Energy states of a quantum system. Populations of quantum levels

The atomic nucleus, like other objects of the microworld, is a quantum system. This means that a theoretical description of its characteristics requires the involvement of quantum theory. In quantum theory, the description of the states of physical systems is based on wave functions, or amplitudes of probabilityψ (α, t). The square of the modulus of this function determines the probability density of detecting the system under study in a state with the characteristic α - ρ (α, t) = | ψ (α, t) | 2. The argument of the wave function can be, for example, the coordinates of a particle.
It is customary to normalize the total probability to one:

Each physical quantity is associated with a linear Hermitian operator acting in the Hilbert space of wave functions ψ. The spectrum of values ​​that a physical quantity can take is determined by the spectrum of the eigenvalues ​​of its operator.
The average value of the physical quantity in the state ψ is

() * = <ψ ||ψ > * = <ψ | + |ψ > = <ψ ||ψ > = .

The states of the nucleus as a quantum system, i.e. functions ψ (t) , obey the Schrödinger equation ("W. Sh.")

(2.4)

Operator - Hermitian Hamilton operator ( Hamiltonian) system. Together with the initial condition on ψ (t), Eq. (2.4) determines the state of the system at any moment of time. If it does not depend on time, then the total energy of the system is an integral of motion. States in which the total energy of the system has a certain value are called stationary. Stationary states are described by the eigenfunctions of the operator (Hamiltonian):

The last of the equations is stationary Schrödinger equation, which determines, in particular, the set (spectrum) of energies of the stationary system.
In stationary states of a quantum system, in addition to energy, other physical quantities can be conserved. The condition for the conservation of the physical quantity F is the equality 0 of the commutator of its operator with the Hamilton operator:

1. Spectra of atomic nuclei

The quantum nature of atomic nuclei is manifested in the pictures of their excitation spectra (see, for example, Fig. 2.1). Spectrum in the range of excitation energies of the 12 C nucleus below (approximately) 16 MeV It has discrete character. Above this energy, the spectrum is continuous. The discrete nature of the excitation spectrum does not mean that the widths of the levels in this spectrum are equal to 0. Since each of the excited levels of the spectrum has a finite average lifetime τ, the width of the level Г is also finite and is related to the average lifetime by a relation that is a consequence of the uncertainty relation for energy and time Δ t ΔE ≥ ћ :

Nuclear spectra diagrams indicate the energies of the nuclear levels in MeV or keV, as well as the spin and parity of states. The diagrams also indicate, if possible, the isospin of the state (since the diagrams of the spectra give excitation energies of levels, the energy of the ground state is taken as the origin). In the excitation energy range E< E отд - т.е. при энергиях, меньших, чем энергия отделения нуклона, спектры ядер - discrete... It means that the width of the spectral levels is less than the distance between the levels G< Δ E.

Quantum systems and their properties.

Distribution of probabilities over energies in space.

Boson statistics. Fermi-Einstein distribution.

Fermion statistics. Fermi-Dirac distribution.

Quantum systems and their properties

In classical statistics, it is assumed that the particles that make up the system obey the laws of classical mechanics. But for many phenomena, when describing microobjects, it is necessary to use quantum mechanics. If a system consists of particles obeying quantum mechanics, then we will call it a quantum system.

The fundamental differences between a classical system and a quantum one include:

1) Corpuscular-wave dualism of microparticles.

2) Discreteness of physical quantities describing micro-objects.

3) Spin properties of microparticles.

From the first it follows that it is impossible to accurately determine all the parameters of the system that determine its state from the classical point of view. This fact is reflected in the Heisendberg uncertainty relation:

In order to mathematically describe these features of micro-objects in quantum physics, the quantity is associated with a linear Hermitian operator that acts on the wave function.

The eigenvalues ​​of the operator determine the possible numerical values ​​of this physical quantity, the average over which coincides with the value of the quantity itself.

Since the impulses and coefficients of the microparticles of the system cannot be measured simultaneously, the wave function is represented either as a function of coordinates:

Or, as a function of impulses:

The square of the modulus of the wave function determines the probability of detecting a microparticle per unit volume:

The wave function describing a specific system is found as an eigenfunction of the Hamelton operator:

Stationary Schrödinger equation.

Nonstationary Schrödinger Equation.

The principle of indistinguishability of microparticles operates in the microcosm.

If the wave function satisfies the Schrödinger equation, then the function also satisfies this equation. The state of the system will not change when the 2 particles are swapped.

Let the first particle be in state a, and the second in state b.

The state of the system is described:

If the particles are interchanged, then: since the movement of the particle should not affect the behavior of the system.

This equation has 2 solutions:

It turned out that the first function is realized for particles with integer spin, and the second with half-integer.

In the first case, 2 particles can be in the same state:

In the second case:

Particles of the first type are called bosons (spin is whole), particles of the second type are called femions (the Pauli principle is valid for them.)

Fermions: electrons, protons, neutrons ...

Bosons: photons, deuterons ...

Fermions and bosons obey non-classical statistics. To see the differences, let's count the number of possible states of a system consisting of two particles with the same energy in two cells in phase space.

1) Classical particles are different. It is possible to trace each particle separately.

Classic particles.

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Dimensional quantization principle The whole complex of phenomena, usually understood by the words "electronic properties of low-dimensional electronic systems" is based on a fundamental physical fact: a change in the energy spectrum of electrons and holes in structures with very small dimensions. Let us demonstrate the basic idea of ​​dimensional quantization using the example of electrons in a very thin metal or semiconductor film of thickness a.

ELECTRONIC PROPERTIES OF LOW-SIZE ELECTRONIC SYSTEMS Dimensional quantization principle Electrons in a film are in a potential well with a depth equal to the work function. The depth of the potential well can be considered infinitely large, since the work function is several orders of magnitude higher than the thermal energy of the carriers. Typical values ​​of the work function in most solids are W = 4 -5 Oe. B, which is several orders of magnitude higher than the characteristic thermal energy of carriers, and has an order of magnitude of k. T, equal to 0.026 Oe at room temperature. C. According to the laws of quantum mechanics, the energy of electrons in such a well is quantized, that is, it can take only some discrete values ​​of En, where n can take integer values ​​1, 2, 3,…. These discrete energy values ​​are called dimensional quantization levels.

ELECTRONIC PROPERTIES OF LOW-SIZE ELECTRONIC SYSTEMS Dimensional quantization principle For a free particle with an effective mass m *, the motion of which in the crystal in the direction of the z axis is limited by impenetrable barriers (i.e., barriers with infinite potential energy), the energy of the ground state in comparison with the state without restriction increases by the amount This increase in energy is called the dimensional quantization energy of the particle. Dimensional quantization energy is a consequence of the uncertainty principle in quantum mechanics. If a particle is limited in space along the z-axis within the distance a, the uncertainty of the z-component of its momentum increases by an amount of the order of ħ / a. Correspondingly, the kinetic energy of the particle increases by an amount E 1. Therefore, the considered effect is often called the quantum size effect.

ELECTRONIC PROPERTIES OF LOW-SIZE ELECTRONIC SYSTEMS Dimensional quantization principle The conclusion about the quantization of the energy of electron motion refers only to the motion across the potential well (along the z-axis). The motion in the xy plane (parallel to the film boundaries) is not affected by the potential of the well. In this plane, the carriers move as free and are characterized, as in a massive sample, by a continuous energy spectrum quadratic in momentum with an effective mass. The total energy of carriers in a quantum-well film has a mixed discrete continuous spectrum

ELECTRONIC PROPERTIES OF LOW-SIZE ELECTRONIC SYSTEMS Dimensional quantization principle In addition to increasing the minimum energy of a particle, the quantum-size effect also leads to quantization of the energies of its excited states. Energy spectrum of a quantum-well film - momentum of charge carriers in the plane of the film

ELECTRONIC PROPERTIES OF LOW-SIZE ELECTRONIC SYSTEMS Dimensional quantization principle Let the electrons in the system have energies less than E 2, and therefore belong to the lower level of dimensional quantization. Then no elastic process (for example, scattering by impurities or acoustic phonons), as well as scattering of electrons by each other, can change the quantum number n, transferring the electron to an overlying level, since this would require additional energy consumption. This means that in elastic scattering electrons can only change their momentum in the plane of the film, i.e., they behave like purely two-dimensional particles. Therefore, quantum-dimensional structures in which only one quantum level is filled are often called two-dimensional electronic structures.

ELECTRONIC PROPERTIES OF LOW-SIZE ELECTRONIC SYSTEMS Dimensional quantization principle There are other possible quantum structures, where the movement of carriers is limited not in one, but in two directions, as in a microscopic wire or filament (quantum filaments or wires). In this case, carriers can move freely only in one direction, along the thread (let's call it the x-axis). In the cross section (yz plane), the energy is quantized and takes discrete values ​​Emn (like any two-dimensional motion, it is described by two quantum numbers, m and n). In this case, the full spectrum is also discrete-continuous, but with only one continuous degree of freedom:

ELECTRONIC PROPERTIES OF LOW-SIZE ELECTRONIC SYSTEMS Dimensional quantization principle It is also possible to create quantum structures resembling artificial atoms, where the movement of carriers is limited in all three directions (quantum dots). In quantum dots, the energy spectrum no longer contains a continuous component, i.e., it does not consist of subbands, but is purely discrete. As in the atom, it is described by three discrete quantum numbers (not counting the spin) and can be written as E = Elmn, and, as in the atom, the energy levels can be degenerate and depend on only one or two numbers. A common feature of low-dimensional structures is the fact that if at least along one direction the movement of carriers is limited to a very small region comparable in size to the de Broglie wavelength of carriers, their energy spectrum changes noticeably and becomes partially or completely discrete.

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Definitions Quantum dots - structures in which in all three directions the dimensions are several interatomic distances (zero-dimensional structures). Quantum wires (filaments) - quantum wires - structures in which in two directions the sizes are equal to several interatomic distances, and in the third - a macroscopic value (one-dimensional structures). Quantum wells - quantum wells - structures in which in one direction the size is several interatomic distances (two-dimensional structures).

ELECTRONIC PROPERTIES OF LOW-SIZE ELECTRONIC SYSTEMS Minimum and maximum sizes The lower limit of dimensional quantization is determined by the critical size Dmin, at which at least one electronic level exists in a quantum-dimensional structure. Dmin depends on the gap of the conduction band DEc in the corresponding heterojunction used to obtain quantum-well structures. In a quantum well, at least one electronic level exists if DEc exceeds the value of h - Planck's constant, me * is the effective mass of an electron, DE 1 QW is the first level in a rectangular quantum well with infinite walls.

ELECTRONIC PROPERTIES OF LOW-SIZE ELECTRONIC SYSTEMS Minimum and maximum dimensions If the distance between energy levels becomes comparable to thermal energy k. BT, then the population of high levels increases. For a quantum dot, the condition under which the population of higher-lying levels can be neglected is written as E 1 QD, E 2 QD are the energies of the first and second dimensional quantization levels, respectively. This means that the benefits of dimensional quantization can be fully realized if this condition sets upper limits for the dimensional quantization. For Ga. As –Alx. Ga 1 -x. As this value is 12 nm.

ELECTRONIC PROPERTIES OF LOW-SIZE ELECTRONIC SYSTEMS Distribution of quantum states in structures of reduced dimension An important characteristic of any electronic system, along with its energy spectrum, is the density of states g (E) (the number of states per unit energy interval E). For three-dimensional crystals, the density of states is determined using the cyclic Born-Karman boundary conditions, from which it follows that the components of the electron wave vector do not change continuously, but take a number of discrete values ​​here ni = 0, ± 1, ± 2, ± 3, and are the dimensions crystal (in the form of a cube with side L). The volume of k-space per one quantum state is (2) 3 / V, where V = L 3 is the volume of the crystal.

ELECTRONIC PROPERTIES OF LOW-SIZE ELECTRONIC SYSTEMSDistribution of quantum states in structures of reduced dimension The number of states per unit volume in reciprocal space, i.e., the density of states) does not depend on the wave vector. In other words, in reciprocal space, allowed states are distributed with constant density.

ELECTRONIC PROPERTIES OF LOW-SIZE ELECTRONIC SYSTEMS Distribution of quantum states in structures of reduced dimension In the general case, it is practically impossible to calculate the energy density function of states, since isoenergy surfaces can have a rather complex shape. In the simplest case of an isotropic parabolic dispersion law, which is valid for the edges of the energy bands, one can find the number of quantum states per volume of a spherical layer enclosed between two close isoenergy surfaces corresponding to energies E and E + d. E.

ELECTRONIC PROPERTIES OF LOW-SIZE ELECTRONIC SYSTEMS Distribution of quantum states in structures of reduced dimension Volume of a spherical layer in k-space. dk - layer thickness. This volume will account for d. N states Taking into account the relationship between E and k according to the parabolic law, we obtain From this, the energy density of states will be equal to m * - the effective mass of the electron

ELECTRONIC PROPERTIES OF LOW-SIZE ELECTRONIC SYSTEMSDistribution of quantum states in structures of reduced dimension Thus, in three-dimensional crystals with a parabolic energy spectrum with increasing energy, the density of allowed energy levels (density of states) will increase in proportion to the density of levels in the conduction band and in the valence band. The area of ​​the shaded areas is proportional to the number of levels in the energy range d. E

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in structures of reduced dimension Let us calculate the density of states for a two-dimensional system. The total carrier energy for an isotropic parabolic dispersion law in a quantum-well film, as shown above, has a mixed discrete continuous spectrum.In a two-dimensional system, the states of a conduction electron are determined by three numbers (n, kx, ky). The energy spectrum is divided into separate two-dimensional subzones En, corresponding to fixed values ​​of n.

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in structures of reduced dimension Curves of constant energy are circles in reciprocal space. Each discrete quantum number n corresponds to the absolute value of the z-component of the wave vector. Therefore, the volume in reciprocal space bounded by a closed surface of a given energy E in the case of a two-dimensional system is divided into a number of sections.

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in structures of reduced dimension Let us determine the dependence of the density of states on energy for a two-dimensional system. For this, for a given n, we find the area S of the ring bounded by two isoenergy surfaces corresponding to the energies E and E + d. E: Here The magnitude of the two-dimensional wave vector corresponding to the given n and E; dkr is the width of the ring. Since one state in the plane (kxky) corresponds to the area where L 2 is the area of ​​a two-dimensional film of thickness a, the number of electronic states in the ring, calculated per unit volume of the crystal, will be, taking into account the electron spin,

ELECTRONIC PROPERTIES OF LOW-SIZE ELECTRONIC SYSTEMS Distribution of quantum states in structures of reduced dimension Since here is the energy corresponding to the bottom of the n-th subband. Thus, the density of states in a two-dimensional film is where Q (Y) is the unit Heaviside function, Q (Y) = 1 for Y ≥ 0 and Q (Y) = 0 for Y

ELECTRONIC PROPERTIES OF LOW-SIZE ELECTRONIC SYSTEMS Distribution of quantum states in structures of reduced dimension The density of states in a two-dimensional film can also be represented as an integer part equal to the number of subbands whose bottom is below the energy E. Thus, for two-dimensional films with a parabolic dispersion law, the density of states in any subzone is constant and does not depend on energy. Each subband makes the same contribution to the total density of states. For a fixed film thickness, the density of states changes abruptly when it does not change by unity.

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in structures of reduced dimension Dependence of the density of states of a two-dimensional film on energy (a) and thickness a (b).

ELECTRONIC PROPERTIES OF LOW-SIZE ELECTRONIC SYSTEMSDistribution of quantum states in structures of reduced dimension

ELECTRONIC PROPERTIES OF LOW-SIZE ELECTRONIC SYSTEMS Distribution of quantum states in structures of reduced dimension Let us calculate the density of states for a one-dimensional structure - a quantum wire. In this case, the isotropic parabolic dispersion law can be written as x directed along the quantum wire, d is the thickness of the quantum wire along the y and z axes, kx is a one-dimensional wave vector. m, n are positive integers characterizing where the axis is the quantum subbands. The energy spectrum of a quantum wire is thus split into separate overlapping one-dimensional subbands (parabolas). The motion of electrons along the x axis turns out to be free (but with an effective mass), and along the other two axes, motion is limited.

ELECTRONIC PROPERTIES OF LOW-SIZE ELECTRONIC SYSTEMS Distribution of quantum states in structures of reduced dimension Energy spectrum of electrons for a quantum wire

ELECTRONIC PROPERTIES OF LOW-SIZE ELECTRONIC SYSTEMS Distribution of quantum states in structures of reduced dimension Density of states in a quantum wire from energy The number of quantum states per interval dkx, calculated per unit volume where is the energy corresponding to the bottom of the subband with given n and m.

ELECTRONIC PROPERTIES OF LOW-SIZE ELECTRONIC SYSTEMS Distribution of quantum states in structures of reduced dimension Density of states in a quantum wire from energy Thus, when deriving this formula, the spin degeneracy of states and the fact that one interval d is taken into account. E corresponds to two intervals ± dkx of each subband, for which (E-En, m)> 0. The energy E is measured from the bottom of the conduction band of the bulk sample.

ELECTRONIC PROPERTIES OF LOW-SIZE ELECTRONIC SYSTEMS Distribution of quantum states in structures of reduced dimension Density of states in a quantum wire on energy Dependence of the density of states of a quantum wire on energy. The numbers on the curves show the quantum numbers n and m. Degeneracy factors for the levels of the subbands are indicated in parentheses.

ELECTRONIC PROPERTIES OF LOW-SIZE ELECTRONIC SYSTEMS Distribution of quantum states in structures of reduced dimension Density of states in a quantum wire from energy Within a separate subband, the density of states decreases with increasing energy. The total density of states is a superposition of identical decreasing functions (corresponding to individual subbands) shifted along the energy axis. For E = E m, n, the density of states is equal to infinity. Subbands with quantum numbers n m turn out to be doubly degenerate (only for Ly = Lz d).

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in structures of reduced dimension Density of states in a quantum dot from energy With a three-dimensional restriction of particle motion, we arrive at the problem of finding allowed states in a quantum dot or a zero-dimensional system. Using the effective mass approximation and the parabolic dispersion law, for the edge of the isotropic energy band, the spectrum of allowed states of a quantum dot with the same size d along all three coordinate axes will have the form n, m, l = 1, 2, 3 ... - positive numbers numbering subbands. The energy spectrum of a quantum dot is a set of discrete allowed states corresponding to fixed n, m, l.

ELECTRONIC PROPERTIES OF LOW-SIZE ELECTRONIC SYSTEMS Distribution of quantum states in structures of reduced dimension Density of states in a quantum dot from energy The number of states in subbands corresponding to one set of n, m, l, calculated per unit volume, The total number of states having the same energy, calculated per unit volume Degeneration of levels is primarily determined by the symmetry of the problem. g - level degeneracy factor

ELECTRONIC PROPERTIES OF LOW-SIZE ELECTRONIC SYSTEMS Distribution of quantum states in structures of reduced dimension Density of states in a quantum dot from energy The degeneracy of levels is primarily determined by the symmetry of the problem. For example, for the considered case of a quantum dot with the same dimensions in all three dimensions, the levels will be threefold degenerate if two quantum numbers are equal to each other and not equal to the third, and sixfold degenerate if all quantum numbers are not equal to each other. The specific form of the potential can also lead to an additional, so-called random degeneracy. For example, for the considered quantum dot, to the threefold degeneracy of the levels E (5, 1, 1); E (1, 5, 1); E (1, 1, 5), associated with the symmetry of the problem, a random degeneration E (3, 3, 3) (n 2 + m 2 + l 2 = 27 in both the first and second cases) is added, associated with the form limiting potential (infinite rectangular potential well).

ELECTRONIC PROPERTIES OF LOW-SIZE SYSTEMS Distribution of quantum states in structures of reduced dimension Density of states in a quantum dot versus energy Distribution of the number of allowed states N in the conduction band for a quantum dot with the same dimensions in all three dimensions. The numbers represent quantum numbers; the level degeneracy factors are indicated in parentheses.

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL SYSTEMS Carrier statistics in low-dimensional structures Three-dimensional electron systems The properties of equilibrium electrons in semiconductors depend on the Fermi distribution function, which determines the probability that an electron will be in a quantum state with energy E EF - Fermi level or electrochemical potential, T - absolute temperature , k is the Boltzmann constant. The calculation of various statistical quantities is greatly simplified if the Fermi level lies in the energy gap and is significantly removed from the bottom of the conduction band Ec (Ec - EF)> k. T. Then, in the Fermi-Dirac distribution, the unit in the denominator can be neglected and it goes over into the Maxwell-Boltzmann distribution of classical statistics. This is the case of a non-degenerate semiconductor

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL SYSTEMS Carrier statistics in low-dimensional structures Three-dimensional electronic systems The distribution function of the density of states in the conduction band g (E), the Fermi-Dirac function for three temperatures, and the Maxwell-Boltzmann function for a three-dimensional electron gas. At T = 0, the Fermi-Dirac function has the form of a discontinuous function. For E EF, the function is equal to zero and the corresponding quantum states are completely free. At T> 0, the Fermi function. Dirac is smeared out in the vicinity of the Fermi energy, where it rapidly changes from 1 to 0, and this smearing is proportional to k. T, i.e., the higher the higher the temperature. (Fig. 1. 4. Gurtov)

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL SYSTEMS Carrier statistics in low-dimensional structures Three-dimensional electronic systems The concentration of electrons in the conduction band is found by summing over all states. But since the Fermi-Dirac function for energies E> EF exponentially decreases rapidly with increasing energy, then replacing the upper limit with infinity does not change the value of the integral. Substituting the values ​​of the functions into the integral, we obtain the effective density of states in the conduction band

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL SYSTEMS Carrier statistics in low-dimensional structures Two-dimensional electronic systems Let us determine the concentration of a charge carrier in a two-dimensional electron gas. Since the density of states of a two-dimensional electron gas We obtain Here also the upper integration limit is taken equal to infinity, taking into account the sharp dependence of the Fermi-Dirac distribution function on energy. By integrating where

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL SYSTEMS Carrier statistics in low-dimensional structures Two-dimensional electron systems For a nondegenerate electron gas, when In the case of ultrathin films, when only the lower subband filling can be taken into account When the electron gas is strongly degenerate, when where n 0 is an integer part

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL SYSTEMS Carrier statistics in low-dimensional structures It should be noted that in quantum-dimensional systems, due to the lower density of states, the condition of complete degeneracy does not require extremely high concentrations or low temperatures and is quite often realized in experiments. For example, in n-Ga. As at N 2 D = 1012 cm-2, the degeneracy will take place already at room temperature. In quantum wires, the integral for calculation, in contrast to the two-dimensional and three-dimensional cases, is not calculated analytically for an arbitrary degeneracy, and simple formulas can be written only in limiting cases. In a nondegenerate one-dimensional electron gas in the case of hyperfine filaments, when it is possible to take into account the filling of only the lowest level with energy E 11, the concentration of electrons is where the one-dimensional effective density of states is

Energy levels (atomic, molecular, nuclear)

1. Characteristics of the state of a quantum system
2. Energy levels of atoms
3. Energy levels of molecules
4. Energy levels of nuclei

Characteristics of the state of a quantum system

At the heart of the explanation of the sv-in atoms, molecules and atomic nuclei, i.e. phenomena occurring in volume elements with linear scales of 10 -6 -10 -13 cm, is quantum mechanics. According to quantum mechanics, any quantum system (i.e. a system of microparticles, which obeys quantum laws) is characterized by a certain set of states. In the general case, this set of states can be either discrete (discrete spectrum of states) or continuous (continuous spectrum of states). Characteristics of the state of an isolated system yavl. internal energy of the system (hereinafter, just energy), total angular momentum (MCM) and parity.

System energy.
A quantum system, being in different states, has, generally speaking, different energies. The energy of a bound system can take on any values. This set of possible energy values ​​is called. discrete energy spectrum, and energy is said to be quantized. An example is energetic. spectrum of an atom (see below). An unbound system of interacting particles has a continuous energy spectrum, and the energy can take arbitrary values. An example of such a system is. free electron (E) in the Coulomb field of an atomic nucleus. A continuous energy spectrum can be represented as a set of an infinitely large number of discrete states, between which energetic. the gaps are infinitely small.

The state, which corresponds to the smallest energy possible for a given system, is called. main: all other states are called. excited. It is often convenient to use a conventional scale of energy, in which the energy is basic. state is considered to be the origin, i.e. is assumed to be zero (in this conventional scale, in what follows, the energy is denoted by the letter E). If the system, being in a state n(moreover, the index n= 1 is assigned to the main. state), has energy E n, then they say that the system is at the energy level E n... Number n, numbering U.E., called. quantum number. In general, each U.e. can be characterized not by one quantum number, but by their totality; then the index n means the collection of these quantum numbers.

If states n 1, n 2, n 3,..., n k corresponds to the same energy, i.e. one U.e., then this level is called degenerate, and the number k- the multiplicity of degeneracy.

For any transformations of a closed system (as well as a system in a constant external field), its total energy energy remains unchanged. Therefore, energy belongs to the so-called. conserved values. The law of conservation of energy follows from the homogeneity of time.

Full moment of momentum.
This value is vector and is obtained by adding the MCD of all particles included in the system. Each particle has as its own. MCD - spin and orbital angular momentum due to the motion of a particle relative to the general center of mass of the system. Quantization of the ICD leads to the fact that its abs. magnitude J takes strictly defined values:, where j- a quantum number, which can take non-negative integer and half-integer values ​​(the quantum number of an orbital MCD is always an integer). Projection of the MKD on the K.-L. the axis is called. magn. quantum number and can take 2j + 1 values: m j = j, j-1,...,-j... If K.-L. moment J yavl. the sum of two other moments, then, according to the rules of addition of moments in quantum mechanics, the quantum number j can take the following values: j=|j 1 -j 2 |, |j 1 -j 2 -1|, ...., |j 1 +j 2 -1|, j 1 +j 2, a. The summation of a larger number of moments is carried out in a similar way. It is accepted for brevity to talk about the MKD system j, implying the moment, abs. the value of which is; about magn. the quantum number is simply spoken of as a projection of the moment.

During various transformations of a system in a centrally symmetric field, the total MCD is conserved, i.e., like energy, it belongs to conserved quantities. The MCD conservation law follows from the isotropy of space. In an axially symmetric field, only the projection of the complete MCD onto the axis of symmetry is retained.

State parity.
In quantum mechanics, the states of a system are described by the so-called. wave functions. Parity characterizes the change in the wave function of the system during the operation of spatial inversion, i.e. change of signs of coordinates of all particles. With such an operation, the energy does not change, while the wave function can either remain unchanged (even state), or change its sign to the opposite (odd state). Parity P takes on two values, respectively. If the system operates nuclear or el.-magn. forces, parity is conserved in atomic, molecular and nuclear transformations, i.e. this quantity also refers to conserved quantities. Parity conservation law a consequence of the symmetry of space in relation to mirror reflections and is violated in those processes in which weak interactions are involved.

Quantum transitions
- system transitions from one quantum state to another. Such transitions can lead to both a change in energy. the state of the system, and to its qualities. changes. These are bound-bound, free-bound, free-free transitions (see Interaction of radiation with matter), for example, excitation, deactivation, ionization, dissociation, and recombination. It is also chemical. and nuclear reactions. Transitions can occur under the action of radiation — radiative (or radiative) transitions, or when a given system collides with a c.-l. by another system or particle - nonradiative transitions. An important characteristic of the quantum transition is yavl. its probability in units. time, showing how often the transition will occur. This value is measured in s -1. Radiation probabilities transitions between levels m and n (m> n) with the emission or absorption of a photon, the energy of which is equal, are determined by the coeff. Einstein A mn, B mn and B nm... Transition from level m to the level n can occur spontaneously. Photon emission probability B mn in this case is equal to A mn... Transitions of the type under the action of radiation (induced transitions) are characterized by the probabilities of photon emission and photon absorption, where is the radiation energy density with frequency.

The possibility of a quantum transition from a given U.e. on K.-L. another W.E. means that the characteristic cf. the time during which the system can be on this UE, of course. It is defined as the reciprocal of the total probability of decay of a given level, i.e. the sum of the probabilities of all possible transitions from the considered level to all others. For radiation. transitions the total probability is, and. The finiteness of time, according to the uncertainty relation, means that the energy of the level cannot be determined absolutely precisely, i.e. W.E. has a certain width. Therefore, the emission or absorption of photons during a quantum transition occurs not at a strictly defined frequency, but within a certain frequency interval lying in the vicinity of the value. The intensity distribution within this interval is specified by the spectral line profile, which determines the probability that the frequency of a photon emitted or absorbed during a given transition is:
(1)
where is the half-width of the line profile. If the broadening of the U.e. and spectral lines caused only by spontaneous transitions, then this broadening is called. natural. If collisions of the system with other particles play a certain role in the broadening, then the broadening has a combined character and the quantity must be replaced by a sum, where it is calculated similarly, but radiaz. transition probabilities must be replaced by collision probabilities.

Transitions in quantum systems obey certain selection rules, i.e. rules that establish how the quantum numbers characterizing the state of the system (MCD, parity, etc.) can change during the transition. The selection rules are most simply formulated for radiacs. transitions. In this case, they are determined by the st-you of the initial and final states, as well as the quantum characteristics of the emitted or absorbed photon, in particular, its MCD and parity. The most likely are the so-called. electrical dipole transitions. These transitions are carried out between levels of opposite parity, the total MCD of which differ by an amount (the transition is impossible). In the current terminology, these transitions are called. permitted. All other types of transitions (magnetic dipole, electric quadrupole, etc.) are called. prohibited. The meaning of this term consists only in the fact that their probabilities turn out to be much less than the probabilities of electric dipole transitions. However, they are not. absolutely forbidden.

In the first and second parts of the textbook, it was assumed that the particles that make up macroscopic systems obey the laws of classical mechanics. However, it turned out that to explain many of the properties of micro-objects, instead of classical mechanics, we must use quantum mechanics. The properties of particles (electrons, photons, etc.) in quantum mechanics are qualitatively different from the usual classical properties of particles. The quantum properties of micro-objects that make up a certain physical system are also manifested in the properties of a macroscopic system.

As such quantum systems, we will consider electrons in a metal, a photon gas, etc. In what follows, the word quantum system or particle will mean a certain material object described by the apparatus of quantum mechanics.

Quantum mechanics describes the properties and features inherent in the particles of the microworld, which we often cannot explain in terms of classical concepts. These features include, for example, the wave-particle dualism of micro-objects in quantum mechanics, discovered and confirmed by numerous experimental facts, the discreteness of various physical parameters, "spin" properties, etc.

The special properties of micro-objects do not allow one to describe their behavior by the usual methods of classical mechanics. For example, the presence of a microparticle manifesting at the same time and wave and corpuscular properties

does not allow one to simultaneously accurately measure all the parameters that determine the state of a particle from the classical point of view.

This fact is reflected in the so-called uncertainty relation, discovered in 1925 by Heisenberg, which consists in the fact that inaccuracies in determining the coordinate and momentum of a microparticle turn out to be related by the ratio:

The consequence of this relationship is a number of other relationships between various parameters and, in particular:

where the uncertainty in the value of the energy of the system and the uncertainty in time.

Both of the above relations show that if one of the quantities is determined with high accuracy, then the second quantity turns out to be determined with low accuracy. Inaccuracies here are determined through the Planck constant, which practically does not limit the accuracy of measurements of various quantities for macroscopic objects. But for microparticles with low energies, small sizes and momenta, the accuracy of the simultaneous measurement of the noted parameters is already insufficient.

Thus, the state of a microparticle in quantum mechanics cannot be simultaneously described using coordinates and momenta, as is done in classical mechanics (Hamilton's canonical equations). In the same way, one cannot speak about the value of the particle's energy at a given moment. States with a certain energy can be obtained only in stationary cases, that is, they are not determined exactly in time.

Possessing corpuscular-wave properties, any microparticle does not have an absolutely precisely defined coordinate, but turns out to be, as it were, "smeared" over space. In the presence of a certain area of ​​space of two or more particles, we cannot distinguish them from each other, since we cannot trace the movement of each of them. This implies the fundamental indistinguishability or identity of particles in quantum mechanics.

Further, it turns out that the quantities characterizing some parameters of microparticles can change only in certain portions, quanta, hence the name of quantum mechanics. This discreteness of many parameters that determine the state of microparticles also cannot be described in classical physics.

According to quantum mechanics, in addition to the energy of the system, discrete values ​​can take the angular momentum of the system or spin, the magnetic moment and their projections to any preferred direction. So, the square of the angular momentum can take only the following values:

Spin can only take values

where could it be

The projection of the magnetic moment onto the direction of the external field can take the values

where the Bohr magneton and the magnetic quantum number, which takes on the value:

In order to mathematically describe these features of physical quantities, it was necessary to associate a certain operator with each physical quantity. In quantum mechanics, thus, physical quantities are represented by operators, and their values ​​are determined as averages over the eigenvalues ​​of operators.

When describing the properties of micro-objects, it was necessary, in addition to the properties and parameters encountered in the classical description of microparticles, to introduce new, purely quantum parameters and properties. These include the "spin" of a particle, which characterizes the proper angular momentum, "exchange interaction", Pauli's principle, etc.

These features of microparticles do not allow them to be described using classical mechanics. As a result, micro-objects are described by quantum mechanics, which takes into account the noted features and properties of micro-particles.