Thermal Radiation Characteristics:
The glow of bodies, that is, the radiation of electromagnetic waves by bodies, can be carried out due to various mechanisms.
Thermal radiation is the emission of electromagnetic waves due to the thermal motion of molecules and atoms. During thermal motion, atoms collide with each other, transfer energy, and at the same time pass into an excited state, and upon transition to the ground state, they emit an electromagnetic wave.
Thermal radiation is observed at all temperatures other than 0 degrees. Kelvin, long infrared waves are emitted at low temperatures, and visible waves and UV waves are emitted at high temperatures. All other types of radiation are called luminescence.
We place the body in a shell with an ideal reflective surface and pump out the air from the shell. (Fig. 1). Radiations coming out of 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 in units. time is equal to the energy absorbed by the body. If the balance is disturbed, then there are processes that restore it. For example: if a body begins to radiate more energy than it absorbs, then the internal energy and temperature of the body decrease, which means it radiates less and the body temperature decreases until the amount of radiated energy becomes equal to the amount received. Only thermal radiation is in equilibrium.
Energy luminosity - , where 
shows what depends 
- temperature).
Energy luminosity is the energy emitted with units. area in units time. 
. The radiation can be different according to the spectral analysis, therefore 
- spectral density of energy luminosity: 
is the energy radiated in the frequency range 
is the energy emitted in the wavelength range 
per unit area per unit time.
Then 
;
- is used in theoretical conclusions, and 
- experimental dependence. 
corresponds 
, That's why 
Then 
, because 
, That 
. The “-” sign indicates that if the frequency increases, then the wavelength decreases. Therefore, “-” is discarded when substituting 
.
- spectral absorbance is the energy absorbed by the body. It shows what fraction of the energy of the incident radiation of a given frequency (or wavelength) is absorbed by the surface. 
.
Absolutely black body –
This is a body that absorbs all radiation falling on it at any frequency and temperature. 
. A gray body is a body whose absorbance 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 on temperature.
Kirchhoff's law.
Between the spectral density of energy luminosity ( 
) and spectral absorbance ( 
) for any body there is a connection.
We 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, i.e. 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 energy luminosity and spectral absorbance for all bodies is the same function of frequency and temperature - this is the Kirchhoff function. The physical meaning of the function: for a completely black body 
so it follows from Kirchhoff's law that 
for a blackbody, i.e. the Kirchhoff function is the spectral density of the energy luminosity of a blackbody. The energy 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 the 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 get a model of a black body experimentally, for this we take a shell with a small hole, the 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 as an absolutely black body, and according to the Kirchhoff law, it radiates as a black body, i.e. by experimentally heating or maintaining the shell at a certain temperature, we can observe the radiation emerging from 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 experimentally determined 
(gr. 1). Features: 1) The spectrum is continuous, i.e. all possible wavelengths are observed. 2) The curve passes through a maximum, i.e. the energy is distributed unevenly. 3) As the temperature rises, the maximum shifts towards shorter wavelengths.
Let us explain the model of a black body with examples, i.e. if the shell is illuminated from the outside, the hole appears black against the background of luminous walls. Even if the walls are black, the hole is still darker. Let the surface of white porcelain be heated and a hole will clearly stand out against the background of weakly luminous 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 obtained that the energy luminosity of a black body is proportional to 
and wrote it down. 
- f-la Stefan-Boltzmann.
Boltzmann's constant. 
.
Wine's Law.
In 1893 Win received - 
- Wien's law. 
;
;
;
, That 
. We substitute: 
;
;
.
, Then 
,
- function from 
, i.e. 
is the solution of this equation with respect to 
there will be some number 
;
from the experiment determined that 
- constant guilt.
Wien's displacement law.
wording: this is the wavelength corresponding to the maximum spectral density of the energy luminosity of a completely black body is inversely proportional to temperature.
Rayleigh formula-Jeans.
Definitions: Energy flow is the energy transferred through the site per unit time. 
. The energy flux density is the energy transferred through a single 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 falling 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 the infinitesimal 
radiates a radiation flux into the cavity into the hemisphere 
.
.
The energy luminosity of a black body is the energy radiated from a unit area per unit time, which means that the energy radiation flux 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 an 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 
you get the volumetric energy density per frequency interval 
.
. Thus 
from here we will find 
for this 
And 
. Substitute 
. Substitute 
V 
, Then 
- Rayleigh-Jeans formula. The formula describes well the experimental data in the region of long waves.
(gr. 2) 
;
and experiment shows that 
. According to the Rayleigh-Jeans formula, the body only radiates and there is no thermal interaction between the body and the radiation.
Planck formula.
Planck, like Rayleigh-Jeans, considered thermal radiation in a cavity as a superposition of standing waves. Also 
,
,
, but Planck postulated that the radiation does not occur continuously, but is determined by portions - quanta. The energy of each quantum takes on the values 
,those 
or the energy of the harmonic oscillator takes discrete values. A harmonic oscillator is understood not only as a particle that performs harmonic oscillation, but also as a standing wave.
For determining 
the average value of energy take into account that the energy is distributed depending on the frequency according to the Boltzmann law, i.e. the probability that a wave with a frequency 
takes on the energy value 
is equal to 
,
, Then 
.
;
,
.
- Planck's formula. 
;
;
. The formula completely describes the experimental dependence 
and all the laws of thermal radiation follow from it.
Consequences from Planck's formula.
;
1)
Low frequencies and high temperatures 
;
;
- Rayleigh Jeans.
2)
High frequencies and low temperatures 
;
and it's almost 
- Wine's Law. 3) 
- Stefan-Boltzmann law.
4)
;
;
;
is a transcendental equation by solving it numerical methods we get the root of the equation 
;
- Wien's displacement law.
Thus, the formula fully describes the dependence 
and all the laws of thermal radiation do not follow from this.
Application of the laws of thermal radiation.
It is used to determine the temperatures of incandescent and self-luminous bodies. For this, 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 fraction of energy attributable to the visible part of the spectrum is much larger than for a black body at the same temperature.
THERMAL RADIATION. QUANTUM OPTICS
thermal radiation
Radiation of electromagnetic waves by bodies can be carried out due to various kinds energy. The most common is thermal radiation, i.e., the emission of electromagnetic waves due to the internal energy of the body. All other types of radiation are combined under the general name "luminescence". Thermal radiation occurs at any temperature, however, at low temperatures, practically only infrared electromagnetic waves are emitted.
Let us surround the radiating body with a shell, the inner surface of which reflects all the radiation incident on it. The air from the shell is removed. The radiation reflected by the shell is partially or completely absorbed by the body. Consequently, there will be a continuous exchange of energy between the body and the radiation filling the shell.
Equilibrium state of the "body-radiation" system corresponds to the condition when the distribution of energy between the body and the radiation remains unchanged for each wavelength. Such radiation is called equilibrium radiation. Experimental studies show that the only type of radiation that can be in equilibrium with radiating bodies is thermal radiation. All other types of radiation are non-equilibrium. The ability of thermal radiation to be in equilibrium with radiating bodies is due to the fact that its intensity increases with increasing temperature.
Let us assume that the balance between the body and radiation is disturbed and the body radiates more energy than it absorbs. Then the internal energy of the body will decrease, which will lead to a decrease in temperature. This, in turn, will lead to a decrease in the energy emitted by the body. If the equilibrium is disturbed in the other direction, i.e., the radiated energy turns out to be less than the absorbed, the temperature of the body will increase until equilibrium is established again.
Of all types of radiation only thermal radiation can be in equilibrium. The laws of thermodynamics apply to equilibrium states and processes. Therefore, thermal radiation obeys general patterns arising from the principles of thermodynamics. It is to the consideration of these regularities that we turn.
Planck formula
In 1900, the German physicist Max Planck managed to find the form of the function exactly corresponding to the experimental data. To do this, he had to make an assumption completely alien to classical ideas, namely, to admit that electromagnetic radiation emitted in the form of separate portions of energy (quanta) proportional to the radiation frequency:
where n is the radiation frequency; h is the coefficient of proportionality, called Planck's constant, h= 6.625 × 10-34 J × s; = h/2p=
= 1.05 × 10–34 J × s = 6.59 × 10-14 eV × s; w = 2pn is the circular frequency. In this case, if radiation is emitted by quanta, then its energy e n must be a multiple of this value:
The distribution density of radiative oscillators was classically calculated by Planck. According to the Boltzmann distribution, the number of particles N n, the energy of each of which is equal to e n, is determined by the formula
, n = 1, 2, 3… (4.2)
Where A is the normalization factor; k is the Boltzmann constant. Using the definition of the mean discrete quantities, we obtain an expression for the average energy of particles, which is equal to the ratio of the total energy of particles to the total number of particles:
where is the number of particles with energy . Taking into account (4.1) and (4.2), the expression for the average particle energy has the form
.
Subsequent transformations lead to the relation
.
Thus, the Kirchhoff function, taking into account (3.4), has the form
. (4.3)
Formula (4.3) is called Planck's formula. This formula agrees with experimental data over the entire frequency range from 0 to . In the region of low frequencies, according to the rules of approximate calculations, for (): 
» and expression (4.3) is transformed into the Rayleigh-Jeans formula.
Bothe experience. Photons
To explain the distribution of energy in the spectrum of equilibrium thermal radiation, it is sufficient, as Planck showed, to assume that light is emitted in quanta. To explain the photoelectric effect, it suffices to assume that light is absorbed in the same portions. Einstein put forward the hypothesis that light propagates in the form of discrete particles, originally called light quanta. Subsequently, these particles were called photons(1926). Einstein's hypothesis was directly confirmed by Bothe's experiment (Fig. 6.1).
A thin metal foil (F) was placed between two gas-discharge counters (SC). The foil was illuminated by a beam x-rays with low intensity, under the influence of which she herself became a source of x-rays.
Due to the low intensity of the primary beam, the number of quanta emitted by the foil was small. When x-rays hit the counter, a special mechanism (M) was launched, making a mark on the moving tape (L). If the radiated energy were distributed uniformly in all directions, as follows from wave representations, both counters would have to work simultaneously and the marks on the tape would fall one against the other.
In fact, there was a completely random arrangement of marks. This can only be explained by the fact that in separate acts of emission, light particles arise, flying first in one direction, then in the other. So the existence of special light particles - photons was proved.
The energy of a photon is determined by its frequency
. (6.1)
An electromagnetic wave, as you know, has momentum. Accordingly, the photon must also have momentum ( p). From relation (6.1) and general principles relativity, it follows that
. (6.2)
Such a relationship between momentum and energy is possible only for particles with zero rest mass moving at the speed of light. Thus: 1) the rest mass of a photon is equal to zero; 2) the photon moves at the speed of light. This means that a photon is a particle of a special kind, different from particles such as an electron, a proton, etc., which can exist by moving at speeds less than With, and even rest. Expressing in (6.2) the frequency w in terms of the wavelength l, we obtain:
,
where is the modulus of the wave vector k. A photon flies in the direction of propagation of an electromagnetic wave. Therefore, the direction of momentum R and wave vector k match up:
Let on completely absorbing surface the flux of photons flying along the normal to the surface decreases. If the photon density is N, then per unit surface falls per unit time Nc photons. When absorbed, each photon imparts momentum to the wall R = E/With. Impulse imparted per unit time to unit surface, i.e. pressure R light on the wall
.
Work NE is equal to the energy of photons contained in a unit volume, i.e., the density of electromagnetic energy w. Thus, the pressure exerted by light on an absorbing surface is equal to the volumetric density of electromagnetic energy P = w.
When reflected from mirror surface photon gives it momentum 2 R. Therefore, for a perfectly reflective surface P = 2w.
Compton effect
The momentum of a photon is too small and cannot be measured directly. However, when a photon collides with a free electron, the momentum transferred can already be measured. Process scattering of a photon by a free electron is called the Compton effect. Let us derive a relation relating the wavelength of the scattered photon to the scattering angle and the wavelength of the photon before the collision. Let a photon with momentum R and energy E = pc collides with a stationary electron whose energy is . After the collision, the momentum of the photon is equal and directed at an angle Q, as shown in Fig. 8.1.
The momentum of the recoil electron will be , and the total relativistic energy . Here we use relativistic mechanics, since the speed of an electron can reach values close to the speed of light.
According to the law of conservation of energy 
or 
, is converted to the form
. (8.1)
Let's write the momentum conservation law:
Let's square (8.2): 
and subtract this expression from (8.1):
. (8.3)
Considering that the relativistic energy 
, it can be shown that the right side of expression (8.2) is equal to . Then, after the transformation, the momentum of the photon is equal to
.
Moving on to wavelengths p = = h/l, Dl = l - l¢, we get:
,
or finally:
The quantity is called the Compton wavelength. For an electron, the Compton wavelength l c= 0.00243 nm.
In his experiment, Compton used X-rays with a known wavelength and found that the scattered photons had an increased wavelength. On fig. 8.1 shows the results pilot study monochromatic scattering x-ray radiation on graphite. The first curve (Q = 0°) characterizes the primary radiation. The remaining curves refer to different scattering angles Q, the values of which are shown in the figure. The ordinate shows the radiation intensity, the abscissa shows the wavelength. All graphs have an unshifted radiation component (left peak). Its presence is explained by the scattering of primary radiation by the bound electrons of the atom.
The Compton effect and the external photoelectric effect confirmed the hypothesis of the quantum nature of light, i.e., light really behaves as if it consisted of particles whose energy h n and momentum h/l. At the same time, the phenomena of interference and diffraction of light can be explained from the standpoint of the wave nature. Both of these approaches are currently appear to be complementary to each other.
Uncertainty principle
IN classical mechanics state material point is determined by specifying coordinate and momentum values. The peculiarity of the properties of microparticles is manifested in the fact that certain values are not obtained for all variables during measurements. So, for example, an electron (and any other microparticle) cannot simultaneously have exact values of the coordinate X and momentum components. Value uncertainties X and satisfy the relation
. (11.1)
From (11.1) it follows that the smaller the uncertainty of one of the variables ( X or ), the greater the uncertainty of the other. It is possible that one of the variables has an exact value, while the other variable turns out to be completely undefined.
A relation analogous to (11.1) holds for at And , z and , as well as for a number of other pairs of quantities (such pairs of quantities are called canonically conjugate). Denoting the canonically conjugate quantities by the letters A And IN, you can write
. (11.2)
Relation (11.2) is called the uncertainty principle for the quantities A And IN. This relation was formulated by W. Heisenberg in 1927. The statement that the product of the uncertainties of the values of two canonically conjugate variables cannot be less than Planck's constant in order of magnitude, called the uncertainty principle .
Energy and time are also canonically conjugate quantities
This relation means that the definition of energy with an accuracy of D E should take a time interval equal to at least .
The uncertainty relation can be illustrated by the following example. Let's try to determine the value of the coordinate X free-flying microparticle by placing a slot of width D in its path X located perpendicular to the direction of motion of the particle.
Before the particle passes through the slit, its momentum component has the exact value zero(the gap is perpendicular to the momentum direction by the condition), so that , but the coordinate X particles is completely indeterminate (Fig. 11.1).
As the particle passes through the slit, the position changes. Instead of the complete uncertainty of the coordinate X there is uncertainty D X, but this comes at the cost of losing the definition of the value. Indeed, due to diffraction, there is some probability that the particle will move within the angle 2j, where j is the angle corresponding to the first diffraction maximum (higher-order maxima can be neglected, since their intensity is small compared to the intensity of the central maximum). Thus, there is uncertainty
.
The edge of the central diffraction maximum (the first minimum) resulting from a slit of width D X, corresponds to the angle j, for which
Hence, 
, and we get
.
Movement along the trajectory is characterized by well-defined values of coordinates and speed at each moment of time. Substituting in (11.1) instead of the product , we obtain the relation
.
It is obvious that the greater the mass of a particle, the less the uncertainty of its coordinates and velocity, and, consequently, the more accurate the concept of a trajectory is applicable. Already for a macroparticle with a size of 1 μm, the uncertainties in the values X and turn out to be beyond the accuracy of measuring these quantities, so that its movement will be practically indistinguishable from the movement along the trajectory.
The uncertainty principle is one of the fundamental provisions of quantum mechanics.
Schrödinger equation
In developing de Broglie's idea of the wave properties of matter, the Austrian physicist E. Schrödinger obtained in 1926 an equation later named after him. In quantum mechanics, the Schrödinger equation plays the same fundamental role as Newton's laws in classical mechanics and Maxwell's equations in classical theory electromagnetism. It allows one to find the form of the wave function of particles moving in various force fields. The form of the wave function or Y-function is obtained by solving the equation, which looks like this
Here m is the particle mass; i is the imaginary unit; D is the Laplace operator, the result of which action on some function is the sum of second derivatives with respect to coordinates
letter U equation (12.1) denotes the function of coordinates and time, the gradient of which, taken with the opposite sign, determines the force acting on the particle.
The Schrödinger equation is the basic equation of non-relativistic quantum mechanics. It cannot be derived from other equations. If the force field in which the particle moves is stationary (i.e., constant in time), then the function U does not depend on time and has the meaning of potential energy. In this case, the solution to the Schrödinger equation consists of two factors, one of which depends only on the coordinates, the other depends only on time
Here E is the total energy of the particle, which remains constant in the case of a stationary field; is the coordinate part of the wave function. To verify the validity of (12.2), we substitute it into (12.1):
As a result, we get
Equation (12.3) is called Schrödinger equation for stationary states.In what follows, we will deal only with this equation and, for brevity, will simply call it the Schrödinger equation. Equation (12.3) is often written as
In quantum mechanics, the concept of an operator plays an important role. An operator is a rule by which one function, let's denote it, is associated with another function, let's denote it f. Symbolically, this is written as follows
here - a symbolic designation of the operator (you could take any other letter with a “hat” above it, for example, etc.). In formula (12.1), the role is played by D, the role is played by the function , and the role f is the right side of the formula. For example, the symbol D means twofold differentiation in three coordinates, X,at,z, followed by summing the resulting expressions. The operator can, in particular, represent the multiplication of the original function by some function U. Then , hence, . If we consider the function U in equation (12.3) as an operator whose action on the Y-function is reduced to multiplication by U, then equation (12.3) can be written as follows:
In this equation, the symbol denotes an operator equal to the sum of the operators and U:
.
The operator is called Hamiltonian (or Hamiltonian operator). The Hamiltonian is the energy operator E. In quantum mechanics, operators are also associated with other physical quantities. Accordingly, the operators of coordinates, momentum, angular momentum, etc. are considered. For each physical quantity an equation similar to (12.4) is compiled. It looks like
where is the operator to match g. For example, the momentum operator is defined by the relations
; 
; 
,
or in vector form , where Ñ is the gradient.
In sec. 10 we have already discussed physical meaning Y-functions: module square Y -function (wave function) determines the probability dP that the particle will be detected within the volume dV:
, (12.5)
Since the square of the modulus of the wave function is equal to the product of the wave function and the complex conjugate value , then
.
Then the probability of finding a particle in the volume V
.
For the one-dimensional case
.
The integral of expression (12.5), taken over the entire space from to , is equal to one:
Indeed, this integral gives the probability that the particle is located at one of the points in space, i.e., the probability of a certain event, which is equal to 1.
In quantum mechanics, it is assumed that the wave function can be multiplied by an arbitrary nonzero complex number WITH, and WITH Y describe the same state of the particle. This allows one to choose the wave function so that it satisfies the condition
Condition (12.6) is called the normalization condition. Functions satisfying this condition are called normalized. In what follows, we will always assume that the Y-functions we are considering are normalized. In the case of a stationary force field, the relation
i.e., the probability density of the wave function is equal to the probability density of the coordinate part of the wave function and does not depend on time.
Properties Y -function: it must be single-valued, continuous and finite (with the possible exception of singular points) and have a continuous and finite derivative. The combination of these requirements is called standard conditions.
The Schrödinger equation includes as a parameter the total energy of the particle E. In the theory of differential equations, it is proved that equations of the form have solutions that satisfy standard conditions, not for any, but only for certain specific values of the parameter (i.e., the energy E). These values are called eigenvalues. Solutions corresponding to eigenvalues are called own functions. Finding eigenvalues and eigenfunctions is usually a very difficult task. math problem. Let us consider some of the simplest special cases.
Particle in a potential well
Let us find the energy eigenvalues and the corresponding wavefunctions for a particle located in an infinitely deep one-dimensional potential well (Fig. 13.1, A). Let's assume that the particle
can only move along the axis X. Let the motion be limited by walls impenetrable for the particle: X= 0 and X = l. Potential energy U= 0 inside the well (at 0 £ X £ l) and outside the well (at X < 0 и X > l).
Consider the stationary Schrödinger equation. Since the Y-function only depends on the coordinate X, then the equation has the form
The particle cannot fall outside the potential well. Therefore, the probability of detecting a particle outside the well is zero. Consequently, the function y outside the well is also equal to zero. It follows from the continuity condition that y must also be equal to zero at the boundaries of the well, i.e.
. (13.2)
Solutions of equation (13.1) must satisfy this condition.
In area II (0 £ X £ l), Where U= 0 equation (13.1) has the form
Using the notation 
, we arrive at the wave equation known from the theory of oscillations
.
The solution of such an equation has the form
Condition (14.2) can be satisfied by an appropriate choice of constants k and a. From the equality we get 
Þ a = 0.
(n = 1, 2, 3, ...), (13.4)
n= 0 is excluded, because in this case º 0, i.e., the probability of finding a particle in the well is zero.
From (13.4) we get 
(n= 1, 2, 3, ...), therefore,
(n = 1, 2, 3, ...).
Thus, we obtain that the energy of a particle in a potential well can take only discrete values. In Fig.13.1, b a diagram of the energy levels of a particle in a potential well is shown. This example implements general rule quantum mechanics: if the particle is localized in a limited region of space, then the spectrum of particle energy values is discrete; in the absence of localization, the energy spectrum is continuous.
Substitute the values k from condition (13.4) in (13.3) and obtain
To find a constant A Let us use the normalization condition, which in this case has the form
.
At the ends of the integration interval, the integrand vanishes. Therefore, the value of the integral can be obtained by multiplying the average value (which is known to be equal to 1/2) by the length of the gap. Thus, we get . Finally, the eigenfunctions have the form
(n = 1, 2, 3, ...).
Graphs of the eigenvalues of functions for various n shown in fig. 13.2. The same figure shows the probability density yy * of detecting a particle at different distances from the walls of the well.
The graphs show that in the state with n= 2 the particle cannot be detected in the middle of the well and, at the same time, it occurs equally often in both the left and right half of the well. This behavior of a particle is incompatible with the idea of a trajectory. Note that, according to classical concepts, all positions of the particle in the well are equally probable.
Free particle motion
Consider the motion of a free particle. total energy E moving particle is kinetic energy(potential energy U= 0). The Schrödinger equation for the stationary state (12.3) has in this case the solution
defines the behavior of a free particle. Thus, a free particle in quantum mechanics is described by a plane monochromatic de Broglie wave with a wave number
.
The probability of detecting a particle at any point in space is found as
,
i.e. the probability of finding a particle along the x-axis is constant everywhere.
Thus, if the momentum of a particle has a certain value, then it, in accordance with the uncertainty principle, can be at any point in space with equal probability. In other words, if the momentum of a particle is exactly known, we know nothing about its location.
In the process of measuring the coordinate, the particle is localized by the measuring device, so the domain of definition of the wave function (17.1) for a free particle is limited to the segment X. A plane wave can no longer be considered monochromatic, having one specific value of the wavelength (momentum).
Harmonic oscillator
In conclusion, consider the problem of oscillations quantum harmonic oscillator. Such an oscillator are particles that make small oscillations around the equilibrium position.
On fig. 18.1, A pictured classical harmonic oscillator in the form of a ball of mass m suspended on a spring with a stiffness coefficient k. The force acting on the ball and responsible for its oscillations is related to the coordinate X formula . The potential energy of the ball is
.
If the ball is taken out of equilibrium, then it oscillates with a frequency. The dependence of potential energy on the coordinate X shown in fig. 18.1, b.
The Schrödinger equation for a harmonic oscillator has the form
The solution of this equation leads to the quantization of the oscillator energy. The oscillator energy eigenvalues are determined by the expression
As in the case of a potential well with infinitely high walls, the minimum energy of the oscillator is nonzero. The lowest possible energy value at n= 0 is called zero point energy. For a classical harmonic oscillator at a point with coordinate x= 0 the energy is zero. The existence of zero-point energy is confirmed by experiments on the study of light scattering by crystals at low temperatures. The particle energy spectrum turns out to be equidistant, i.e., the distance between the energy levels is equal to the energy of oscillations of the classical oscillator is the turning point of the particle during oscillations, i.e. 
.
The graph of the "classical" probability density is shown in Fig. 18.3 dotted curve. It can be seen that, as in the case of a potential well, the behavior of a quantum oscillator differs significantly from that of a classical one.
The probability for a classical oscillator is always maximum near the turning points, while for a quantum oscillator the probability is maximum at the antinodes of the Y-functions eigenfunctions. In addition, the quantum probability turns out to be non-zero even beyond the turning points that limit the motion of the classical oscillator.
On the example of a quantum oscillator, the previously mentioned correspondence principle is again traced. On fig. 18.3 shows graphs for classical and quantum probability densities for a large quantum number n. It is clearly seen that the averaging of the quantum curve leads to the classical result.
Content
THERMAL RADIATION. QUANTUM OPTICS
1. Thermal radiation ............................................................... ............................................... 3
2. Kirchhoff's law. Absolutely black body .............................................. 4
3. Stefan-Boltzmann law and Wien's law. Rayleigh-Jeans formula. 6
4. Planck's formula.............................................. ....................................... 8
5. The phenomenon of the external photoelectric effect ............................................... ............... 10
6. Bothe's experience. Photons................................................. .............................. 12
7. Vavilov-Cherenkov radiation .............................................. ............ 14
8. Compton effect.................................................... .................................... 17
MAIN PROPOSITIONS OF QUANTUM MECHANICS
9. De Broglie's hypothesis. Experience of Davisson and Germer .............................. 19
10. Probabilistic nature of de Broglie waves. Wave function ......... 21
11. Uncertainty principle .......................................................... ................. 24
12. Schrödinger equation............................................... ......................... 26
Light- electromagnetic radiation with wave and quantum properties.
Quantum- particle (corpuscle).
Wave properties.
Light is a transverse electromagnetic wave ().
, E 0 ,H 0 - amplitude values, 
- circle. Cycle. frequency, 
- frequency. Fig.1.
V - speed Distribution waves in a given medium. V=C/n, where C is the speed of light (in vacuum C=3*10 8 m/s), n is the refractive index of the medium (depends on the properties of the medium).
,
- the dielectric constant, 
- magnetic permeability.
is the phase of the wave.
The sensation of light is due to the electromagnetic component of the wave ( 
).
- wavelength, equal to the path traveled by the wave for the period ( 
;
).
Visible range: 
=0,4
0.75 µm.
;
4000 - short (purple); 7500 - long (red).
Quantum properties of light.
From point of view quantum theory light is emitted, propagated and absorbed by separate portions - quanta.
Photon characteristics.
1. Mass.
; m 0 - rest mass.
If m0 
0 (photon), then because V=C, m= 
- nonsense, therefore m 0 =0 - a moving photon. Therefore, the light cannot be stopped.
Therefore, the photon mass must be calculated from relativistic formula for energy. E=mC 2 , m=E/C 2 .
2. Photon energy.E=mC 2 .
In 1900, Max Planck, a German physicist, derived the following formula for the energy of a photon: 
.
h=6.62*10 -34 J*sis Planck's constant.
3. Impulse.
p=mV=mC=mC 2 /C=E/C=h/ 
; p-characteristic of the particle, 
is the characteristic of the wave.
Wave optics. Interference - redistribution. Light in space.
The superposition of light waves, as a result of which in some places of space there is an increase in the intensity of light, and in others - attenuation. That is, there is a redistribution of light intensity in space.
The condition for observing interference is the coherence of light waves (waves that satisfy the condition: -monochromatic waves; 
- the phase of the wave is constant at a given point in space over time).
CALCULATION OF INTERFERENCE PATTERNS.
Sources are coherent waves. 
; * - point. source.
Dark and light band. 
1.
If l ~ d, then 
the picture is indistinguishable, therefore, in order to see something, it is necessary 2.
l<
.
At point M, two coherent waves are superimposed.
, d1,d2 - meters passed by the waves; 
- phase difference.
Darker / lighter - intensity. 
(proportional). 
If the waves are not coherent: 
(average value for the period).
(superposition, overlay).
If are coherent: 
;
;
-there is an interference of light (redistribution of light).
; If 
(optical difference in the course of waves); n-refractive index; (d2-d1) - geometrical difference in the course of the waves; 
-wavelength (the path that the wave travels in a period).
is the basic formula of interference.
Depending on the path 
, they come with different 
. Ires depends on the latter.
1.
Ires.
max.
This condition maximum interference of light, because in this case the waves come in the same phase and therefore reinforce each other.
n-multiplicity factor; 
- means that the interference pattern is symmetrical about the center of the screen.
If the phases coincide, then the amplitudes do not depend on the phases.
-
Also the maximum condition.
2
.
Ires.
min.
; k=0,1,2…; 
.
- This condition minimum, because the waves arrive in antiphase and cancel each other out.
Methods for obtaining coherent waves.
The principle of receiving.
To obtain coherent waves, it is necessary to take one source and divide the light wave coming from it into two parts, which are then forced to meet. These waves will be coherent, because will belong to the same moment of radiation, therefore. .
Phenomena used to split a light wave in two.
1. Phenomenon light reflections(Fresnel bimirror). Fig.4.
2 . Phenomenon refraction of light(Fresnel biprism). Fig.5.
3 . Phenomenon light diffraction.
This is the deviation of light from rectilinear propagation when light passes through small holes or near opaque obstacles, if their dimensions (both) d are commensurate with the wavelength 
(d~ 
). Then: Fig.6. - Young's installation.
In all these cases, the real light source was a point. In real life, light can be extended - a section of the sky.
4.
, n is the refractive index of the film.
Two cases are possible:
H=const, then 
. In this case, the interference pattern is called a fringe of equal slope.
H 
const. A parallel beam of rays falls. 
.
stripes of equal thickness.
Installation of "Newton's ring".
It is necessary to consider the interference pattern in reflected and refracted light.
Definition 1
Quantum optics is a branch of optics whose main task is to study phenomena in which the quantum properties of light can manifest themselves.
These events can 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 that it touches upon 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 in special (specific conditions).
A more accurate solution of such problems requires the description of both matter (including the propagation medium) and light solely from the standpoint of the existence of quanta. At the same time, when describing, scientists often simplify the task 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 turns out to be 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 particles-corpuscles). According to the hypothesis of M. Planck on 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 $m_f$, energy $E$ and momentum $p_f$. Then the formula is written:
Where $h$ represents Planck's constant.
$v=\frac(c)(\lambda)$
Where $\lambda$ is the frequency of the light
$c$ will be the speed of light in vacuum.
The main optical phenomena explained by quantum theory are the pressure of light and the photoelectric effect.
Photoelectric effect and light pressure in quantum optics
Definition 2
The photoelectric effect is such a phenomenon of interactions between photons of light and matter, in which the radiation energy will be transferred to the electrons of matter. There are such varieties of the 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 is carried out independently of the others.
An increase in the radiation intensity is accompanied by an increase in the number of incident and absorbed photons. When the energy is absorbed by the matter of frequency $ν$, each of the electrons is able to absorb 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 (expression of the law of conservation of energy):
$hv=A_(out)+\frac(mv^2)(2)$
$A_(out)$ is the work function of an electron from the metal.
The kinetic energy of the emitted electron is obtained by the formula:
$E_k=\frac(mv^2)(2)$
From the Einstein equation it turns out that if $E_k=0$, then it is possible to obtain the very minimum frequency (the red border 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, like particles, photons have a certain momentum, which they transfer to the body in the process of absorption and reflection:
Such a phenomenon as light pressure also explains the wave theory, according to which (if we refer to the de Broglie hypothesis), any particle also has wave properties. The relationship between momentum $P$ and wavelength $\lambda$ shows the equation:
$P=\frac(h)(\lambda)$
Compton effect
Remark 1
The Compton effect is characterized by incoherent scattering of photons by free electrons. The very concept of incoherence means non-interference of photons before and after scattering. The effect changes the frequency of photons, while after scattering, the electrons receive part of the energy.
The Compton effect is an experimental proof 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 an important proof of the quantum concepts of light. At the same time, such phenomena as diffraction, interference, polarization of light serve as confirmation of the wave nature of light.
The Compton effect is one of the proofs of the corpuscular-wave dualism 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 a higher energy than the photon. In this interaction, energy is transferred to the photon from the electron. The energy of scattered photons is determined by the expression: 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.
