Quantum systems and their properties.
Probability distribution 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 micro-objects, it is necessary to use quantum mechanics. If a system consists of particles that obey 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.
The first implies the impossibility of accurately determining all the parameters of the system that determine its state from the classical point of view. This fact is reflected in the Heisandberg uncertainty relation:
In order to mathematically describe these features of micro-objects in quantum physics, a linear Hermitian operator is assigned to the quantity, which 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 momenta and coefficients of the microparticles of the system cannot be measured simultaneously, the wave function is presented 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 particular system is found as an eigenfunction of the Hamelton operator:
Stationary Schrödinger equation.
Non-stationary Schrödinger equation.
The principle of indistinguishability of microparticles operates in the microworld.
If the wave function satisfies the Schrödinger equation, then the function also satisfies this equation. The state of the system will not change when 2 particles are swapped.
Let the first particle be in state a and the second particle be in state b.
The system state is described by:
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 for 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 spin integer bosons, 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 over two cells in the phase space.
1) Classical particles are different. It is possible to trace each particle separately.
classical particles.
Quantum systems of identical particles
Quantum features of the behavior of microparticles, which distinguish them from the properties of macroscopic objects, appear not only when considering the motion of a single particle, but also when analyzing the behavior systems microparticles . This is most clearly seen in the example of physical systems consisting of identical particles - systems of electrons, protons, neutrons, etc.
For a 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
magnitude
w =
determines the probability that one particle is in the volume dV 1 , another in volume dV 2 etc.
Thus, knowing the wave function of a system of particles, one can find the probability of any spatial configuration of a system of microparticles, as well as the probability of any mechanical quantity, both for the system as a whole and for an individual particle, and also calculate the average value of the mechanical quantity.
The wave function of a system of particles is found from the Schrödinger equation
, Where
Hamilton function operator for a system of particles
+
.
force function for i-
th particle in an external field, and
Interaction energy i- oh and j- oh particles.
The indistinguishability of identical particles in the quantum
mechanics
Particles that have the same mass, electric charge, spin, etc. will behave in exactly the same way under the same conditions.
The Hamiltonian of such a system of particles with the same masses m oi and the same force functions U i can be written as above.
If the system changes i- oh and j- th particle, then, due to the identity of identical particles, the state of the system should not change. The total energy of the system remains unchanged, as well as all physical quantities describing her condition.
The principle of identity of identical particles: in a system of identical particles, only such states are realized that do not change when the particles are rearranged.
Symmetric and antisymmetric states
Let us introduce the particle permutation operator in the system under consideration - . The effect of this operator is that it swaps i- wow Andj- th particle of the system.
The principle of identity of identical particles in quantum mechanics leads to the fact that all possible states of a system formed by identical particles are divided into two types:
symmetrical, for which
antisymmetric, for 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 is symmetric (antisymmetric) at some point in time, then this type of symmetry persists at any other point in time.
Bosons and fermions
Particles whose states are described by symmetric wave functions are called bosons Bose–Einstein statistics . Bosons are photons, π- And To- mesons, phonons solid body, excitons in semiconductors and dielectrics. All bosons havezero or integer spin .
Particles whose states are described by antisymmetric wave functions are called fermions . Systems consisting of such particles obey Fermi–Dirac statistics . Fermions include electrons, protons, neutrons, neutrinos and All elementary particles and antiparticleshalf back.
The connection between the particle spin and the type of statistics remains valid in the case of complex particles consisting of elementary ones. If the total spin of a complex particle is equal to an integer or zero, then this particle is a boson, and if it is equal to a half-integer, then the particle is a fermion.
Example: α-particle() consists of two protons and two neutrons i.e. four fermions with spins +. Therefore, the spin of the nucleus is 2 and this nucleus is a boson.
The nucleus of a light isotope consists of two protons and one neutron (three fermions). The spin of this nucleus is . Hence the core is a fermion.
Pauli principle (Pauli prohibition)
In the system of identicalfermions no two particles can be in the same quantum state.
As for the system consisting of bosons, the principle of symmetry of wave functions does not impose any restrictions on the states of the system. can be in the same state any number of identical bosons.
Periodic system of elements
At first glance, it seems that in an atom, all electrons should fill the level with the lowest possible energy. Experience shows that this is not so.
Indeed, in accordance with the Pauli principle, in the atom there cannot be electrons with the same values of all four quantum numbers.
Each value of the principal quantum number P corresponds 2 P 2 states that differ from each other by the values of quantum numbers l , m And m S .
The set of electrons of an atom with the same values of the quantum number P forms the so-called shell. according to the number P
Shells are divided into subshells, differing in quantum number l . The number of states in a subshell is 2(2 l + 1).
Different states in a subshell differ in their quantum numbers T And m S .
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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 St. in atoms, molecules and atomic nuclei, i.e. phenomena occurring in volume elements with linear scales of 10 -6 -10 -13 cm lies quantum mechanics. According to quantum mechanics, any quantum system (ie, a system of microparticles, which obeys quantum laws) is characterized by a certain set of states. IN 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. the internal energy of the system (everywhere below, just energy), the total angular momentum (MKD) and parity.
System energy.
A quantum system, being in different states, generally speaking, has different energies. The energy of the bound system can take any value. This set of possible energy values is called. discrete energy spectrum, and energy is said to be quantized. An example would be energy. 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 the atomic nucleus. The continuous energy spectrum can be represented as a set of an infinitely large number of discrete states, between which the energy. gaps are infinitely small.
The state, to-rum corresponds to the lowest energy possible for a given system, called. basic: all other states are called. excited. It is often convenient to use a conditional scale of energy, in which the energy is basic. state is considered the starting point, i.e. is assumed to be zero (in this conditional scale, everywhere below the energy is denoted by the letter E). If the system is in the state n(and the index n=1 is assigned to main. state), has energy E n, then the system is said to be at the energy level E n. Number n, numbering U.e., called. quantum number. In the general case, each U.e. can be characterized not by one quantum number, but by their combination; then the index n means the totality of these quantum numbers.
If the states n 1, n 2, n 3,..., nk corresponds to the same energy, i.e. one U.e., then this level is called degenerate, and the number k- multiplicity of degeneration.
During any transformations of a closed system (as well as a system in a constant external field), its total energy, energy, remains unchanged. Therefore, energy refers to the so-called. conserved values. The law of conservation of energy follows from the homogeneity of time.
Total angular momentum.
This value is yavl. vector and is obtained by adding the MCD of all particles in the system. Each particle has both its own MCD - spin, and orbital moment, due to the motion of the particle relative to the common center of mass of the system. The quantization of the MCD leads to the fact that its abs. magnitude J takes strictly defined values: , where j- quantum number, which can take on non-negative integer and half-integer values (the quantum number of an orbital MCD is always an integer). The projection of the MKD on the c.-l. axis name 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 for adding 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 . Similarly, the summation of a larger number of moments is performed. It is customary for brevity to talk about the MCD system j, implying the moment, abs. the value of which is ; about magn. The quantum number is simply spoken of as the projection of the momentum.
During various transformations of a system in a centrally symmetric field, the total MCD is conserved, i.e., like energy, it is a conserved quantity. The MKD conservation law follows from the isotropy of space. In an axially symmetric field, only the projection of the full MCD onto the axis of symmetry is preserved.
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 the coordinates of all particles. In 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 two values, respectively. If nuclear or el.-magnets operate in the system. forces, parity is preserved in atomic, molecular and nuclear transformations, i.e. this quantity also applies to conserved quantities. Parity conservation law yavl. a consequence of the symmetry of space with respect to mirror reflections and is violated in those processes in which weak interactions are involved.
Quantum transitions
- transitions of the system from one quantum state to another. Such transitions can lead both to a change in energy. the state of the system, and to its qualities. changes. These are bound-bound, freely-bound, free-free transitions (see Interaction of radiation with matter), for example, excitation, deactivation, ionization, dissociation, recombination. It is also a chem. and nuclear reactions. Transitions can occur under the influence of radiation - radiative (or radiative) transitions, or when a given system collides with a c.-l. other system or particle - non-radiative transitions. An important characteristic of the quantum transition yavl. its probability in units. time, indicating how often this 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 to, are determined by the coefficient. Einstein A mn , B mn And B nm. Level transition m to the level n may occur spontaneously. Probability of emitting a photon Bmn in this case equals Amn. Type transitions under the action of radiation (induced transitions) are characterized by the probabilities of photon emission and photon absorption , where is the energy density of radiation with frequency .
The possibility of implementing a quantum transition from a given R.e. on k.-l. another w.e. means that the characteristic cf. time , during which the system can be at this UE, of course. It is defined as the reciprocal of the total decay probability of a given level, i.e. the sum of the probabilities of all possible transitions from the level in question to all others. For the radiation transitions, the total probability is , and . The finiteness of time , according to the uncertainty relation , means that the level energy cannot be determined absolutely exactly, i.e. U.e. has a certain width. Therefore, the emission or absorption of photons during a quantum transition does not occur 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 given by the spectral line profile , which determines the probability that the frequency of a photon emitted or absorbed in a given transition is equal to:
(1)
where is the half-width of the line profile. If the broadening of W.e. and spectral lines is caused only by spontaneous transitions, then such a 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 the sum , where is calculated similarly to , but the radiat. transition probabilities should 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 (MKD, parity, etc.) can change during the transition. The most simple selection rules are formulated for radiats. transitions. In this case, they are determined by the properties 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 so-called. electric dipole transitions. These transitions are carried out between levels of opposite parity, the complete MCD to-rykh differ by an amount (the transition is impossible). In the framework of 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 is only that their probabilities turn out to be much less than the probabilities of electric dipole transitions. However, they are not yavl. absolutely prohibited.
Bohr's model of the atom was an attempt to reconcile the ideas of classical physics with the emerging laws of the quantum world.
E. Rutherford, 1936:
How are the electrons arranged in the outer part of the atom? I regard Bohr's original quantum theory of the spectrum as one of the most revolutionary that has ever been made in science; and I don't know of any other theory that has more success. He was at that time in Manchester and, firmly believing in the nuclear structure of the atom, which became clear in experiments on scattering, he tried to understand how the electrons should be arranged in order to obtain the known spectra of atoms. The basis of his success lies in the introduction of completely new ideas into the theory. He introduced into our minds the idea of a quantum of action, as well as the idea, alien to classical physics, that an electron can orbit around a nucleus without emitting radiation. When putting forward the theory of the nuclear structure of the atom, I was fully aware that, according to the classical theory, electrons should fall on the nucleus, and Bohr postulated that for some unknown reason this does not happen, and on the basis of this assumption, as you know, he was able to explain the origin of the spectra. Using quite reasonable assumptions, he solved step by step the problem of the arrangement of electrons in all atoms of the periodic table. There were many difficulties here, since the distribution had to correspond to the optical and x-ray spectra of the elements, but in the end Bohr managed to propose an arrangement of electrons that showed the meaning of the periodic law.
As a result of further improvements, mainly introduced by Bohr himself, and modifications made by Heisenberg, Schrödinger and Dirac, the whole mathematical theory was changed and the ideas of wave mechanics were introduced. Quite apart from these further improvements, I regard Bohr's work as the greatest triumph of human thought.
To realize the significance of his work, one should only consider the extraordinary complexity of the spectra of the elements and imagine that within 10 years all the main characteristics of these spectra have been understood and explained, so that now the theory of optical spectra is so complete that many consider this an exhausted question, similar to how it was a few years ago with sound.
By the middle of the 1920s, it became obvious that N. Bohr's semiclassical theory of the atom could not give an adequate description of the properties of the atom. In 1925–1926 In the works of W. Heisenberg and E. Schrödinger, a general approach was developed for describing quantum phenomena - quantum theory.
The quantum physics |
|---|
(x,y,z,p x ,p y ,p z)
=∂H/∂p, = -∂H/∂t,
x, y, z, p x , p y , p z
ΔхΔp x ~
∆y∆p y ~
∆z∆p z ~
Determinism
|(x,y,z)| 2
The state of a classical particle at any moment of time is described by setting its coordinates and momenta (x,y,z,p x ,p y ,p z ,t). Knowing these values at the time t, it is possible to determine the evolution of the system under the action of known forces at all subsequent moments of time. The coordinates and momenta of the particles are themselves quantities that can be directly measured experimentally. In quantum physics, the state of a system is described by the wave function ψ(x, y, z, t).
Because for a quantum particle, it is impossible to accurately determine the values of its coordinates and momentum at the same time, then it makes no sense to talk about the movement of the particle along a certain trajectory, you can only determine the probability of the particle being at a given point at a given time, which is determined by the square of the modulus of the wave function W ~ |ψ( x,y,z)| 2.
The evolution of a quantum system in the nonrelativistic case is described by a wave function that satisfies the Schrödinger equation
where is the Hamilton operator (the operator of the total energy of the system).
In the nonrelativistic case − 2 /2m + (r), where t
is the mass of the particle, is the momentum operator, (x,y,z) is the operator of the potential energy of the particle. To set the law of motion of a particle in quantum mechanics means to determine the value of the wave function at every moment of time at every point in space. In the stationary state, the wave function ψ(x, y, z) is a solution to the stationary Schrödinger equation ψ = Eψ. Like any bound system in quantum physics, the nucleus has a discrete spectrum of energy eigenvalues.
The state with the highest binding energy of the nucleus, i.e., with the lowest total energy E, is called the ground state. States with higher total energy are excited states. The lowest energy state is assigned a zero index and the energy E 0
=
0.
E0 → Mc 2 = (Zm p + Nm n)c 2 − W 0 ;
W 0 is the binding energy of the nucleus in the ground state.
Energies E i (i = 1, 2, ...) of excited states are measured from the ground state.
Scheme of the lower levels of the 24 Mg nucleus.
The lower levels of the kernel are discrete. As the excitation energy increases, the average distance between the levels decreases.
An increase in the level density with increasing energy is a characteristic property of many-particle systems. It is explained by the fact that with an increase in the energy of such systems, the number various ways distribution of energy between nucleons.
quantum numbers- integer or fractional numbers that determine the possible values of physical quantities characterizing a quantum system - an atom, an atomic nucleus. Quantum numbers reflect the discreteness (quantization) of physical quantities characterizing the microsystem. A set of quantum numbers that exhaustively describe a microsystem is called complete. So the state of the nucleon in the nucleus is determined by four quantum numbers: the main quantum number n (can take values 1, 2, 3, ...), which determines the energy E n of the nucleon; orbital quantum number l = 0, 1, 2, …, n, which determines the value L
the orbital angular momentum of the nucleon (L = ћ 1/2); the quantum number m ≤ ±l, which determines the direction of the orbital momentum vector; and the quantum number m s = ±1/2, which determines the direction of the nucleon spin vector.
quantum numbers
| n | Principal quantum number: n = 1, 2, … ∞. |
| j | The quantum number of the total angular momentum. j is never negative and can be integer (including zero) or half-integer depending on the properties of the system in question. The value of the total angular momentum of the system J is related to j by the relation J 2 = ћ 2 j(j+1). = + where and are the orbital and spin angular momentum vectors. |
| l | Quantum number of orbital angular momentum. l can only take integer values: l= 0, 1, 2, … ∞, The value of the orbital angular momentum of the system L is related to l relation L 2 = ћ 2 l(l+1). |
| m | The projection of the total, orbital, or spin angular momentum onto a preferred axis (usually the z-axis) is equal to mћ. For the total moment m j = j, j-1, j-2, …, -(j-1), -j. For the orbital moment m l = l, l-1, l-2, …, -(l-1), -l. For the spin moment of an electron, proton, neutron, quark m s = ±1/2 |
| s | Quantum number of spin angular momentum. s can be either integer or half-integer. s is a constant characteristic of the particle, determined by its properties. The value of the spin moment S is related to s by the relation S 2 = ћ 2 s(s+1) |
| P | Spatial parity. It is equal to either +1 or -1 and characterizes the behavior of the system under mirror reflection P = (-1) l . |
Along with this set of quantum numbers, the state of the nucleon in the nucleus can also be characterized by another set of quantum numbers n, l, j, jz . The choice of a set of quantum numbers is determined by the convenience of describing a quantum system.
The existence of conserved (invariant in time) physical quantities for a given system is closely related to the symmetry properties of this system. So, if an isolated system does not change during arbitrary rotations, then it retains the orbital angular momentum. This is the case for the hydrogen atom, in which the electron moves in the spherically symmetric Coulomb potential of the nucleus and is therefore characterized by a constant quantum number l. An external perturbation can break the symmetry of the system, which leads to a change in the quantum numbers themselves. A photon absorbed by a hydrogen atom can transfer an electron to another state with different values of quantum numbers. The table lists some quantum numbers used to describe atomic and nuclear states.
In addition to quantum numbers, which reflect the space-time symmetry of the microsystem, the so-called internal quantum numbers of particles play an important role. Some of them, such as spin and electric charge, are conserved in all interactions, others are not conserved in some interactions. So the strangeness quantum number, which is conserved in the strong and electromagnetic interactions, is not conserved in the weak interaction, which reflects the different nature of these interactions.
The atomic nucleus in each state is characterized by the total angular momentum. This moment in the rest frame of the nucleus is called nuclear spin.
The following rules apply to the kernel:
a) A is even J = n (n = 0, 1, 2, 3,...), i.e. an integer;
b) A is odd J = n + 1/2, i.e. half-integer.
In addition, one more rule has been experimentally established: for even-even nuclei in the ground state Jgs
= 0. This indicates the mutual compensation of the moments of nucleons in the ground state of the nucleus, which is a special property of the internucleon interaction.
The invariance of the system (hamiltonian) with respect to spatial reflection - inversion (replacement → -) leads to the parity conservation law and the quantum number parity R. This means that the nuclear Hamiltonian has the corresponding symmetry. Indeed, the nucleus exists due to the strong interaction between nucleons. In addition, the electromagnetic interaction plays a significant role in nuclei. Both of these types of interactions are invariant to spatial inversion. This means that nuclear states must be characterized by a certain parity value P, i.e., be either even (P = +1) or odd (P = -1).
However, weak forces that do not preserve parity also act between nucleons in the nucleus. The consequence of this is that a (usually insignificant) admixture of a state with the opposite parity is added to the state with a given parity. The typical value of such an impurity in nuclear states is only 10 -6 -10 -7 and in most cases can be ignored.
The parity of the nucleus P as a system of nucleons can be represented as the product of the parities of individual nucleons p i:
P \u003d p 1 p 2 ... p A ,
moreover, the parity of the nucleon p i in the central field depends on the orbital moment of the nucleon , where π i is the internal parity of the nucleon, equal to +1. Therefore, the parity of a nucleus in a spherically symmetric state can be represented as the product of the orbital parities of nucleons in this state:
Nuclear level diagrams usually indicate the energy, spin, and parity of each level. The spin is indicated by a number, and the parity is indicated by a plus sign for even levels and a minus sign for odd levels. This sign is placed to the right of the top of the number indicating the spin. For example, the symbol 1/2 + denotes an even level with spin 1/2, and the symbol 3 - denotes an odd level with spin 3.
Isospin of atomic nuclei. Another characteristic of nuclear states is isospin I. Core (A, Z) consists of A nucleons and has a charge Ze, which can be represented as the sum of nucleon charges q i , expressed in terms of projections of their isospins (I i) 3
is the projection of the isospin of the nucleus onto axis 3 of the isospin space.
Total isospin of the nucleon system A
All states of the nucleus have the value of the isospin projection I 3 = (Z - N)/2. In a nucleus consisting of A nucleons, each of which has isospin 1/2, isospin values are possible from |N - Z|/2 to A/2
|N - Z|/2 ≤ I ≤ A/2.
The minimum value I = |I 3 |. The maximum value of I is equal to A/2 and corresponds to all i directed in the same direction. It has been experimentally established that the higher the excitation energy of the nuclear state, the greater the value of isospin. Therefore, the isospin of the nucleus in the ground and low-excited states has a minimum value
I gs = |I 3 | = |Z - N|/2.
The electromagnetic interaction breaks the isotropy of the isospin space. The interaction energy of a system of charged particles changes during rotations in isospace, since during rotations the charges of particles change and in the nucleus part of the protons passes into neutrons or vice versa. Therefore, the actual isospin symmetry is not exact, but approximate.
Potential well. The concept of a potential well is often used to describe the bound states of particles. Potential hole - a limited region of space with a reduced potential energy of a particle. The potential well usually corresponds to the forces of attraction. In the area of action of these forces, the potential is negative, outside - zero.
The particle energy E is the sum of its kinetic energy T ≥ 0 and potential energy U (it can be both positive and negative). If the particle is inside the well, then its kinetic energy T 1 is less than the depth of the well U 0, the energy of the particle E 1 = T 1 + U 1 = T 1 - U 0 In quantum mechanics, the energy of a particle in a bound state can take only certain discrete values, i.e. there are discrete levels of energy. In this case, the lowest (main) level always lies above the bottom of the potential well. In order of magnitude, the distance Δ E between the levels of a particle of mass m in a deep well of width a is given by
ΔE ≈ ћ 2 / ma 2.
An example of a potential well is the potential well of an atomic nucleus with a depth of 40-50 MeV and a width of 10 -13 -10 -12 cm, in which nucleons with an average kinetic energy of ≈ 20 MeV are located at different levels.
Under such boundary conditions, the particle, being inside the potential well 0< x < l, не может выйти за ее пределы, т. е.
ψ(x) = 0, x ≤ 0, x ≥ L.
Using the stationary Schrödinger equation for the region where U = 0,
we obtain the position and energy spectrum of the particle inside the potential well.
For an infinite one-dimensional potential well, we have the following:
The wave function of a particle in an infinite rectangular well (a), the square of the modulus of the wave function (b) determines the probability of finding a particle at various points in the potential well.
The Schrödinger equation plays the same role in quantum mechanics as Newton's second law plays in classical mechanics.
The most striking feature of quantum physics turned out to be its probabilistic nature.
The probabilistic nature of the processes occurring in the microworld is a fundamental property of the microworld.
E. Schrödinger:
“The usual quantization rules can be replaced by other provisions that no longer introduce any “whole numbers”. Integrity is obtained in this case in a natural way by itself, just as the integer number of knots is obtained by itself when considering a vibrating string. This new representation can be generalized and, I think, is closely related to the true nature of quantization.
It is quite natural to associate the function ψ with some oscillatory process in the atom, in which the reality of electronic trajectories has recently been repeatedly questioned. At first, I also wanted to substantiate the new understanding of quantum rules using the indicated comparatively clear way, but then I preferred a purely mathematical method, since it makes it possible to better clarify all the essential aspects of the issue. It seems to me essential that quantum rules are no longer introduced as a mysterious " integer requirement”, but are determined by the need for the boundedness and uniqueness of some specific spatial function.
I do not consider it possible, until more complex problems are successfully calculated in a new way, to consider in more detail the interpretation of the introduced oscillatory process. It is possible that such calculations will lead to a simple coincidence with the conclusions of conventional quantum theory. For example, when considering the relativistic Kepler problem according to the above method, if we act according to the rules indicated at the beginning, a remarkable result is obtained: half-integer quantum numbers(radial and azimuth)…
First of all, it is impossible not to mention that the main initial impetus that led to the appearance of the arguments presented here was de Broglie's dissertation, which contains many deep ideas, as well as reflections on the spatial distribution of "phase waves", which, as shown by de Broglie, each time corresponds to periodic or quasi-periodic motion of an electron, if only these waves fit on the trajectories integer once. The main difference from de Broglie's theory, which speaks of a rectilinearly propagating wave, lies here in the fact that we are considering, if we use the wave interpretation, standing natural oscillations.
M. Laue:
“The achievements of quantum theory accumulated very quickly. It had a particularly striking success in its application to radioactive decay by the emission of α-rays. According to this theory, there is a "tunnel effect", i.e. penetration through a potential barrier of a particle whose energy, according to the requirements of classical mechanics, is insufficient to pass through it.
G. Gamov gave in 1928 an explanation of the emission of α-particles, based on this tunnel effect. According to Gamow's theory, the atomic nucleus is surrounded by a potential barrier, but α-particles have a certain probability of "stepping over" it. Empirically found by Geiger and Nettol, the relationship between the radius of action of an α-particle and the half-period of decay was satisfactorily explained on the basis of Gamow's theory.
Statistics. Pauli principle. The properties of quantum mechanical systems consisting of many particles are determined by the statistics of these particles. Classical systems consisting of identical but distinguishable particles obey the Boltzmann distribution
In a system of quantum particles of the same type, new features of behavior appear that have no analogues in classical physics. Unlike particles in classical physics, quantum particles are not only the same, but also indistinguishable - identical. One reason is that, in quantum mechanics, particles are described in terms of wave functions, which allow us to calculate only the probability of finding a particle at any point in space. If the wave functions of several identical particles overlap, then it is impossible to determine which of the particles is at a given point. Since only the square of the modulus of the wave function has physical meaning, it follows from the particle identity principle that when two identical particles are interchanged, the wave function either changes sign ( antisymmetric state), or does not change sign ( symmetrical state).
Symmetric wave functions describe particles with integer spin - bosons (pions, photons, alpha particles ...). Bosons obey Bose-Einstein statistics
An unlimited number of identical bosons can be in one quantum state at the same time.
Antisymmetric wave functions describe particles with half-integer spin - fermions (protons, neutrons, electrons, neutrinos). Fermions obey Fermi-Dirac statistics
The relationship between the symmetry of the wave function and spin was first pointed out by W. Pauli.
For fermions, the Pauli principle is valid - two identical fermions cannot simultaneously be in the same quantum state.
The Pauli principle determines the structure of the electron shells of atoms, the filling of nucleon states in nuclei, and other features of the behavior of quantum systems.
With the creation of the proton-neutron model of the atomic nucleus, the first stage in the development of nuclear physics can be considered completed, in which the basic facts of the structure of the atomic nucleus were established. The first stage began in the fundamental concept of Democritus about the existence of atoms - indivisible particles of matter. The establishment of the periodic law by Mendeleev made it possible to systematize atoms and raised the question of the reasons underlying this systematics. The discovery of electrons in 1897 by J. J. Thomson destroyed the concept of the indivisibility of atoms. According to Thomson's model, electrons are the building blocks of all atoms. The discovery by A. Becquerel in 1896 of the phenomenon of uranium radioactivity and the subsequent discovery by P. Curie and M. Sklodowska-Curie of the radioactivity of thorium, polonium and radium showed for the first time that chemical elements are not eternal formations, they can spontaneously decay, turn into other chemical elements . In 1899, E. Rutherford found that as a result of radioactive decay, atoms can eject α-particles from their composition - ionized helium atoms and electrons. In 1911, E. Rutherford, generalizing the results of the experiment of Geiger and Marsden, developed a planetary model of the atom. According to this model, atoms consist of a positively charged atomic nucleus with a radius of ~10 -12 cm, in which the entire mass of the atom and negative electrons rotating around it is concentrated. The size of the electron shells of an atom is ~10 -8 cm. In 1913, N. Bohr developed a representation of the planetary model of the atom based on quantum theory. In 1919, E. Rutherford proved that protons are part of the atomic nucleus. In 1932, J. Chadwick discovered the neutron and showed that neutrons are part of the atomic nucleus. The creation in 1932 by D. Ivanenko and W. Heisenberg of the proton-neutron model of the atomic nucleus completed the first stage in the development of nuclear physics. All the constituent elements of the atom and the atomic nucleus have been established.
1869 Periodic system of elements D.I. Mendeleev
By the second half of the 19th century, chemists had accumulated extensive information on the behavior of chemical elements in various chemical reactions. It was found that only certain combinations of chemical elements form a given substance. Some chemical elements have been found to have roughly the same properties while their atomic weights vary greatly. D. I. Mendeleev analyzed the relationship between chemical properties elements and their atomic weight and showed that the chemical properties of elements arranged as atomic weights increase are repeated. This served as the basis for the periodic system of elements he created. When compiling the table, Mendeleev found that the atomic weights of some chemical elements fell out of the regularity he had obtained, and pointed out that the atomic weights of these elements were determined inaccurately. Later precise experiments showed that the originally determined weights were indeed incorrect and the new results corresponded to Mendeleev's predictions. Leaving some places blank in the table, Mendeleev pointed out that there should be new yet undiscovered chemical elements and predicted their chemical properties. Thus, gallium (Z = 31), scandium (Z = 21) and germanium (Z = 32) were predicted and then discovered. Mendeleev left the task of explaining the periodic properties of chemical elements to his descendants. The theoretical explanation of Mendeleev's periodic system of elements, given by N. Bohr in 1922, was one of the convincing proofs of the correctness of the emerging quantum theory.
Atomic nucleus and the periodic system of elements
The basis for the successful construction of the periodic system of elements by Mendeleev and Logar Meyer was the idea that atomic weight can serve as a suitable constant for the systematic classification of elements. Modern atomic theory, however, has approached the interpretation of the periodic system without touching upon atomic weight at all. The place number of any element in this system and, at the same time, its chemical properties are uniquely determined by the positive charge of the atomic nucleus, or, what is the same, by the number of negative electrons located around it. The mass and structure of the atomic nucleus play no part in this; thus, at the present time, we know that there are elements, or rather types of atoms, which, with the same number and arrangement of outer electrons, have vastly different atomic weights. Such elements are called isotopes. So, for example, in a galaxy of zinc isotopes, the atomic weight is distributed from 112 to 124. On the contrary, there are elements with significantly different chemical properties that exhibit the same atomic weight; they are called isobars. An example is the atomic weight of 124 found for zinc, tellurium and xenon.
To determine a chemical element, one constant is sufficient, namely, the number of negative electrons located around the nucleus, since all chemical processes take place among these electrons.
Number of protons n 2
, located in the atomic nucleus, determine its positive charge Z, and thereby the number of external electrons that determine the chemical properties of this element; some number of neutrons n 1
enclosed in the same core, in total with n 2
gives its atomic weight
A=n 1
+n 2
. Conversely, the serial number Z gives the number of protons contained in the atomic nucleus, and from the difference between the atomic weight and the charge of the nucleus A - Z, the number of nuclear neutrons is obtained.
With the discovery of the neutron, the periodic system received some replenishment in the region of small serial numbers, since the neutron can be considered an element with an ordinal number equal to zero. In the region of high ordinal numbers, namely from Z = 84 to Z = 92, all atomic nuclei are unstable, spontaneously radioactive; therefore, it can be assumed that an atom with a nuclear charge even higher than that of uranium, if it can only be obtained, should also be unstable. Fermi and his collaborators recently reported on their experiments, in which, when uranium was bombarded with neutrons, the appearance of a radioactive element with the serial number 93 or 94 was observed. It is quite possible that the periodic system has a continuation in this region as well. It only remains to add that Mendeleev's ingenious foresight provided for the framework of the periodic system so broadly that each new discovery, remaining within its scope, further strengthens it.
The atomic nucleus, like other objects of the microworld, is a quantum system. This means that the 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 probability amplitudesψ(α,t). The square of the modulus of this function determines the probability density of detecting the system under study in a state with characteristic α – ρ (α,t) = |ψ(α,t)| 2. The argument of the wave function can be, for example, the coordinates of the particle.
The total probability is usually normalized 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 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 ("u. Sh.")
 |
(2.4) |
The operator is the Hermitian Hamilton operator ( Hamiltonian) systems. Together with initial condition on ψ(t), equation (2.4) determines 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 motion. The 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):
| ψ(α,t) = Eψ(α,t); ψ (α ) = Eψ( α ). |
(2.5) |
The last of the equations - stationary Schrödinger equation, which determines, in particular, the set (spectrum) of energies of the stationary system.
In the stationary states of a quantum system, in addition to energy, other physical quantities can also 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:
| [,] ≡ – = 0. | (2.6) |
1. Spectra of atomic nuclei
The quantum nature of atomic nuclei is manifested in the patterns of their excitation spectra (see, for example, Fig. 2.1). Spectrum in the region 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 level widths in this spectrum are equal to 0. Since each of the excited levels of the spectrum has a finite average lifetime τ, the level width 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 ∆t ∆E ≥ ћ :
The diagrams of the spectra of nuclei indicate the energies of the levels of the nucleus in MeV or keV, as well as the spin and parity of the states. The diagrams also indicate, if possible, the isospin of the state (since the diagrams of the spectra give level excitation energy, the energy of the ground state is taken as the origin). In the region of excitation energies 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.
