Constant plank or avogadro. Plank constant

Planck's constant defines the boundary between the macroworld, where Newton's laws of mechanics apply, and the microworld, where the laws of quantum mechanics apply.

Max Planck - one of the founders of quantum mechanics - came to the ideas of quantization of energy, trying to theoretically explain the process of interaction between recently discovered electromagnetic waves (see Maxwell's equations) and atoms and, thereby, solve the problem of black body radiation. He realized that to explain the observed emission spectrum of atoms, it is necessary to take for granted that atoms emit and absorb energy in portions (which the scientist called quanta) and only at individual wave frequencies. The energy transferred by one quantum is equal to:

where v is the frequency of radiation, and h is the elementary quantum of action, which is a new universal constant, soon called Planck’s constant. Planck was the first to calculate its value based on experimental data h = 6.548 x 10–34 J s (in the SI system); according to modern data, h = 6.626 x 10–34 J s. Accordingly, any atom can emit a wide spectrum of interconnected discrete frequencies, which depends on the orbits of the electrons in the atom. Niels Bohr would soon create a coherent, albeit simplified, model of the Bohr atom, consistent with the Planck distribution.

Having published his results at the end of 1900, Planck himself - and this is clear from his publications - at first did not believe that quanta were a physical reality, and not a convenient mathematical model. However, when five years later Albert Einstein published a paper explaining the photoelectric effect based on the quantization of radiation energy, in scientific circles Planck's formula was no longer perceived as a theoretical game, but as a description of a real physical phenomenon at the subatomic level, proving the quantum nature of energy.

Planck's constant appears in all equations and formulas of quantum mechanics. In particular, it determines the scale from which the Heisenberg uncertainty principle comes into force. Roughly speaking, Planck's constant shows us the lower limit of spatial quantities beyond which quantum effects cannot be ignored. For grains of sand, say, the uncertainty in the product of their linear size and speed is so insignificant that it can be neglected. In other words, Planck’s constant draws the boundary between the macrocosm, where Newton’s laws of mechanics apply, and the microcosm, where the laws of quantum mechanics come into force. Having been obtained only for a theoretical description of a single physical phenomenon, Planck’s constant soon became one of the fundamental constants of theoretical physics, determined by the very nature of the universe.

Max Karl Ernst Ludwig PLANCK

Max Karl Ernst Ludwig Plank, 1858–1947

German physicist. Born in Kiel in the family of a law professor. Being a virtuoso pianist, Planck in his youth was forced to make a difficult choice between science and music (they say that before the First World War, in his spare time, pianist Max Planck often formed a very professional classical duet with violinist Albert Einstein. - Translator's note) Doctoral dissertation on the second Planck defended the law of thermodynamics in 1889 at the University of Munich - and in the same year he became a teacher, and from 1892 - a professor at the University of Berlin, where he worked until his retirement in 1928. Planck is rightfully considered one of the fathers of quantum mechanics. Today, a whole network of German research institutes bears his name.

Material from the free Russian encyclopedia “Tradition”

Values h	Units
6,626   070  040(81) 10 −34	J∙c
4,135   667  662(25) 10 −15	eV∙c
6,626   070  040(81) 10 −27	erg∙c

Planck's constant , denoted as h, is a physical constant used to describe the magnitude of the quantum of action in quantum mechanics. This constant first appeared in M. Planck’s works on thermal radiation, and is therefore named after him. It is present as a coefficient between energy E and frequency ν photon in Planck's formula:

Speed ​​of light c related to frequency ν and wavelength λ ratio:

Taking this into account, Planck’s relation is written as follows:

The value is often used

J c,

Erg c,

EV c,

called the reduced (or rationalized) Planck constant or.

The Dirac constant is convenient to use when angular frequency is used ω , measured in radians per second, instead of the usual frequency ν , measured by the number of cycles per second. Because ω = 2π ν , then the formula is valid:

According to Planck's hypothesis, which was later confirmed, the energy of atomic states is quantized. This leads to the fact that the heated substance emits electromagnetic quanta or photons of certain frequencies, the spectrum of which depends on the chemical composition of the substance.

In Unicode, Planck's constant is U+210E (h), and Dirac's constant is U+210F (ħ).

Content 1 Magnitude 2 Origin of Planck's constant 2.1 Black body radiation 2.2 Photo effect 2.3 Atomic structure 2.4 The Uncertainty Principle 2.5 Bremsstrahlung X-ray spectrum 3 Physical constants related to Planck's constant 3.1 Electron rest mass 3.2 Avogadro's constant 3.3 Elementary charge 3.4 Bohr magneton and nuclear magneton 4 Determination from experiments 4.1 Josephson constant 4.2 Power balance 4.3 Magnetic resonance 4.4 Faraday's constant 4.5 5 Planck's constant in SI units 6 Planck's constant in the theory of infinite nesting of matter 7 See also 8 Links 9 Literature 10 External links

Magnitude

Planck's constant has the dimension of energy times time, just like the dimension of action. In the international SI system of units, Planck's constant is expressed in units of J s. The product of impulse and distance in the form N m s, as well as angular momentum, has the same dimension.

The value of Planck's constant is:

J s eV s.

The two digits between the brackets indicate the uncertainty in the last two digits of the value of Planck's constant (data are updated approximately every 4 years).

Origin of Planck's constant

Black body radiation

Main article: Planck's formula

At the end of the 19th century, Planck investigated the problem of black body radiation, which Kirchhoff had formulated 40 years earlier. Heated bodies glow the more strongly, the higher their temperature and the greater the internal thermal energy. Heat is distributed among all the atoms of the body, causing them to move relative to each other and to excite the electrons in the atoms. As electrons transition to stable states, photons are emitted, which can be reabsorbed by atoms. At each temperature, a state of equilibrium between radiation and matter is possible, and the fraction of radiation energy in total energy system depends on temperature. In a state of equilibrium with radiation, an absolutely black body not only absorbs all the radiation incident on it, but also emits the same amount of energy, according to a certain law of energy distribution over frequencies. The law relating body temperature to the power of total radiated energy per unit surface area of ​​the body is called the Stefan-Boltzmann law and was established in 1879–1884.

When heated, not only does the total amount of emitted energy increase, but the composition of the radiation also changes. This can be seen by the fact that the color of heated bodies changes. According to Wien's displacement law of 1893, based on the principle of adiabatic invariant, for each temperature it is possible to calculate the wavelength of radiation at which the body glows most strongly. Wien made a fairly accurate estimate of the shape of the black body energy spectrum at high frequencies, but was unable to explain either the shape of the spectrum or its behavior at low frequencies.

Planck proposed that the behavior of light is similar to the motion of a set of many identical harmonic oscillators. He studied the change in entropy of these oscillators depending on temperature, trying to substantiate Wien's law, and found a suitable mathematical function for the black body spectrum.

However, Planck soon realized that in addition to his solution, others were possible, leading to other values ​​of the entropy of the oscillators. As a result, he was forced to use statistical physics, which he had previously rejected, instead of a phenomenological approach, which he described as “an act of desperation ... I was ready to sacrifice any previous beliefs in physics.” One of Planck's new conditions was:

interpret U N ( vibration energy of N oscillators ) not as a continuous infinitely divisible quantity, but as a discrete quantity consisting of a sum of limited equal parts. Let us denote each such part in the form of an energy element by ε;

With this new condition, Planck actually introduced the quantization of oscillator energy, saying that it was “a purely formal assumption... I haven’t really thought about it deeply...”, but it led to a real revolution in physics. Application of a new approach to Wien's displacement law showed that the "energy element" must be proportional to the frequency of the oscillator. This was the first version of what is now called "Planck's formula":

Planck was able to calculate the value h from experimental data on black body radiation: its result was 6.55 10 −34 J s, with an accuracy of 1.2% of the currently accepted value. He was also able to determine for the first time k B from the same data and his theory.

Before Planck's theory, it was assumed that the energy of a body could be anything, being a continuous function. This is equivalent to the fact that the energy element ε (the difference between allowed energy levels) is zero, therefore must be zero and h. Based on this, one should understand the statements that “Planck’s constant is equal to zero in classical physics” or that “classical physics is the limit of quantum mechanics when Planck’s constant tends to zero.” Due to the smallness of Planck's constant, it almost does not appear in ordinary human experience and was invisible before Planck's work.

The black body problem was revised in 1905, when Rayleigh and Jeans on the one hand, and Einstein on the other, independently proved that classical electrodynamics could not justify the observed radiation spectrum. This led to the so-called "ultraviolet catastrophe", so designated by Ehrenfest in 1911. The efforts of theorists (together with Einstein's work on the photoelectric effect) led to the recognition that Planck's postulate about the quantization of energy levels was not a simple mathematical formalism, but an important element of understanding about physical reality. The first Solvay Congress in 1911 was dedicated to the “theory of radiation and quanta.” Max Planck received the Nobel Prize in Physics in 1918 “for recognition of his services to the development of physics and the discovery of the energy quantum.”

Photo effect

Main article: Photo effect

The photoelectric effect involves the emission of electrons (called photoelectrons) from a surface when light is illuminated. It was first observed by Becquerel in 1839, although it is usually mentioned by Heinrich Hertz, who published an extensive study on the subject in 1887. Stoletov in 1888–1890 made several discoveries in the field of the photoelectric effect, including the first law of the external photoelectric effect. Another important study of the photoelectric effect was published by Lenard in 1902. Although Einstein did not conduct experiments on the photoelectric effect himself, his 1905 work examined the effect based on light quanta. This earned Einstein a Nobel Prize in 1921 when his predictions were confirmed by Millikan's experimental work. At this time, Einstein's theory of the photoelectric effect was considered more significant than his theory of relativity.

Before Einstein's work, each electromagnetic radiation was considered as a set of waves with their own "frequency" and "wavelength". The energy transferred by a wave per unit time is called intensity. Other types of waves, such as a sound wave or a water wave, have similar parameters. However, the transfer of energy associated with the photoelectric effect is not consistent with the wave pattern of light.

The kinetic energy of photoelectrons appearing in the photoelectric effect can be measured. It turns out that it does not depend on the light intensity, but depends linearly on frequency. In this case, an increase in light intensity does not lead to an increase in the kinetic energy of photoelectrons, but to an increase in their number. If the frequency is too low and the kinetic energy of photoelectrons is about zero, then the photoelectric effect disappears, despite the significant intensity of light.

According to Einstein's explanation, these observations reveal the quantum nature of light; Light energy is transferred in small "packets" or quanta, rather than as a continuous wave. The magnitude of these "packets" of energy, which were later called photons, was the same as those of Planck's "elements of energy". This led to modern look Planck's formula for photon energy:

Einstein's postulate was proven experimentally: the constant of proportionality between the frequency of light ν and photon energy E turned out to be equal to Planck's constant h.

Atomic structure

Main article: Bohr's postulates

Niels Bohr presented the first quantum model of the atom in 1913, trying to get rid of the difficulties of Rutherford's classical model of the atom. According to classical electrodynamics, a point charge, when rotating around a stationary center, should radiate electromagnetic energy. If such a picture is true for an electron in an atom as it rotates around the nucleus, then over time the electron will lose energy and fall onto the nucleus. To overcome this paradox, Bohr proposed to consider, similarly to what is the case with photons, that the electron in a hydrogen-like atom should have quantized energies E n:

Where R∞ is an experimentally determined constant (Rydberg constant in units of reciprocal length), With– speed of light, n– integer ( n = 1, 2, 3, …), Z– the serial number of a chemical element in the periodic table, equal to one for the hydrogen atom. An electron that reaches the lower energy level ( n= 1), is in the ground state of the atom and can no longer, due to reasons not yet defined in quantum mechanics, reduce its energy. This approach allowed Bohr to arrive at the Rydberg formula, which empirically describes the emission spectrum of the hydrogen atom, and to calculate the value of the Rydberg constant R∞ through other fundamental constants.

Bohr also introduced the quantity h/2π , known as the reduced Planck constant or ħ, as the quantum of angular momentum. Bohr assumed that ħ determines the angular momentum of each electron in an atom. But this turned out to be inaccurate, despite improvements to Bohr's theory by Sommerfeld and others. The quantum theory turned out to be more correct, in the form of Heisenberg’s matrix mechanics in 1925 and in the form of the Schrödinger equation in 1926. At the same time, the Dirac constant remained the fundamental quantum of angular momentum. If J is the total angular momentum of the system with rotational invariance, and Jz is the angular momentum measured along the selected direction, then these quantities can only have the following values:

The Uncertainty Principle

Planck's constant is also contained in the expression for Werner Heisenberg's uncertainty principle. If you take a large number of particles in the same state, then the uncertainty in their position Δ x, and the uncertainty in their momentum (in the same direction), Δ p, obey the relation:

where uncertainty is specified as the standard deviation of the measured value from its mathematical expectation. There are other similar pairs of physical quantities for which the uncertainty relation is valid.

In quantum mechanics, Planck's constant appears in the expression for the commutator between the position operator and the momentum operator:

where δ ij is the Kronecker symbol.

Bremsstrahlung X-ray spectrum

When electrons interact with the electrostatic field of atomic nuclei, bremsstrahlung radiation appears in the form of X-ray quanta. It is known that the frequency spectrum of bremsstrahlung X-rays has a precise upper limit, called the violet limit. Its existence follows from the quantum properties of electromagnetic radiation and the law of conservation of energy. Really,

where is the speed of light,

– wavelength of X-ray radiation,

– electron charge,

– accelerating voltage between the electrodes of the X-ray tube.

Then Planck's constant will be equal to:

Physical constants related to Planck's constant

The list of constants below is based on 2014 data CODATA. . Approximately 90% of the uncertainty in these constants is due to uncertainty in the determination of Planck's constant, as can be seen from the square of the Pearson correlation coefficient ( r 2 > 0,99, r> 0.995). Compared with other constants, Planck's constant is known to an accuracy of the order of with measurement uncertainty 1 σ .This accuracy is significantly better than that of the universal gas constant.

Electron rest mass

Typically, the Rydberg constant R∞ (in reciprocal length units) is determined in terms of mass m e and other physical constants:

The Rydberg constant can be determined very precisely ( ) from the spectrum of a hydrogen atom, while there is no direct way of measuring the electron mass. Therefore, to determine the mass of an electron, the formula is used:

Where c is the speed of light and α There is . The speed of light is determined quite accurately in SI units, as is the fine structure constant ( ). Therefore, the inaccuracy in determining the electron mass depends only on the inaccuracy of Planck’s constant ( r 2 > 0,999).

Avogadro's constant

Main article: Avogadro's number

Avogadro's number N A is defined as the ratio of the mass of one mole of electrons to the mass of one electron. To find it, you need to take the mass of one mole of electrons in the form of the “relative atomic mass” of the electron A r(e), measured in Penning trap (), multiplied by unit molar mass M u, which in turn is defined as 0.001 kg/mol. The result is:

Dependence of Avogadro's number on Planck's constant ( r 2 > 0.999) is repeated for other constants related to the amount of matter, for example, for the atomic mass unit. Uncertainty in the value of Planck's constant limits the values ​​of atomic masses and particles in SI units, that is, in kilograms. At the same time, the particle mass ratios are known with better accuracy.

Elementary charge

Sommerfeld originally determined the fine structure constant α So:

Where e there is an elementary electric charge, ε 0 – (also called dielectric constant of vacuum), μ 0 – magnetic constant or magnetic permeability of vacuum. The last two constants have fixed values ​​in the SI system of units. Meaning α can be determined experimentally by measuring the g-factor of the electron g e and subsequent comparison with the value resulting from quantum electrodynamics.

Currently, the most accurate value of the elementary electric charge is obtained from the above formula:

Bohr magneton and nuclear magneton

Main articles: Bohr magneton , Nuclear magneton

The Bohr magneton and nuclear magneton are units used to describe the magnetic properties of the electron and atomic nuclei, respectively. The Bohr magneton is the magnetic moment that would be expected for an electron if it behaved like a rotating charged particle according to classical electrodynamics. Its value is derived through the Dirac constant, the elementary electric charge and the mass of the electron. All these quantities are derived through Planck’s constant, the resulting dependence on h ½ ( r 2 > 0.995) can be found using the formula:

A nuclear magneton has a similar definition, with the difference that the proton is much more massive than the electron. The ratio of electron relative atomic mass to proton relative atomic mass can be determined with great accuracy ( ). For the connection between both magnetons, we can write:

Determination from experiments

Method	Meaning h, 10 –34 J∙s	Accuracy definitions
Power balance	6,626 068 89(23)	3,4∙10 –8
X-ray crystal density	6,626 074 5(19)	2,9∙10 –7
Josephson constant	6,626 067 8(27)	4,1∙10 –7
Magnetic resonance	6,626 072 4(57)	8,6∙10 –7	[ 20 ]
Faraday's constant	6,626 065 7(88)	1,3∙10 –6
CODATA 20 10 accepted value	6,626 06 9 57 (29 )	4 , 4 ∙10 –8	[ 22 ]
Nine recent measurements of Planck's constant are listed for five different methods. If there is more than one measurement, the weighted average is indicated h according to the CODATA method.

Planck's constant can be determined from the spectrum of a radiating black body or the kinetic energy of photoelectrons, as was done in the early twentieth century. However, these methods are not the most accurate. Meaning h according to CODATA based on the basis of three measurements by the power balance method of the product of quantities K J2 R K and one interlaboratory measurement of the molar volume of silicon, mainly by the power balance method until 2007 in the USA at the National Institute of Standards and Technology (NIST). Other measurements listed in the table did not affect the result due to lack of accuracy.

There are both practical and theoretical difficulties in determining h. Thus, the most accurate methods for balancing the power and X-ray density of a crystal do not fully agree with each other in their results. This may be a consequence of the overestimation of accuracy in these methods. Theoretical difficulties arise from the fact that all methods, except for X-ray crystal density, are based on the theoretical basis of the Josephson effect and the quantum Hall effect. With some possible inaccuracy of these theories, there will also be an inaccuracy in determining Planck's constant. In this case, the obtained value of Planck’s constant can no longer be used as a test to test these theories in order to avoid a vicious logical circle. The good news is that there are independent statistical ways to test these theories.

Josephson constant

Main article: Josephson effect

Josephson constant K J relates the potential difference U, arising in the Josephson effect in "Josephson contacts", with a frequency ν microwave radiation. The theory quite strictly follows the expression:

The Josephson constant can be measured by comparison with the potential difference across a bank of Josephson contacts. To measure the potential difference, compensation of the electrostatic force by the force of gravity is used. From the theory it follows that after replacing the electric charge e to its value through fundamental constants (see above Elementary charge ), expression for Planck's constant through K J:

Power balance

This method compares two types of power, one of which is measured in SI units in watts, and the other is measured in conventional electrical units. From the definition conditional watt W 90, it gives the measure for the product K J2 R K in SI units, where R K is the Klitzing constant, which appears in the quantum Hall effect. If the theoretical interpretation of the Josephson effect and the quantum Hall effect is correct, then R K= h/e 2, and measurement K J2 R K leads to the definition of Planck's constant:

Magnetic resonance

Main article: Gyromagnetic ratio

Gyromagnetic ratio γ is the proportionality coefficient between frequency ν nuclear magnetic resonance (or electron paramagnetic resonance for electrons), and an applied magnetic field B: ν = γB. Although there is difficulty in determining the gyromagnetic ratio due to measurement inaccuracy B, for protons in water at 25 °C it is known with better accuracy than 10 –6. Protons are partially “screened” from the applied magnetic field by the electrons of water molecules. The same effect leads to chemical shift in nuclear magnetic spectroscopy, and is indicated by a prime next to the gyromagnetic ratio symbol, γ′ p. The gyromagnetic ratio is related to the magnetic moment of the shielded proton μ′ p, spin quantum number S (S=1/2 for protons) and the Dirac constant:

Screened proton magnetic moment ratio μ′ p to the magnetic moment of the electron μ e can be measured independently with high accuracy, since the inaccuracy of the magnetic field has little effect on the result. Meaning μ e, expressed in Bohr magnetons, is equal to half the electron g-factor g e. Hence,

Further complication arises from the fact that to measure γ′ p measurement of electric current is required. This current is independently measured in conditional amperes, so a conversion factor is required to convert to SI amperes. Symbol Γ′ p-90 denotes the measured gyromagnetic ratio in conventional electrical units (the permitted use of these units began in early 1990). This quantity can be measured in two ways, the “weak field” method and the “strong field” method, and the conversion factor in these cases is different. Typically, the high field method is used to measure Planck's constant and the value Γ′ p-90(hi):

After the replacement, we obtain an expression for Planck’s constant through Γ′ p-90(hi):

Faraday's constant

Main article: Faraday's constant

Faraday's constant F is the charge of one mole of electrons equal to Avogadro's number N A multiplied by the elementary electric charge e. It can be determined by careful electrolysis experiments, by measuring the amount of silver transferred from one electrode to another in a given time at a given electric current. In practice, it is measured in conventional electrical units, and is designated F 90. Substituting values N A and e, and moving from conventional electrical units to SI units, we obtain the relation for Planck’s constant:

X-ray crystal density

The X-ray crystal density method is the main method for measuring Avogadro's constant N A, and through it Planck’s constant h. To find N A is the ratio between the volume of the unit cell of a crystal, measured by X-ray diffraction analysis, and the molar volume of the substance. Silicon crystals are used because they are available in high quality and purity thanks to technology developed in semiconductor manufacturing. The unit cell volume is calculated from the space between two crystal planes, denoted d 220. Molar volume V m(Si) is calculated through the density of the crystal and the atomic weight of the silicon used. Planck's constant is given by:

Planck's constant in SI units

Main article: Kilogram

As stated above, the numerical value of Planck's constant depends on the system of units used. Its value in the SI system of units is known with an accuracy of 1.2∙10 –8, although it is determined in atomic (quantum) units exactly(in atomic units, by choosing the units of energy and time, it is possible to ensure that the Dirac constant as a reduced Planck constant is equal to 1). The same situation occurs in conventional electrical units, where Planck’s constant (written h 90 in contrast to the designation in SI) is given by the expression:

Where K J–90 and R K–90 are precisely defined constants. Atomic units and conventional electrical units are convenient to use in the relevant fields, since the uncertainties in the final result depend only on the uncertainties of measurements, without requiring an additional and inaccurate conversion factor into the SI system.

There are a number of proposals to modernize the values ​​of the existing system of basic SI units using fundamental physical constants. This has already been done for the meter, which is determined through a given value of the speed of light. A possible next unit for revision is the kilogram, whose value has been fixed since 1889 by the mass of a small cylinder of platinum-iridium alloy stored under three glass bells. There are about 80 copies of these mass standards, which are periodically compared with the international unit of mass. The accuracy of secondary standards varies over time through their use, down to values ​​in the tens of micrograms. This roughly corresponds to the uncertainty in the determination of Planck's constant.

At the 24th General Conference on Weights and Measures on October 17-21, 2011, a resolution was unanimously adopted, in which, in particular, it was proposed that in a future revision of the International System of Units (SI) the SI units of measurement should be redefined so that Planck's constant would be equal to exactly 6.62606X 10 −34 J s, where X stands for one or more significant figures to be determined based on the best CODATA recommendations. . The same resolution proposed to determine in the same way the exact values ​​of Avogadro's constant, and .

Planck's constant in the theory of infinite nesting of matter

Unlike atomism, the theory does not contain material objects—particles with minimal mass or size. Instead, it is assumed that matter is endlessly divisible into ever smaller structures, and at the same time the existence of many objects significantly larger in size than our Metagalaxy. In this case, matter is organized into separate levels according to mass and size, for which it arises, manifests itself and is realized.

Just like Boltzmann's constant and a number of other constants, Planck's constant reflects the properties inherent in the level of elementary particles (primarily nucleons and components that make up matter). On the one hand, Planck's constant relates the energy of photons and their frequency; on the other hand, it, up to a small numerical coefficient 2π, in the form ħ, specifies the unit of orbital momentum of an electron in an atom. This connection is not accidental, since when emitted from an atom, an electron reduces its orbital angular momentum, transferring it to the photon during the period of existence of the excited state. During one period of revolution of the electron cloud around the nucleus, the photon receives such a fraction of energy that corresponds to the fraction of angular momentum transferred by the electron. The average frequency of a photon is close to the frequency of rotation of the electron near the energy level where the electron goes during radiation, since the radiation power of the electron increases rapidly as it approaches the nucleus.

Mathematically it can be described as follows. The equation rotational movement has the form:

Where K - moment of power, L – angular momentum. If we multiply this ratio by the increment in the rotation angle and take into account that there is a change in the electron rotation energy, and there is the angular frequency of the orbital rotation, then it will be:

In this ratio the energy dE can be interpreted as an increase in the energy of an emitted photon when its angular momentum increases by the amount dL . For the total photon energy E and the total angular momentum of the photon, the value ω should be understood as the average angular frequency of the photon.

In addition to correlating the properties of emitted photons and atomic electrons through angular momentum, atomic nuclei also have angular momentum expressed in units of ħ. It can therefore be assumed that Planck's constant describes the rotational motion of elementary particles (nucleons, nuclei and electrons, orbital motion of electrons in an atom), and the conversion of the energy of rotation and vibrations of charged particles into radiation energy. In addition, based on the idea of ​​particle-wave dualism, in quantum mechanics all particles are assigned an accompanying material de Broglie wave. This wave is considered in the form of a wave of the amplitude of the probability of finding a particle at a particular point in space. As for photons, the Planck and Dirac constants in this case become proportionality coefficients for a quantum particle, entering the expressions for the particle momentum, for energy E and for action S :

CONSTANT BARh, one of the universal numerical constants of nature, included in many formulas and physical laws that describe the behavior of matter and energy on a microscopic scale. The existence of this constant was established in 1900 by M. Planck, a professor of physics at the University of Berlin, in a work that laid the foundations of quantum theory. He also gave a preliminary estimate of its size. The currently accepted value of Planck's constant is (6.6260755 ± 0.00023)H 10 –34 JH s.

Planck made this discovery while trying to find a theoretical explanation for the spectrum of radiation emitted by heated bodies. Such radiation is emitted by all bodies consisting of a large number of atoms at any temperature above absolute zero, but it becomes noticeable only at temperatures close to the boiling point of water 100 ° C and above it. In addition, it covers the entire spectrum of frequencies from radio frequency to infrared, visible and ultraviolet regions. In the region of visible light, radiation becomes sufficiently bright only at approximately 550° C. The dependence of radiation intensity per unit time on frequency is characterized by the spectral distributions presented in Fig. 1 for several temperature values. The radiation intensity at a given frequency is the amount of energy emitted in a narrow frequency band in the vicinity of a given frequency. The area of ​​the curve is proportional to the total energy emitted at all frequencies. As is easy to see, this area increases rapidly with increasing temperature.

Planck wanted to theoretically derive the spectral distribution function and find an explanation for two simple experimentally established patterns: the frequency corresponding to the brightest glow of a heated body is proportional to the absolute temperature, and the total energy emitted over 1 unit area of ​​the surface of an absolutely black body is the fourth power of its absolute temperature .

The first pattern can be expressed by the formula

Where n m– frequency corresponding to the maximum radiation intensity, T– absolute body temperature, and a– constant, depending on the properties of the emitting object. The second pattern is expressed by the formula

Where E– total energy emitted by a unit surface area in 1 s, s is a constant characterizing the emitting object, and T– absolute body temperature. The first formula is called Wien's displacement law, and the second is called Stefan-Boltzmann's law. Based on these laws, Planck sought to derive an exact expression for the spectral distribution of emitted energy at any temperature.

The universal nature of the phenomenon could be explained from the standpoint of the second law of thermodynamics, according to which thermal processes occurring spontaneously in physical system, always go in the direction of establishing thermal equilibrium in the system. Let's imagine that two hollow bodies A And IN different shapes, different sizes and different materials with the same temperature facing each other, as shown in Fig. 2. Assuming that from A V IN more radiation comes in than from IN V A, then the body IN would inevitably become warmer due to A and the balance would spontaneously be disrupted. This possibility is excluded by the second law of thermodynamics, and therefore, both bodies must emit the same amount of energy, and, therefore, the quantity s in formula (2) does not depend on the size and material of the emitting surface, provided that the latter is a kind of cavity. If the cavities were separated by a color screen that would filter and reflect back all radiation, except radiation with any one frequency, then everything said would remain true. This means that the amount of radiation emitted by each cavity in each part of the spectrum is the same, and the spectral distribution function for the cavity has the character of a universal law of nature, and the value a in formula (1), similar to the quantity s, is a universal physical constant.

Planck, who was well versed in thermodynamics, preferred this particular solution to the problem and, through trial and error, found a thermodynamic formula that made it possible to calculate the spectral distribution function. The resulting formula was consistent with all available experimental data and, in particular, with empirical formulas (1) and (2). To explain this, Planck used a clever trick suggested by the second law of thermodynamics. Rightly believing that the thermodynamics of matter was better studied than the thermodynamics of radiation, he focused his attention primarily on the substance of the walls of the cavity, and not on the radiation inside it. Since the constants included in the Wien and Stefan-Boltzmann laws do not depend on the nature of the substance, Planck had the right to make any assumptions regarding the material of the walls. He chose a model in which the walls consisted of a huge number of tiny electrically charged oscillators, each with a different frequency. Oscillators can oscillate under the influence of radiation incident on them, emitting energy. The whole process could be studied based on the well-known laws of electrodynamics, i.e. the spectral distribution function could be found by calculating the average energy of oscillators with different frequencies. Reversing the sequence of reasoning, Planck, based on the correct spectral distribution function he guessed, found a formula for the average energy U oscillator with frequency n in a cavity in equilibrium at absolute temperature T:

Where b is a value determined experimentally, and k– a constant (called Boltzmann’s constant, although it was first introduced by Planck), which appears in thermodynamics and the kinetic theory of gases. Since this constant usually comes with a multiplier T, it is convenient to introduce a new constant h= b k. Then b = h/k and formula (3) can be rewritten as

New constant h and represents Planck's constant; its value calculated by Planck was 6.55H 10 –34 JH s, which is only about 1% different from the modern value. Planck's theory made it possible to express the quantity s in formula (2) through h,k and the speed of light With:

This expression agreed with experiment to the extent of the accuracy with which the constants were known; Later, more precise measurements revealed no discrepancies.

Thus, the problem of explaining the spectral distribution function has been reduced to a “simple” problem. It was necessary to explain what the physical meaning of the constant h or rather, works hn. Planck's discovery was that its physical meaning can be explained only by introducing into mechanics a completely new concept of “energy quantum.” On December 14, 1900, at a meeting of the German Physical Society, Planck showed in his report that formula (4), and thus the other formulas, can be explained if we assume that an oscillator with a frequency n exchanges energy with the electromagnetic field not continuously, but in steps, gaining and losing its energy in discrete portions, quanta, each of which is equal hn. HEAT; THERMODYNAMICS. The consequences of Planck's discovery are presented in the articles: PHOTOELECTRIC EFFECT; COMPTON EFFECT; ATOM; ATOMIC STRUCTURE; QUANTUM MECHANICS.

Quantum mechanics is a general theory of phenomena on a microscopic scale. Planck's discovery now appears as an important consequence of a special nature arising from the equations of this theory. In particular, it turned out that it is valid for everyone energy exchange processes that occur during oscillatory motion, for example in acoustics and electromagnetic phenomena. This explains the high penetrating power of X-ray radiation, whose frequencies are 100–10,000 times higher than those characteristic of visible light, and whose quanta have a correspondingly higher energy. Planck's discovery serves as the basis for the entire wave theory of matter, which deals with the wave properties of elementary particles and their combinations.

between the characteristics of a wave and a particle. This hypothesis was confirmed, making Planck's constant a universal physical constant. Her role turned out to be much more significant than one might have expected from the very beginning.

Memorial sign to Max Planck in honor of his discovery of Planck's constant, on the facade of the Humboldt University, Berlin. The inscription reads: “Max Planck, who invented the elementary quantum of action, taught in this building h, from 1889 to 1928." – an elementary quantum of action, a fundamental physical quantity that reflects the quantum nature of the Universe. The total angular momentum of a physical system can only change in multiples of Planck's constant. Just like in quantum mechanics, physical quantities are expressed through Planck’s constant.
Planck's constant is denoted by the Latin letter h. It has the dimension of energy times time.
More often used summary Planck's constant

In addition to the fact that it is convenient for use in the formulas of quantum mechanics, it has a special designation that cannot be confused with anything.
In the SI system, Planck's constant has the following meaning:
For calculations in quantum physics, it is more convenient to use the value of the summary Planck constant, expressed in terms of electron volts.
Max Planck introduced his constant to explain the radiation spectrum of a completely black body, suggesting that the body emits electromagnetic waves in portions (quanta) with energy proportional to the frequency (h?). In 1905, Einstein used this assumption to explain the phenomenon of the photoelectric effect, postulating that electromagnetic waves are absorbed in bursts of energy proportional to the frequency. This is how quantum mechanics was born, the validity of which both Nobel Prize laureates doubted all their lives.

· Mixed state · Measurement · Uncertainty · Pauli's principle · Dualism · Decoherence · Ehrenfest's theorem · Tunnel effect

Experiments
Davisson-Germer experiment · Popper experiment · Stern-Gerlach experiment · Young experiment · Bell's inequalities test · Photoelectric effect · Compton effect

Formulations
Schrödinger representation · Heisenberg representation · Interaction representation · Matrix quantum mechanics · Path integrals · Feynman diagrams

Equations
Schrödinger equation · Pauli equation · Klein-Gordon equation · Dirac equation · Schwinger-Tomonaga equation · Von Neumann equation · Bloch equation · Lindblad equation · Heisenberg equation

Interpretations
Copenhagen · Hidden parameter theory · Many-worlds · De Broglie-Bohm theory

Development of the theory
Quantum field theory · Quantum electrodynamics · Glashow-Weinberg-Salam theory · Quantum chromodynamics · Standard Model · Quantum gravity

Famous scientists
Planck · Einstein · Schrödinger · Heisenberg · Jordan · Bohr · Pauli · Dirac · Fock · Born · de Broglie · Landau · Feynman · Bohm · Everett

See also: Portal:Physics

Physical meaning

In quantum mechanics, impulse has the physical meaning of a wave vector, energy - frequency, and action - wave phase, but traditionally (historically) mechanical quantities are measured in other units (kg m/s, J, J s) than the corresponding wave ones (m −1, s −1, dimensionless phase units). Planck's constant plays the role of a conversion factor (always the same) connecting these two systems of units - quantum and traditional:

$\backslash mathbf\; p\; =\; \backslash hbar\; \backslash mathbf\; k$(pulse) $(|\backslash mathbf\; p|=\; 2\; \backslash pi\; \backslash hbar\; /\; \backslash lambda)$ $E\; =\; \backslash hbar\backslash omega$(energy) $S\; =\; \backslash hbar\backslash phi$(action)

If the system of physical units had been formed after the advent of quantum mechanics and had been adapted to simplify the basic theoretical formulas, Planck's constant would probably simply have been made equal to one, or, in any case, to a more round number. In theoretical physics, a system of units with $\backslash hbar\; =\; 1$, in it

$\backslash mathbf\; p\; =\; \backslash mathbf\; k$ $(|\backslash mathbf\; p|=\; 2\; \backslash pi\; /\; \backslash lambda)$ $E\; =\; \backslash omega$ $S\; =\; \backslash phi$ $(\backslash hbar\; =\; 1)$.

Planck's constant also has a simple evaluative role in delimiting the areas of applicability of classical and quantum physics: in comparison with the magnitude of the action or angular momentum characteristic of the system under consideration, or the product of a characteristic impulse by a characteristic size, or a characteristic energy by a characteristic time, it shows how applicable classical mechanics to this physical system. Namely, if $S$- the action of the system, and $M$ is its angular momentum, then at $\backslash frac(S)(\backslash hbar)\backslash gg1$ or $\backslash frac(M)(\backslash hbar)\backslash gg1$ The behavior of the system is described with good accuracy by classical mechanics. These estimates are fairly directly related to the Heisenberg uncertainty relations.

History of discovery

Planck's formula for thermal radiation

Planck's formula is an expression for the spectral power density of a black body radiation, which was obtained by Max Planck for the equilibrium radiation density $u(\backslash omega,\; T)$. Planck's formula was obtained after it became clear that the Rayleigh-Jeans formula satisfactorily describes radiation only in the long-wave region. In 1900, Planck proposed a formula with a constant (later called Planck's constant), which agreed well with experimental data. At the same time, Planck believed that this formula was just a successful mathematical trick, but had no physical meaning. That is, Planck did not assume that electromagnetic radiation is emitted in the form of individual portions of energy (quanta), the magnitude of which is related to the cyclic frequency of the radiation by the expression:

$\backslash varepsilon\; =\; \backslash hbar\; \backslash omega.$

Proportionality factor $\backslash hbar$ later named Planck's constant, $\backslash hbar$= 1.054·10 −34 J·s.

Photo effect

The photoelectric effect is the emission of electrons by a substance under the influence of light (and, generally speaking, any electromagnetic radiation). In condensed substances (solid and liquid) there is an external and internal photoelectric effect.

The same photocell is then irradiated with monochromatic light at a frequency $\backslash nu\_2$ and in the same way they lock it with tension $U\_2:$

$h\backslash nu\_2=A+eU\_2.$

Subtracting the second expression term by term from the first, we get

$h(\backslash nu\_1-\backslash nu\_2)=e(U\_1-U\_2),$

whence follows

$h=\backslash frac\; (e(U\_1-U\_2))((\backslash nu\_1-\backslash nu\_2)).$

Analysis of the X-ray bremsstrahlung spectrum

This method is considered the most accurate of the existing ones. It takes advantage of the fact that the frequency spectrum of bremsstrahlung X-rays has a precise upper limit, called the violet limit. Its existence follows from the quantum properties of electromagnetic radiation and the law of conservation of energy. Really,

$h\backslash frac(c)(\backslash lambda)=eU,$

Where $c$- speed of light,

$\backslash lambda$- x-ray wavelength, $e$- electron charge, $U$- accelerating voltage between the electrodes of the X-ray tube.

Then Planck's constant is

$h=\backslash frac((\backslash lambda)(Ue))(c).$

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Notes

Literature

• John D. Barrow. The Constants of Nature; From Alpha to Omega - The Numbers that Encode the Deepest Secrets of the Universe. - Pantheon Books, 2002. - ISBN 0-37-542221-8.
• Steiner R.// Reports on Progress in Physics. - 2013. - Vol. 76. - P. 016101.

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Excerpt characterizing Planck's Constant

“This is my cup,” he said. - Just put your finger in, I’ll drink it all.
When the samovar was all drunk, Rostov took the cards and offered to play kings with Marya Genrikhovna. They cast lots to decide who would be Marya Genrikhovna's party. The rules of the game, according to Rostov’s proposal, were that the one who would be king would have the right to kiss Marya Genrikhovna’s hand, and that the one who would remain a scoundrel would go and put a new samovar for the doctor when he woke up.
- Well, what if Marya Genrikhovna becomes king? – Ilyin asked.
- She’s already a queen! And her orders are law.
The game had just begun when the doctor’s confused head suddenly rose from behind Marya Genrikhovna. He had not slept for a long time and listened to what was said, and, apparently, did not find anything cheerful, funny or amusing in everything that was said and done. His face was sad and despondent. He did not greet the officers, scratched himself and asked permission to leave, as his way was blocked. As soon as he came out, all the officers burst into loud laughter, and Marya Genrikhovna blushed to tears and thereby became even more attractive in the eyes of all the officers. Returning from the yard, the doctor told his wife (who had stopped smiling so happily and was looking at him, fearfully awaiting the verdict) that the rain had passed and that she had to go spend the night in the tent, otherwise everything would be stolen.
- Yes, I’ll send a messenger... two! - said Rostov. - Come on, doctor.
– I’ll watch the clock myself! - said Ilyin.
“No, gentlemen, you slept well, but I didn’t sleep for two nights,” said the doctor and gloomily sat down next to his wife, waiting for the end of the game.
Looking at the gloomy face of the doctor, looking askance at his wife, the officers became even more cheerful, and many could not help laughing, for which they hastily tried to find plausible excuses. When the doctor left, taking his wife away, and settled into the tent with her, the officers lay down in the tavern, covered with wet overcoats; but they didn’t sleep for a long time, either talking, remembering the doctor’s fright and the doctor’s amusement, or running out onto the porch and reporting what was happening in the tent. Several times Rostov, turning over his head, wanted to fall asleep; but again someone’s remark entertained him, a conversation began again, and again causeless, cheerful, childish laughter was heard.

At three o'clock no one had yet fallen asleep when the sergeant appeared with the order to march to the town of Ostrovne.
With the same chatter and laughter, the officers hastily began to get ready; again they put the samovar on dirty water. But Rostov, without waiting for tea, went to the squadron. It was already dawn; the rain stopped, the clouds dispersed. It was damp and cold, especially in a wet dress. Coming out of the tavern, Rostov and Ilyin, both in the twilight of dawn, looked into the doctor’s leather tent, shiny from the rain, from under the apron of which the doctor’s legs stuck out and in the middle of which the doctor’s cap was visible on the pillow and sleepy breathing could be heard.
- Really, she’s very nice! - Rostov said to Ilyin, who was leaving with him.
- What a beauty this woman is! – Ilyin answered with sixteen-year-old seriousness.
Half an hour later the lined up squadron stood on the road. The command was heard: “Sit down! – the soldiers crossed themselves and began to sit down. Rostov, riding forward, commanded: “March! - and, stretching out into four people, the hussars, sounding the slap of hooves on the wet road, the clanking of sabers and quiet talking, set off along the large road lined with birches, following the infantry and battery walking ahead.
Torn blue-purple clouds, turning red at sunrise, were quickly driven by the wind. It became lighter and lighter. The curly grass that always grows along country roads, still wet from yesterday’s rain, was clearly visible; The hanging branches of the birches, also wet, swayed in the wind and dropped light drops to their sides. The faces of the soldiers became clearer and clearer. Rostov rode with Ilyin, who did not lag behind him, on the side of the road, between a double row of birch trees.
During the campaign, Rostov took the liberty of riding not on a front-line horse, but on a Cossack horse. Both an expert and a hunter, he recently got himself a dashing Don, a large and kind game horse, on which no one had jumped him. Riding this horse was a pleasure for Rostov. He thought about the horse, about the morning, about the doctor, and never thought about the upcoming danger.
Before, Rostov, going into business, was afraid; Now he did not feel the slightest sense of fear. It was not because he was not afraid that he was accustomed to fire (you cannot get used to danger), but because he had learned to control his soul in the face of danger. He was accustomed, when going into business, to think about everything, except for what seemed to be more interesting than anything else - about the upcoming danger. No matter how hard he tried or reproached himself for cowardice during the first period of his service, he could not achieve this; but over the years it has now become natural. He now rode next to Ilyin between the birches, occasionally tearing leaves from branches that came to hand, sometimes touching the horse’s groin with his foot, sometimes, without turning around, giving his finished pipe to the hussar riding behind, with such a calm and carefree look, as if he was riding ride. He felt sorry to look at Ilyin’s agitated face, who spoke a lot and restlessly; he knew from experience the painful state of waiting for fear and death in which the cornet was, and knew that nothing except time would help him.
The sun had just appeared on a clear streak from under the clouds when the wind died down, as if it did not dare spoil this lovely summer morning after the thunderstorm; the drops were still falling, but vertically, and everything became quiet. The sun came out completely, appeared on the horizon and disappeared into a narrow and long cloud standing above it. A few minutes later the sun appeared even brighter on the upper edge of the cloud, breaking its edges. Everything lit up and sparkled. And along with this light, as if answering it, gun shots were heard ahead.
Before Rostov had time to think about and determine how far these shots were, the adjutant of Count Osterman Tolstoy galloped up from Vitebsk with orders to trot along the road.
The squadron drove around the infantry and battery, who were also in a hurry to go faster, went down the mountain and, passing through some empty village without inhabitants, climbed the mountain again. The horses began to lather, the people became flushed.
- Stop, be equal! – the division commander’s command was heard ahead.
- Left shoulder forward, step march! - they commanded from the front.
And the hussars along the line of troops went to the left flank of the position and stood behind our lancers who were in the first line. On the right stood our infantry in a thick column - these were reserves; above it on the mountain, our guns were visible in the clean, clear air, in the morning, oblique and bright light, right on the horizon. Ahead, behind the ravine, enemy columns and cannons were visible. In the ravine we could hear our chain, already engaged and cheerfully clicking with the enemy.
Rostov, as if hearing the sounds of the most cheerful music, felt joy in his soul from these sounds, which had not been heard for a long time. Tap ta ta tap! – suddenly, then several shots clapped quickly, one after another. Again everything fell silent, and again it was as if firecrackers were cracking as someone walked on them.
The hussars stood in one place for about an hour. The cannonade began. Count Osterman and his retinue rode behind the squadron, stopped, talked with the regiment commander and rode off to the guns on the mountain.
Following Osterman’s departure, the lancers heard a command:
- Form a column, line up for the attack! “The infantry ahead of them doubled their platoons to let the cavalry through. The lancers set off, their pike weather vanes swaying, and at a trot they went downhill towards the French cavalry, which appeared under the mountain to the left.
As soon as the lancers went down the mountain, the hussars were ordered to move up the mountain, to cover the battery. While the hussars were taking the place of the lancers, distant, missing bullets flew from the chain, squealing and whistling.
This sound, not heard for a long time, had an even more joyful and exciting effect on Rostov than the previous sounds of shooting. He, straightening up, looked at the battlefield opening from the mountain, and with all his soul participated in the movement of the lancers. The lancers came close to the French dragoons, something was tangled there in the smoke, and five minutes later the lancers rushed back not to the place where they stood, but to the left. Between the orange lancers on red horses and behind them, in a large heap, were visible blue French dragoons on gray horses.

Rostov, with his keen hunting eye, was one of the first to see these blue French dragoons pursuing our lancers. Closer and closer the lancers and the French dragoons pursuing them moved in frustrated crowds. One could already see how these people, who seemed small under the mountain, collided, overtook each other and waved their arms or sabers.
Rostov looked at what was happening in front of him as if he were being persecuted. He instinctively felt that if he now attacked the French dragoons with the hussars, they would not resist; but if you hit, you had to do it now, this minute, otherwise it will be too late. He looked around him. The captain, standing next to him, did not take his eyes off the cavalry below in the same way.
“Andrei Sevastyanich,” said Rostov, “we will doubt them...
“It would be a dashing thing,” said the captain, “but in fact...
Rostov, without listening to him, pushed his horse, galloped ahead of the squadron, and before he had time to command the movement, the entire squadron, experiencing the same thing as him, set off after him. Rostov himself did not know how and why he did it. He did all this, as he did on the hunt, without thinking, without thinking. He saw that the dragoons were close, that they were galloping, upset; he knew that they could not stand it, he knew that there was only one minute that would not return if he missed it. The bullets screeched and whistled around him so excitedly, the horse begged forward so eagerly that he could not stand it. He touched his horse, gave the command, and at the same moment, hearing behind him the sound of the stomping of his deployed squadron, at full trot, he began to descend towards the dragoons down the mountain. As soon as they went downhill, their trot gait involuntarily turned into a gallop, which became faster and faster as they approached their lancers and the French dragoons galloping behind them. The dragoons were close. The front ones, seeing the hussars, began to turn back, the rear ones stopped. With the feeling with which he rushed across the wolf, Rostov, releasing his bottom at full speed, galloped across the frustrated ranks of the French dragoons. One lancer stopped, one foot fell to the ground so as not to be crushed, one riderless horse got mixed up with the hussars. Almost all the French dragoons galloped back. Rostov, having chosen one of them on a gray horse, set off after him. On the way he ran into a bush; good horse carried him over him, and, barely able to cope in the saddle, Nikolai saw that in a few moments he would catch up with the enemy whom he had chosen as his target. This Frenchman was probably an officer - judging by his uniform, he was bent over and galloping on his gray horse, urging it on with a saber. A moment later, Rostov’s horse hit the rear of the officer’s horse with its chest, almost knocking it down, and at the same moment Rostov, without knowing why, raised his saber and hit the Frenchman with it.