Energy levels (atomic, molecular, nuclear)

1. Characteristics of the status of a quantum system
2. Energy levels of atoms
3. Energy levels of molecules
4. Energy levels of nuclear

Characteristics of the status of a quantum system

At the heart of the explanation of the SV-in atoms, molecules and atomic nuclei, i.e. The phenomena occurring in the elements of a linear scale of 10 -6 -10 -13 cm, lies quantum mechanics. According to quantum mechanics, any quantum system (that is, the microparticle system, K-paradium obeys quantum laws) is characterized by a certain set of states. In the general case, this set of states can be both a discrete (discrete spectrum of states) and continuous (continuous spectrum of states). Characteristics of the state of an isolated yawl system. The internal energy of the system (everywhere there is just energy), the full moment of the amount of movement (MKD) and parity.

Energy system.
The quantum system, being in various states, has, generally speaking, of various energy. The energy of the associated system can take any values. This set of possible energy values \u200b\u200bis called. Discrete energy splock, and about energy they say that it is quantized. An example is energy. Atom's spectrum (see below). An unbound system of interacting particles has a continuous energy spectrum, and energy can make arbitrary values. An example of such a yawl system. Free electron (E) in the Coulomb field of the atomic nucleus. The continuous energy spectrum can be represented as a set of an infinitely large number of discrete states, between K-Rya Energy. The gaps are infinitely small.

Condition, to-ryom corresponds to the lowest energy, possible for this system, called. The main thing: all other states of the name. excited. It is often convenient to use the conditional energy scale, in which the energy of the Osn. States are considered to be the beginning of the reference, i.e. relies equal to zero (in this conditional scale everywhere in the future, the energy is indicated by the letter E.). If the system is in a state n. (and the index n.\u003d 1 is assigned to the main. status), has energy E N.then they say that the system is at the energy level E N.. Number n., Numbered by P.E., Naz. quantum number. In general, each P.E. It can be characterized by a non-one quantum number, but their combination; Then the index n. means the combination of these quantum numbers.

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

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

The full moment of the amount of movement.
This amount of Yavl. Vector and obtained by the addition of MKD of all particles that are in the system. Each particle has both their own. MKD spin and orbital moment, due to the movement of the particle relative to the common center of the mass system. MKD quantization leads to the fact that its abs. Value J. Accepts strictly defined values: where j. - Quantum number, which can take nonnegative integer and half-purpose values \u200b\u200b(the quantum number of orbital MKD is always a whole). Projection of the MKD on K.L. The axis is called. Magn. quantum number and can take 2J + 1. Values: m j \u003d j, j-1,...,-j.. If K.L. moment J. Yawl. the sum of two dr. moments, then, according to the rules of addition of moments in quantum mechanics, 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. Similarly, a higher number of moments are produced. Accepted for brevity to talk about the MKD system j., implying the moment, ABS. there is a quantity; About Magn. Quantum number speak simply as about the projection of the moment.

With different transformations of the system in the central symmetric field, the full MKD is maintained, i.e., as well as energy, it refers to the continuing values. The law of conservation of the MKD follows from the isotropy of space. In an axially symmetric field, only the projection of the full MKD on the symmetry axis remains.

The parity of the state.
In quantum mechanics, the state of the system is described by the so-called. wave fictions. Parity characterizes the change in the wave f-system of the system during the operation of spatial inversion, i.e. Replace the coordinate signs of all particles. With such an operation, the energy does not change, while the wave f │ can either remain unchanged (even state), or change its sign to the opposite (odd state). Parity P. Takes two values, respectively. If the system has nuclear or email-magnet. Forces, parity persisted in atomic, molecular and nuclear transformations, i.e. This value also refers to the continuing values. The law of preserving the parity of the Yavl. The consequence of the symmetry of space in relation to mirror reflections and is violated in those processes, weak interactions are involved in the reasons.

Quantum transitions
- System transitions from one quantum state to another. Such transitions can lead to energetic change. The state of the system and its qualities. Changes. These are related, free-related, free-free transitions (see the interaction of radiation with substance), for example, excitation, deactivation, ionization, dissociation, recombination. It is also chemical. and nuclear reactions. Transitions can occur under the action of radiation - emitting (or radiation) transitions or in a collision of this system with K.L. Dr. System or particle - non-durable transitions. An important characteristic of the Jawl quantum transition. His probability in the unit. Time showing how this transition often will happen. This value is measured in C -1. Probability radiats. Transitions between levels m. and n. (m\u003e N.) With the radiation or absorption of the photon, the energy is equal to the coefficient. Einstein A MN, B Mn and B nm.. Transition from level m. Level n. It can occur spontaneously. The likelihood of photon radiation B Mn. In this case, equal A MN.. Type transitions under the action of radiation (induced transitions) are characterized by probabilities of photon radiation and photon absorption, where - the radiation energy density with a frequency.

The ability to carry out a quantum transition from this W.E. On K.L. Other O. means that the characteristic Wed. Time, during the course, the system may be on this C.E., of course. It is defined as the value, the inverse of the total probability of decay of this level, i.e. The sum of the probabilities of all possible transitions from the level under consideration to all others. For radiats. Transitions The total probability is, but. The limb of time, according to the ratio of uncertainties, means that the level of level cannot be determined absolutely, i.e. W.E. It has some width. Therefore, the radiation or absorption of photons during a quantum transition occurs on a strictly defined frequency, but inside a certain frequency interval lying in the vicinity of the value. The intensity of the intensity inside this interval is defined by the profile of the spectral line, which determines the likelihood that the frequency of the photon emitted or absorbed at this transition is equal to:
(1)
where - half-width line profile. If the broadening of W.E. And the spectral lines are caused only by spontaneous transitions, then such a broadening is called. Natural. If there is a certain role in the broadening of a system with other particles, the broadening has a combination character and the value must be replaced by the amount where the radiaz is calculated. The probabilities of transitions must be replaced by collisional probabilities.

Transitions in quantum systems are subject to certain selection rules, i.e. The rules establishing how the quantum numbers may change in the transition characterizing the state of the system (MKD, parity, etc.). The most simply selection rules are formulated for radiances. transitions. In this case, they are determined by the central and end states, as well as the quantum characteristics of the emitted or absorbed photon, in particular its MKD and parity. The greatest probability is so-called. Electric dipole transitions. These transitions are carried out between the levels of the opposite parity, the full MKD of the to-rye differ by magnitude (the transition is not possible). In the framework of the current terminology, these transitions are called. Allowed. All other types of transitions (magnetic dipole, electric quadrupole, etc.) Naz. prohibited. The meaning of this term consists only that their probabilities turn out to be much less probabilities of dipole electrical transitions. However, they did not yavl. Forbidden absolutely.

Quantum systems from identical particles

Quantum features of the behavior of microparticles that distinguish them from the properties of macroscopic objects are manifested not only when considering the movement of one particle, but also when analyzing behavior systems microphastitz . This is most clearly seen by the example of physical systems consisting of identical particles - electrons, protons, neutrons, etc. systems, etc.

For system from N. particles with masses t. 01 , T. 02 ... T. 0 i. , … m. 0 N. having coordinates ( x. i. , y. i. , z. i.), the wave function can be represented as

Ψ (x. 1 , y. 1 , z. 1 , … x. i. , y. i. , z. i. , … x. N. , y. N. , z. N. , t.) .

For elementary volume

dV i. = dX. i. . dY. i. . dZ. i.

value

w. =

determines the likelihood that one particle is in the volume dV 1, the other in the volume dV 2, etc.

Thus, knowing the wave function of the particle system, one can find the likelihood of any spatial configuration of the microparticle system, as well as the likelihood of any mechanical value of both the system as a whole and in a separate particle, as well as calculate the average mechanical value.

The wave function of the particle system is found from the Schrödinger equation

where

Hamilton function operator for particle system

+ .

Power function for i.- oh particles in the outer field, and

Energy interaction i.- oh I. j.- oh particles.

Indistinguishability of identical particles in quantum

mechanics

Particles possessing the same mass, electric charge, spin, etc. Will behave in the same conditions in exactly the same way.

Hamiltonian such a particle system with the same masses m. oi and identical power functions U. i can be written in the form presented above.

If in the system change i.- uu I. j.- the particles, then, due to the identity of the same particles, the system status should not change. The total energy of the system will remain unchanged, as well as all the physical quantities characterizing its condition.

The principle of identity of the same particles: In the system of identical particles, only such states are implemented that do not change when particles are permutive.

Symmetric and antisymmetric states

We introduce the permutation operator in the system under consideration - . The action of this operator lies in the fact that it rearranges in places i.- yu andj.- system particles system.

The principle of identity of the same particles in quantum mechanics leads to the fact that all possible states of the system formed by the same particles are divided into two types:

symmetricfor which

antisymmetricfor which

(x. 1 , y. 1 ,z. 1 … x. N. , y. N. , z. N. , t.) = - Ψ A. ( x. 1 , y. 1 ,z. 1 … x. N. , y. N. , z. N. , t.).

If the wave function describing the state of the system, at any point of time is symmetric (antisymmetric), then this type of symmetry save and at any other time.

Bosons and Fermions

Particles whose states are described by symmetric wave functions are called bosons bose - Einstein Statistics . The bosons include photons, π- and to-mesons, phonons in a solid body, excitons in semiconductors and dielectrics. All bosons possesszero or integer back .

Particles whose states are described by antisymmetric wave functions are called fermiones . Systems consisting of particles obey fermi Statistics - Dirac . Fermions include electrons, protons, neutrons, neutrinos and all elementary particles and antiparticles withhalf-spin.

The relationship between the spin of the particles and the statistical type remains fair and in the case of complex particles consisting of elementary. If the total spin of the complex particle is equal to an integer or zero, then this particle is a boson, and if it is equal to a half-time number, then the particle is a fermion.

Example: α-particle () consists of two protons and two neutrons. Four fermions with backs +. Therefore, the kernel spin is 2 and this core is a boson.

The core of the light isotope consists of two protons and one neutron (three fermion). Spin of this kernel. Consequently, the kernel of the Fermion.

Powli principle (ban Pauli)

In the system of identityfermionov There can be no two particles in the same quantum state.

As for the system consisting of bosons, the principle of symmetry of wave functions does not observe any restrictions on the system status. In the same condition may be any number of identical bosons.

Periodic system of elements

At first glance, it seems that in the atom all electrons must fill the level with the smallest possible energy. The experience shows that it is not.

Indeed, in accordance with the principle of Pauli, in the atom there can be no electrons with the same values \u200b\u200bof all four quantum numbers.

Each value of the main quantum number p correspond to 2 p 2 states that differ from each other values \u200b\u200bof quantum numbers l. , m. and m. S. .

A combination of an atom electrons with the same quantum number p Forms the so-called shell. In accordance with the number p

The shells are divided by submarine quantum l. . The number of states in the submarine is 2 (2 l. + 1).

Different states in the suburbs are distinguished by the values \u200b\u200bof quantum numbers. T. and m. S. .

In the first and second parts of the textbook, it was assumed that particles that make up macroscopic systems are subject to the laws of classical mechanics. However, it turned out that in order to explain the many properties of micro-lectures 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 classic properties of particles. The quantum properties of microjects constituting a certain physical system are also manifested in the properties of the macroscopic system.

As such quantum systems, we will consider electrons in metal, photon gas, etc. In the future, a quantum system or particle, we will understand a certain material object described by the quantum mechanic apparatus.

Quantum Mehanska describes inherent properties and features inherent in the micrometer, we often cannot explain the classical ideas on the B ^. Such peculiarities include, for example, a corpuscular-wave dualism of micro-lectures in quantum mechanics, open and confirmed on numerous experienced facts, discreteness of various physical parameters, "spin" properties, etc.

Special properties of microjects do not allow them to describe their behavior by conventional methods of classical mechanics. For example, the presence of microparticles manifest themselves at the same time and wave and corpuscular properties

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

This fact was reflected in the so-called uncertainty relation in 1925 by Heisenberg, which is that the inaccuracies in determining the coordinates and pulse of microparticles are related to the relation:

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

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

Both reduced ratios show that if one of the values \u200b\u200bis defined with great accuracy, the second value is determined with low accuracy. Inaccuracies here are determined through a constant plank, which practically does not limit the accuracy of measurements of different quantities for macroscopic objects. But for microparticles with small energies, small size and impulses, the accuracy of the simultaneous measurement of marked parameters is already insufficient.

Thus, the condition of the microparticle in quantum mechanics cannot be simultaneously described using coordinates and pulses, as is done in classical mechanics (canonical Hamilton equations). Similarly, it is impossible to talk about the value of the particle's energy at the moment. Certain energy conditions can only be obtained in stationary cases, that is, they are not defined exactly in time.

Possessing the corpuscular wave properties, any microparticle does not have an absolutely accurate coordinate, but it turns out to be "smeared" in space. If there is a certain area of \u200b\u200bspace of two or more particles, we cannot distinguish them from each other, since we cannot trace the movement of each of them. Hence the fundamental indistinguishability or identity of particles in quantum mechanics.

This turns out that the values \u200b\u200bcharacterizing some parameters of microparticles may vary only by certain portions, quanta, from where the name of quantum mechanics occurred. This discreteness of many parameters that determine the state of microparticles, can also be described in classical physics.

According to quantum mechanics, in addition to the energy of the system, discrete values \u200b\u200bcan take the moment of the amount of system or spin, the magnetic moment and their projection on any highlighted direction. So, the square of the moment of movement can take only the following values:

Spin can only take values

where can be

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

where the magneton boron and the magnetic quantum number that is received:

In order to mathematically describe these features of physical quantities, each physical value put in compliance with a certain operator. In quantum mechanics, thus, physical quantities are depicted by operators, and their values \u200b\u200bare defined as the average for their own values \u200b\u200bof operators.

When describing the properties of microjects, it was possible, except for the properties and parameters encountered in the classical description of microparticles, enter and introduce new, purely quantum parameters and properties. These include "spin" particles characterizing their own moment of the amount of movement, "exchange interaction", principle Pauli, etc.

These features of microparticles do not allow them to describe them with the help of classical mechanics. As a result, microjects are described by quantum mechanics, which takes into account the marked features and properties of microparticles.

Electronic properties of low-dimensional electronic systems The principle of size quantization The whole complex of phenomena, usually understood under the words "electronic properties of low-dimensional electronic systems", has a fundamental physical fact: a change in the energy spectrum of electrons and holes in structures with very small sizes. We will demonstrate the basic idea of \u200b\u200bdimensional quantization on the example of electrons in a very thin metal or semiconductor film thickness a.

Electronic properties of low-dimensional electronic systems The principle of measuring quantization electrons in the film is in a potential pummy of depth equal to the operation of the output. The depth of a potential pit can be considered infinitely large, since the operation of the release of several orders of magnitude exceeds the thermal energy of the carriers. Typical output values \u200b\u200bin most solids have the value w \u003d 4 -5 e. B, several orders of magnitude exceeding the characteristic thermal energy of carriers, having the order of k. T equal to room temperature 0, 026 e. B. According to the laws of quantum mechanics, the energy of electrons in such a hole is quantum, i.e., only some discrete values \u200b\u200bof EN can take, where n can take integer values \u200b\u200bof 1, 2, 3, .... These discrete energy values \u200b\u200bare called dimensional quantization levels.

Electronic properties of low-dimensional electronic systems The principle of dimensional quantization for a free particle with an effective mass M *, the movement of which in the crystal in the direction of the z axis is limited to impermeable barriers (i.e. barriers with infinite potential energy), the energy of the main state compared with the state without limitation increases By magnitude, this increase in energy is called the energy of the particle size quantization. The energy of dimensional quantization is a consequence of the principle of uncertainty in quantum mechanics. If the particle is limited in space along the z axis within the distance A, the uncertainty Zcomponents of its impulse increases by order of ħ / a. Accordingly, the kinetic particle energy is increased by the value of E 1. Therefore, the considered effect is often referred to as a quantum-dimensional effect.

Electronic properties of low-dimensional electronic systems The principle of dimensional quantization The conclusion about the quantization of the electronic motion energy refers only to the movement of the potential pit (along the Z axis). On the movement in the XY plane (parallel to the boundaries of the film), the potential of the pit does not affect. In this plane, the carriers move as free and characterized, as in a massive sample, continuously quadratic pulsed with an energy spectrum with an effective mass. The total energy of the carriers in the quantum-dimensional film is mixed discretely continuous spectrum

Electronic properties of low-dimensional electronic systems The principle of size quantization In addition to increasing the minimum energy of the particle, the quantum separation effect also leads to quantization of the energy of its excited states. Energy spectrum of quantum-dimensional film - impulse charge carriers in the film plane

Electronic properties of low-dimensional electronic systems The principle of size quantization let 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 on impurities or acoustic phonons), as well as the scattering of electrons each other, cannot change the quantum number N by transferring the electron to the overlying level, as it would require additional energy costs. This means that electrons with elastic scattering can only change their impulse in the film plane, i.e., behave like purely two-dimensional particles. Therefore, quantum-dimensional structures in which only one quantum level is filled, is often called two-dimensional electronic structures.

Electronic properties of low-dimensional electronic systems The principle of size quantization There are other possible quantum structures where the movement of carriers is limited not in one, but in two directions, as in microscopic wire or thread (quantum thread or wire). In this case, the carriers can move freely only in one direction, along the threads (let's call it axis x). In cross section (the plane YZ), the energy is quantized and receives discrete values \u200b\u200bof EMN (like any two-dimensional movement, it is described by two quantum numbers, M and N). The full spectrum is also discrete-member, but only with one continuous degree of freedom:

Electronic properties of low-dimensional electronic systems The principle of size quantization is also possible, also, the creation of quantum structures resembling artificial atoms, where the movement of carriers is limited in all three directions (quantum points). In quantum points, the energy spectrum no longer contains the continuous component, i.e. it does not consist of a subzon, but is purely discrete. As in the atom, it is described by three discrete quantum numbers (not counting the back) and can be recorded as an E \u003d ELMn, and, as in the atom, energy levels can be degenerate and depend on one or two numbers. The total feature of low-dimensional structures is the fact that, if at least one direction, the movement of carriers is limited to a very small region, comparable in size with a de-broile wavelength of carriers, their energy spectrum changes significantly and becomes partially or completely discrete.

Electronic properties of low-dimensional electronic definition systems Quantum dots - Quantum Dots - structures in which in all three directions dimensions constitute several interatomic distances (zero-dimensional structures). Quantum wires (threads) - Quantum Wires - structures in which in two directions sizes are equal to several interatomic distances, and in the third - macroscopic value (one-dimensional structures). Quantum pits - Quantum Wells - structures in which in one direction the size is several interatomic distances (two-dimensional structures).

Electronic properties of low-dimensional electronic systems Minimum and maximum dimensions The lower limit of the size quantization is determined by the critical size of the DMIN, in which there is at least one electronic level in the quantum-dimensional structure. DMIN depends on the rupture of the DEC conduction zone in the corresponding heterogerer used to obtain quantum-dimensional structures. In the quantum yam, at least one electronic level, if DEC exceeds the value of H is a strap constant, ME * is an efficient electron mass, DE 1 QW is the first level in a rectangular quantum pit with infinite walls.

The electronic properties of low-dimensional electronic systems are minimal and maximum dimensions if the distance between the energy levels become comparable to the thermal energy k. BT, then the population of high levels increases. For a quantum point, a condition in which the settling of higher lying levels can be neglectedly recorded as E 1 QD, E 2 QD - the energy of the first and second level of dimensional quantization, respectively. This means that the advantages of dimensional quantization can be fully implemented if this condition sets the upper limits for dimensional quantization. For GA. AS -ALX. GA 1 -X. AS This value is 12 nm.

Electronic properties of low-dimensional electronic systems The distribution of quantum states in the structures of the reduced dimension is 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 are determined using cyclic boundary conditions of the Born-pocket, from which it follows that the components of the electron wave vector change not continuously, and take a number of discrete values \u200b\u200bhere ni \u003d 0, ± 1, ± 2, ± 3, and - dimensions Crystal (in the form of Cuba with side L). The volume of the K-space per quantum state is (2) 3 / v, where V \u003d L 3 is the volume of the crystal.

Electronic properties of low-dimensional electronic systems The distribution of quantum states in the structures of the reduced dimension in this way, the number of electronic states per particle DK \u003d DKXDKYDKZ, calculated per unit volume, will be equal to the multiplier 2 takes into account the two possible orientations of the back. The number of states per unit volume in the opposite space, i.e. the density of states) does not depend on the wave vector in other words, in the opposite space, the permitted states are distributed with a constant density.

Electronic properties of low-dimensional electronic systems The distribution of quantum states in the structures of the reduced dimension The function of the energy density in the general case is practically impossible to calculate, since the isoenergy surfaces may have a rather complicated form. In the simplest case of an isotropic parabolic law of the dispersion, which is fair for the edges of the energy zones, you can find the number of quantum states per spherical layer concluded between two close iso-energy surfaces corresponding to the e and E + D energies. E.

Electronic properties of low-dimensional electronic systems The distribution of quantum states in the reduced dimension structures volume of the spherical layer in the to-space. DK - layer thickness. This volume will account for D. N states Considering the relation E and K on a parabolic law we obtain from here the density of energy states will be equal to M * - Effective electron mass

Electronic properties of low-dimensional electronic systems The distribution of quantum states in the structures of the reduced dimension in this way, in three-dimensional crystals with a parabolic energy spectrum, with an increase in energy, the density of the allowed energy levels (state density) will increase in proportion to the level density in the conduction zone and in the valence zone. The area of \u200b\u200bshaded domains 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 reduced dimension structures Calculate the density of states for a two-dimensional system. The total energy of the carriers for an isotropic parabolic dispersion law in the quantum-dimensional film, as shown above, has a mixed discrete continuous spectrum in a two-dimensional conduction electron state system are determined by three numbers (N, KX, KY). The energy spectrum is divided into separate two-dimensional subzones EN, corresponding to fixed values \u200b\u200bn.

Electronic properties of low-dimensional electronic systems The distribution of quantum states in the structures of a reduced dimension The curves of constant energy are in the opposite space of the circle. Each discrete quantum number n corresponds to the absolute value of the Z-component of the wave vector, therefore, the volume in the opposite space limited by the closed surface of this energy E in the case of a two-dimensional system is divided into a number of sections.

Electronic properties of low-dimensional electronic systems The distribution of quantum states in the reduced dimension structures to determine the dependence of the density of state states for the two-dimensional system. To do this, at a given n, we will find the square s ring bounded by two isoenergetic surfaces corresponding to the enexes E and E + D. E: Here the magnitude of the two-dimensional wave vector corresponding to data n and e; DKR - Ring width. Since one state in the plane (Kxky) corresponds to an area where L 2 is the area of \u200b\u200bthe two-dimensional film A, the number of electron states in the ring, calculated per unit of crystal volume, will be equal to the electron spin

Electronic properties of low-dimensional electronic systems The distribution of quantum states in the structures of the reduced dimension Since here is the energy corresponding to the bottom of the N-o bezone. Thus, the density of states in a two-dimensional film where Q (y) is a single function of heviside, Q (y) \u003d 1 at y≥ 0 and q (y) \u003d 0 at y

Electronic properties of low-dimensional electronic systems The distribution of quantum states in the structures of the reduced dimension The density of states in a two-dimensional film can also be represented as a whole part equal to the number of subbands, the bottom of which is located below E. E. Thus, for two-dimensional films with parabolic dispersion, the density of states in Any subzone is constant and does not depend on energy. Each subzone gives the same contribution to the general density of states. With a fixed film thickness, the density of states changes with a jump when it does not change to one.

Electronic properties of low-dimensional electronic systems Distribution of quantum states in reduced dimension structures The dependence of the density of states of the two-dimensional film from energy (a) and thickness A (b).

Electronic properties of low-dimensional electronic systems The distribution of quantum states in the structures of the reduced dimension in the case of an arbitrary dispersion law or with a different form of a potential pension of the density of the state of energy and the thickness of the film may differ from the above, but the main feature is a non-monotonic move - will continue.

Electronic properties of low-dimensional electronic systems The distribution of quantum states in the structures of the reduced dimension calculate the density of states for a one-dimensional structure - a quantum thread. Isotropic parabolic dispersion law In this case, it can be written in the form of x directed along the quantum thread, D is the thickness of the quantum thread along the axes of the Y and Z, KX - one-dimensional wave vector. M, n - whole positive numbers characterizing where the axis quantum subzones. The energy spectrum of the quantum thread is divided, thus, on separate overlapping one-dimensional subzones (parabolas). The movement of electrons along the X axis is free (but with an effective mass), and along the other two axes, the movement is limited.

Electronic properties of low-dimensional electronic systems The distribution of quantum states in the structures of the reduced dimension The energy spectrum of electrons for the quantum thread

Electronic properties of low-dimensional electronic systems The distribution of quantum states in the structures of the reduced dimension The density of states in the quantum filament from the energy number of quantum states per interval DKX, calculated per unit of volume where the energy corresponding to the bottom of the subzone with specified N and M.

Electronic properties of low-dimensional electronic systems The distribution of quantum states in the structures of the reduced dimension The density of states in the quantum filament on the energy in this way therefore, in the output of this formula, the spin degeneration of states and the fact that the same interval d is taken into account. E corresponds to two intervals ± dkx of each subzone, for which (E-EN, M)\u003e 0. Energy E is counted from the bottom of the conduction zone of the massive sample.

Electronic properties of low-dimensional electronic systems The distribution of quantum states in the structures of the reduced dimension The density of states in the quantum thread from the energy dependence of the density of the calf filament states. Figures in curves show quantum numbers n and m. In brackets indicated factors of degeneration levels of the subband.

Electronic properties of low-dimensional electronic systems The distribution of quantum states in the structures of the reduced dimension The density of states in the quantum thread from energy within the separate subband the density of states decreases with increasing energy. The total density of states is a superposition of the same decreasing functions (corresponding to individual subzones), shifted along the energy axis. At e \u003d e m, n, the density of states is equal to infinity. Subzones with quantum numbers N M are twice degenerate (only for LY \u003d LZ D).

Electronic properties of low-dimensional electronic systems The distribution of quantum states in the structures of the reduced dimension The density of states in the quantum point from the energy during the three-dimensional constraint of the particle movement, we arrive at the task of finding the allowed states in the quantum dot or zero-dimensional system. Using the approximation of the effective mass and the parabolic dispersion law, for the edge of the isotropic energy zone, the spectrum of the permitted states of the quantum point with the same dimensions D along all three coordinate axes will be viewed N, M, L \u003d 1, 2, 3 ... - the positive numbers that are numbered subzones. The energy spectrum of the quantum point is a set of discrete permitted states corresponding to a fixed N, M, L.

Electronic properties of low-dimensional electronic systems The distribution of quantum states in the structures of the reduced dimension The density of states in the quantum point from the energy number of states in subzones corresponding to one set N, M, calculated per unit volume, the total number of states having the same energy calculated per unit of volume The degeneracy of the levels is primarily determined by the symmetry of the problem. G - level degeneration factor

Electronic properties of low-dimensional electronic systems The distribution of quantum states in the structures of the reduced dimension The density of states in the quantum point from the energy of the level of levels primarily is determined by the symmetry of the problem. For example, for the case of a quantum point with the same dimensions in all three dimensions, the levels will be threefoldly degenerate if two quantum numbers are equal to each other and are not equal to the third, and six times are degenerate if all quantum numbers are not equal to each other. The specific type of potential can also lead to an additional, so-called random degeneration. For example, for the quantum point under consideration, to three-time degeneration of levels E (5, 1, 1); E (1, 5, 1); E (1, 1, 5) associated with the symmetry of the problem is added random degeneration E (3, 3, 3) (N 2 + m 2 + L 2 \u003d 27 in both the first and second cases) related to the view Limiting potential (infinite rectangular potential pit).

Electronic properties of low-dimensional systems The distribution of quantum states in the structures of the reduced dimension The density of states in the quantum point from the energy distribution of the number of allowed states n in the conduction zone for a quantum point with the same dimensions in all three dimensions. The numbers denote quantum numbers; In brackets indicated factors of degeneration levels.

Electronic properties of low-dimensional systems Statistics of carriers in low-dimensional structures Three-dimensional electronic systems The properties of equilibrium electrons in semiconductors depend on the Fermian distribution function, which determines the likelihood that the electron will be in the quantum state with the EF EF level - the Fermi level or electrochemical potential, T - absolute temperature , k - Boltzmann. The calculation of various statistical quantities is greatly simplified if the Fermi level lies in the prohibited zone of energies and is significantly removed from the bottom of the EU conduction zone (EC EF)\u003e k. T. Then in the distribution of Fermi Dirac unit in the denominator can be neglected and it goes into the distribution of Maxwell-Boltzmann of classical statistics. This is the case of a nondegenerate semiconductor

Electronic properties of low-dimensional systems Statistics of carriers in low-dimensional structures Three-dimensional electronic systems The function of the density distribution of states in the conduction zone G (E), the Fermi Dirac function for three temperatures and the Maxwell-Boltzmann function for three-dimensional electronic gas. When T \u003d 0, the Fermi Dirac feature has a view of a discontinuous function. For E EF, the function is zero and the corresponding quantum states are completely free. When T\u003e 0 FERMI function. Dirac is blurred in the surroundings of the Energy of Fermi, where it quickly changes from 1 to 0 and this blur is proportional to k. T, i.e., the greater the higher the temperature. (Fig. 1. 4. Gurtov)

Electronic properties of low-dimensional systems Statistics of carriers in low-dimensional structures Three-dimensional electronic systems The concentration of electrons in the conduction zone is located by summing in all states, we note that as a top limit in this integral, we would have to take the energy of the upper edge of the conduction zone. But since the Fermi Dirac feature for E\u003e EF energies exponentially decreases with an increase in energy, the replacement of the upper limit on infinity does not change the integral values. Substituting the values \u200b\u200bof functions to the integral, we obtain-an efficient state density in the conduction zone

Electronic properties of low-dimensional systems Statistics of carriers in low-dimensional structures Two-dimensional electronic systems Determine the concentration of the charge carrier in a two-dimensional electronic gas. Since the density of the states of the two-dimensional electron gas, we also obtain the upper limit of integration takes equal infinity, given the sharp dependence of the Fermi Dirac distribution function from energy. Integrating where

Electronic properties of low-dimensional systems statistics of carriers in low-dimensional structures Two-dimensional electronic systems for non-degenerate electron gas, when in the case of ultra-thin films, when the filling of only lower subzones can be taken into account with a strong degeneration of electronic gas, when where N 0 is a whole part.

Electronic properties of low-dimensional systems Statistics of carriers in low-dimensional structures It should be noted that in quantum-dimensional systems due to a smaller density of states, the condition of complete degeneracy does not require extremely high concentrations or low temperatures and is often implemented quite often in experiments. For example, in N-Ga. As for n 2 d \u003d 1012 cm-2, the degeneration will occur at room temperature. In quantum threads, the integral for calculation, in contrast to two-dimensional and three-dimensional cases, is not calculated by analytically arbitrary degeneration, and simple formulas can be written only in limiting cases. In a non-degenerate one-dimensional electronic gas in the case of ultra-thin threads, when it is possible to take into account the filling of only the lowest level with an energy E 11 concentration of electrons where one-dimensional effective density of states

Atomic core, like other micromyr objects, is a quantum system. This means that theoretical description of its characteristics requires the attraction of quantum theory. In quantum theory, the description of the states of physical systems is based on wave featuresor probability amplitudesψ (α, t). The square of the module of this function determines the detection of the detection of the system under study in a state with the characteristic α - ρ (α, t) \u003d | ψ (α, t) | 2. The argument of the wave function can be, for example, the coordinates of the particle.
Completely likely to normalize per unit:

Each physical value is compared to linear Ermites operator acting in the Hilbert space of wave functions ψ. The spectrum of values \u200b\u200bthat the physical value can take is determined by the spectrum of its own values \u200b\u200bof its operator.
The average value of the physical quantity is in a state of ψ

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

The state of the kernel as a quantum system, i.e. Functions ψ (t) , obey the Schrödinger equation ("W. Sh.")

(2.4)

Operator - Hermite operator Hamilton ( hamiltonian) Systems. Together with the initial condition on ψ (t), equation (2.4) defines the state of the system at any time. If it does not depend on time, then the total energy of the system is the integral of the movement.Conditions in which the total energy of the system has a certain meaning called stationary.Stationary states describes with their own functions of the operator (Hamiltonian):

Last of equations - stationary Schrödinger equation, defining, in particular, the set (spectrum) of the energies of the stationary system.
In the stationary states of the quantum system, in addition to energy, other physical quantities can be maintained. The condition of maintaining the physical size F is the equality of the 0 switch 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 region of the excitation energy of the kernel 12 with below (approximately) 16 MeV it has discrete character.Above this energy, the spectrum is continuous. The discrete character 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 spectrum levels has a finite average lifetime τ, the level of level g is also finite and is associated with an average life time by the relationship that is a consequence of the uncertainty ratio for energy and time Δ T · ΔE ≥ ћ :

In the diagrams of the nuclei spectra indicate the energies of the kernel levels in MeV or CEV, as well as the spin and parity of states. The schemes also indicate, if possible, the isospin of the state (since the spectra schemes are given energy excitation levels, the energy of the basic state is taken for the beginning of the reference). In the region of excitation ei< 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.