units. time is equal to the energy absorbed by the body. If the balance is disturbed, processes arise that restore it. For example: if a body begins to emit more energy than it absorbs, then the internal energy and temperature of the body decrease, which means it emits less and the decrease in body temperature occurs until the amount of energy emitted becomes equal to the amount received. Only thermal radiation is equilibrium.

Energy luminosity - , where shows what it depends on ( - temperature).

Energy luminosity is the energy emitted per unit. area in units time.
. The radiation may be different according to spectral analysis, therefore
- spectral density of energy luminosity:
is the energy emitted in the frequency range

is the energy emitted in the wavelength range
per unit area per unit time.

Then
;
- used in theoretical conclusions, and
- experimental dependence.
corresponds
, That's why
Then

, because
, That
. The “-” sign indicates that if the frequency increases, the wavelength decreases. Therefore, we discard “-” when substituting
.

- spectral absorptivity is the energy absorbed by the body. It shows what fraction of the incident radiation energy of a given frequency (or wavelength) is absorbed by the surface.
.

Absolutely black body - This is a body that absorbs all radiation incident on it at any frequency and temperature.
. A gray body is a body whose spectral absorption capacity is less than 1, but is the same for all frequencies
. For all other bodies
, depends on frequency and temperature.

And
depends on: 1) body material 2) frequency or wavelength 3) surface condition and temperature.

Kirchhoff's law.

Between the spectral density of energetic luminosity (
) and spectral absorptivity (
) for any body there is a connection.

Let us place several different bodies in the shell at different temperatures, pump out the air and maintain the shell at a constant temperature T. The exchange of energy between the bodies and the bodies and the shell will occur due to radiation. After some time, the system will go into an equilibrium state, that is, the temperature of all bodies is equal to the temperature of the shell, but the bodies are different, so if one body radiates in units. time, more energy then it must absorb more than the other in order for the temperature of the bodies to be the same, which means
- refers to different bodies.

Kirchhoff's law: the ratio of the spectral density of energetic luminosity and spectral absorptivity for all bodies is the same function of frequency and temperature - this is the Kirchhoff function. Physical meaning of the function: for a completely black body
therefore, from Kirchhoff's law it follows that
for an absolutely black body, that is, the Kirchhoff function is the spectral density of the energy luminosity of an absolutely black body. The energetic luminosity of a black body is denoted by:
, That's why
Since the Kirchhoff function is a universal function for all bodies, the main task is thermal radiation, experimental determination of the type of Kirchhoff function and the determination of theoretical models that describe the behavior of these functions.

There are no absolutely black bodies in nature; soot, velvet, etc. are close to them. You can obtain a black body model experimentally, for this we take a shell with a small hole, light enters it and is repeatedly reflected and absorbed with each reflection from the walls, so the light either does not come out, or a very small amount, i.e. such a device behaves in relation to absorption, it is an absolutely black body, and according to Kirchhoff’s law, it emits as a black body, that is, by experimentally heating or maintaining the shell at a certain temperature, we can observe the radiation coming out of the shell. Using a diffraction grating, we decompose the radiation into a spectrum and, by determining the intensity and radiation in each region of the spectrum, the dependence was determined experimentally
(gr. 1). Features: 1) The spectrum is continuous, i.e. all possible wavelengths are observed. 2) The curve passes through a maximum, that is, the energy is distributed unevenly. 3) With increasing temperature, the maximum shifts towards shorter wavelengths.

Let us explain the black body model with examples, that is, if the shell is illuminated from the outside, the hole appears black against the background of luminous walls. Even if the walls are made black, the hole is still darker. Let the surface of the white porcelain be heated and the hole will clearly stand out against the background of the faintly glowing walls.

Stefan-Boltzmann law

After conducting a series of experiments with various bodies, we determine that the energy luminosity of any body is proportional to
. Boltzmann found that the energy luminosity of a black body is proportional to
and wrote it down.
- Stefan-Boltzmann Faculty.

Boltzmann's constant.
.

Wine's Law.

In 1893 Vin received -
- Wien's law.
;
;
;, That
. Let's substitute:
;


;
.
, Then
,
- function from
, i.e.
- solution of this equation relative to
there will be some number at
;
from the experiment it was determined that
- constant Guilt.

Wien's law of displacement.

formulation: this wavelength corresponding to the maximum spectral density of the energy luminosity of an absolutely black body is inversely proportional to temperature.

Rayleigh formula-Jeans.

Definitions: Energy flow is the energy transferred through the site per unit time.
. Energy flux density is the energy transferred through a unit area per unit time
. Volumetric energy density is the energy per unit volume
. If the wave propagates in one direction, then through the area
during
the energy transferred in the volume of the cylinder is equal to
(Fig. 2) then

. Let's consider thermal radiation in a cavity with absolutely black walls, then 1) all radiation incident on the walls is absorbed. 2) Energy flux density is transferred through each point inside the cavity in any direction
(Fig. 3). Rayleigh and Jeans considered thermal radiation in a cavity as a superposition of standing waves. It can be shown that infinitesimal
emits a radiation flux into the cavity into the hemisphere
.
.

The energetic luminosity of a black body is the energy emitted from a unit area per unit time, which means that the flux of energy radiation is equal to:
,
; Equated

;
is the volumetric energy density per frequency interval
. Rayleigh and Jeans used the thermodynamic law of uniform distribution of energy over degrees of freedom. A standing wave has degrees of freedom and for each oscillating degree of freedom there is energy
. The number of standing waves is equal to the number of standing waves in the cavity. It can be shown that the number of standing waves per unit volume and per frequency interval
equals
here it is taken into account that 2 waves with mutually perpendicular orientation can propagate in one direction
.

If the energy of one wave is multiplied by the number of standing waves per unit volume of the cavity per frequency interval
we get the volumetric energy density per frequency interval
.
. Thus
we'll find it from here
for this
And
. Let's substitute
. Let's substitute
V
, Then
- Rayleigh-Jeans formula. The formula describes well the experimental data in the long wavelength region.

(gr. 2)
;
and the experiment shows that
. According to the Rayleigh-Jeans formula, the body only radiates and thermal interaction between the body and radiation does not occur.

Planck's formula.

Planck, like Rayleigh-Jeans, considered thermal radiation in a cavity as a superposition of standing waves. Also
,
,
, but Planck postulated that radiation does not occur continuously, but is determined in portions - quanta. The energy of each quantum takes on the values
,those
or the energy of a harmonic oscillator takes on discrete values. A harmonic oscillator is understood not only as a particle performing a harmonic oscillation, but also as a standing wave.

For determining
the average value of energy takes into account that energy is distributed depending on frequency according to Boltzmann’s law, i.e. the probability that a wave with frequency takes the energy value equal to
,
, Then







.

;
,
.

- Planck's formula.

;
;


. The formula fully describes the experimental dependence
and all the laws of thermal radiation follow from it.

Corollaries from Planck's formula.

;

1)
Low frequencies and high temperatures

;
;
- Rayleigh Jeans.

2)
High frequencies and low temperatures
;
and that's almost
- Wine's Law. 3)


- Stefan-Boltzmann law.

4)
;
;
;
- this transcendental equation, solving it using numerical methods, we obtain the root of the equation
;
- Wien's law of displacement.

Thus, the formula completely describes the dependence
and all the laws of thermal radiation do not follow.

Application of the laws of thermal radiation.

It is used to determine the temperatures of hot and self-luminous bodies. For this purpose pyrometers are used. Pyrometry is a method that uses the dependence of the energy dependence of bodies on the rate of glow of hot bodies and is used for light sources. For tungsten, the share of energy in the visible part of the spectrum is significantly greater than for a black body at the same temperature.