Physics of Atoms and Molecules

§ 1 The hydrogen atom in quantum mechanics

Consider a system consisting of a fixed nucleus with charge Ze (Z - integer), and moving around the electron. When Z > 1, such a system is called hydrogen-like system, with Z = 1 is a hydrogen atom.

The potential energy of the electron-nucleus



Stationary Schrodinger equation:


Because field in which the electron moves is centrally symmetric, then Ψ is usually written in the form of spherical functions.


Energy eigenvalue (1) have the form:


Thus, the energy spectrum of the hydrogen atom is discrete. For Е< 0 electron inside a hyperbolic "potential well" and he is bound, with Е > 0 - electron free. Ei - ionization energy - energy that must be expended to detach an electron from the nucleus. For hydrogen


n = 1 - ground state; n> 1 excited state.

  • Quantum numbers.

The eigenfunctions contains three integer parameter n, ℓ, m, called quantum numbers. n - principal quantum number determines the energy levels of the electron..

n = 1, 2, 3,...

            Solutions of the Schrödinger equation, it follows that the angular momentum (mechanical orbital angular momentum, which is the result of motion of the electron in the orbit) quantized, i.e. can take discrete values


where - the orbital quantum number, which determines the unit of angular momentum and takes the value

                                               = 0, 1 ... (n-1)          in all n values.

            Solutions of the Schrödinger equation that the vector  of the angular momentum of the electron can only have such orientation, for which its projection LеZ on  direction Z of the external magnetic field takes the quantized values ??are multiples

where m- magnetic quantum number, which determines the projection of the angular momentum on the direction z of the external magnetic field at a given l can take values:

                                               m? = 0; ±1; ±2; ...±l         , in all 2+1  values.

i.e. angular momentum vector can have 2+1 direction in space.

            The presence of the quantum number m leads to the fact that the magnetic field energy levels with principal quantum number n split into 2+1 sublevels - Zeeman effect (Dutch physicist) (1896). Splitting of the energy levels in an electric field - called the Stark effect (German physicist).

            The states with the same energy are called degenerate, and the number of different states of whatever the value of energy is called degeneracy. The degree of degeneracy is calculated from the possible values of ℓ and m. Each of the n values ??of the quantum number l corresponds 2ℓ +1 values ??of the quantum number m then the degeneracy: without a field



            The probability of finding an electron in different parts of the atom is different. Electron in its motion as it is "spread" over the entire volume, forming an electron cloud density (thickness) which characterizes the probability of finding an electron at various points of the atom. The quantum numbers n and characterize the size and shape of the electron cloud, and the quantum number m characterizes the orientation of the electron cloud in space.
In quantum physics the following symbols are used with a variety of electron states of angular momentum = 0 - S-state; = 1 - p - state; = 2 - d - state; = 3 - f - state, etc.

  • The spectrum of the hydrogen atom.

Quantum numbers n, , m can more fully describe the spectrum emission (absorption) of hydrogen produced in the Bohr theory.

In quantum mechanics, the selection rules are introduced to limit the number of possible transitions of electrons in the atom associated with the emission and absorption of light. An electron can only such transitions, for which:

1) changes in the orbital quantum number satisfies

                                                           Δ = ±1

2) The change of the magnetic quantum number satisfies the condition                                                    Δm = 0; ±1

However, there are transitions with Δl = 2, so-called quadrupole transitions.
Then a series of Lyman describes the transition;

                                                           np1S      (n = 2, 3...)

Balmer series

                                                           np2S;    nS2р;    nd2р    (n = 3, 4)

§ 2 the electron spin.

Spin quantum number.

An investigation of the alkali metals with instruments with high resolving power showed that each line of the spectrum is a double (doublet).

Structure of the spectrum, reflecting the line splitting into components called fine structure. Sophisticated lines, consisting of several components, called multiplets - doublets (split into two lines), triplet (three), quartet (4), the quintet (5). but there may be single line - singlet.
The splitting of spectral lines due to the fact that the splitting of the energy levels of the electron. Gaudmsit and Uhlenbeck in 1925 assumed that the electron has an intrinsic angular momentum - Ls, not involving the movement of electrons in space. This angular momentum of its own has been named spin.

Spin - it is an internal inherent property of the electron, such as charge and mass. If the electron is assigned its own mechanical angular momentum (spin) Ls, then an intrinsic magnetic moment рms.

From quantum mechanics;


where S - the spin quantum number.

The projection of the spin can be either 2S +1. Because in the Stern-Gerlach observed only two orientations, then 2S + 1 = 2 S = 1/2  -  spin.

            The projection of the spin in the direction of the field is determined

                                               Lsz = ms

ms magnetic spin quantum number: ms = ±1/2 thus electron in an atom is described by four quantum numbers:

n,ℓ, m, ms.

§ 3 of the Pauli principle.

The distribution of electrons in an atom the states.

The spontaneous (spontaneous) transitions with radiation atoms of transition to the normal or less excited. For atoms with one electron, this means that the electron tends to the ground state. The same should happen in many-electron atoms, i.e. all of the electrons will tend to occupy the lowest of all possible levels.

In 1925 Pauli set of quantum-mechanical law, called the Pauli exclusion principle and the principle of exclusion. The Pauli principle determines the filling of the electronic levels of the electron.
Pauli exclusion principle:

            In the atom, no two electrons with the same set of quantum numbers.

Justification of the Pauli principle is related to the position of the indistinguishability of identical particles. For example, all of the electrons in atoms, molecules or crystals can not be distinguished from each other - they have the same n, ℓ,m, S. Swapping of two electrons in an atom can not change its state.

Pauli principle governs all particles with half-integer spin S = 1/2 - are called fermions obey Fermi - Dirac (electron, proton, neutron, etc.)

Particles with zero or integer spins and obey Bose-Einstein statistics are called bosons (e.g. p meson photon).

For a given value n may n2 different states differ values ??of ℓ and m. ms can take two values ?? (±1/2).The number of items in the states determined by the principal quantum number, as well


Aggregate of electrons in a multielectron atom having the same principal quantum number n, called the electron shell. In each of the shells of electrons are distributed by subshells corresponding to the given ℓ. Since the orbital quantum number ranges from 0 to n - 1, the number of subshells equal to the ordinal number n shell. The number of elements in a subshell by the magnetic  and ms quantum numbers: the number of elements in the subshell with the same ℓ equal to 2(2ℓ+1)

The principal quantum number















symbol of shell














The maximum number of electrons in the shell














Orbital quantum number ?


0     1

0   1   2

0   1   2   3  

0   1   2   3   4 









symbol of subshell

1 S

2 S 2p

3S 3p 3d

4S 4p 4d 4f

5s 5p 5d 5f 5g









The maximum number of electrons in the subshell


2     6

2    6   10

2   6   10   14

2   6  10 14 18











§ 4 Mendeleev periodic system
(§ 37 Savelyev) - self study.

§ 5 The structure of molecules.
Chemical bonds. Molecular spectra

Molecule is the smallest particle of a homogeneous substance, which has its main chemical properties.
Molecules are composed of the same or different atoms bonded together interatomic chemical bonds. Chemical bonds due to the interaction outer valence electrons. Most often in the molecules are two types of communication: ionic and covalent.

            Ionic bond (NaCl, KBr) at the expense of electric interaction forces arising between the outer electrons of the atoms when the electron of one atom to another, i.e. in the formation of positive and negative ions.

            Covalent bond (H2 ,C2 , Cl2 ,CO) is the socialization of the valence electrons in two adjacent atoms (spins of the valence electrons must be antiparallel). The covalent bond is due to the principle of indistinguishability of identical particles (such as electrons in hydrogen H2). Indistinguishability of particles leads to the exchange interaction: an


electron in each atom of the hydrogen molecule spends some time in the nucleus of another atom and, therefore, communicates the two atoms that form the molecule. (alternately, each electron is one that belongs to a different kernel - the exchange of electrons).

            As in the spectra of atoms, a separate spectral line of the molecular spectrum is the result of changes in the energy of the molecule.

The total energy of a molecule is:

                                   W= WTransl.+ Wel. + Wvidr + Wrot + Wnucl

WTransl. - Translational energy of the center of mass of the molecule.

Wel. - Energy of the electrons in the atoms of the molecule:

Wvidr - The energy of the vibrational motion of the nuclei of atoms in the molecule, about their equilibrium positions;

Wrot - Energy of the rotational motion of the molecule as a whole.

Wnucl - The energy of the atomic nuclei in the molecule.

WTransl energy not quantized and its changes can not lead to the molecular spectrum, the influence Wnucl the molecular spectrum can be ignored. Thus, the energy of the molecule W ', which determines the change of the molecular spectrum:

                                      W’ = Wel. + Wvidr + Wrot.


        Wel.  : Wvidr : Wrot =

where m - mass of the electron, М - quantity of the order the masses of the nuclei of atoms in a molecule. m/М 10-5 10-3; Wel.    110 eV; Wvidr 10-210-1 eV; Wrot 10-510-3 eV.

 Spectrum of a molecule has the form:



The molecule is observed:

1) In the field of FIR (l~0,11mm)  rotational lines of the spectrum - the transition from one rotational level to another.

2) in the IR region (l~110 μm) lines of the vibrational spectrum - the transition between the vibrational levels.

3) In the visible and ultraviolet region of the spectrum - the electronic spectra - the transition between different electronic energy levels are also possible

4) electronic-vibronic transitions and

5) vibrational-rotational.

Typical molecular spectra - striped, a collection of more or less narrow bands from UV to FIR regions. The structure of molecular spectra is different for different molecules, and with the number of atoms in a molecule complex. Infrared and Raman spectra of polyatomic molecules have only diatomic without.

§ 6 Absorption.
Spontaneous and stimulated emission

In normal conditions (in the absence of external influences), most of the electrons in the atoms are at the lowest level Е1 unexcited, ie atom has a minimum reserve of internal energy, the other levels of Е2, Е3....Еn corresponding to the excited states have a minimum population of electrons, or even free. If the atom is in its ground state with Е1, then under the action of external radiation induced transition can take place in the excited state with Е2. The probability of such transitions is proportional to the density of radiation that causes these transitions. An atom in the excited state 2, can over time spontaneously (without external influence) to move to a state with lower energy, giving the excess energy in the form of electromagnetic radiation, ie emitting a photon.

            The process of photon emission by an excited atom with no external influences is called spontaneous emission. The greater the probability of spontaneous transitions, the lower the average lifetime of an atom in an excited state. Because spontaneous transitions are mutually not linked, the spontaneous emission is not coherent.

If the atom in an excited state 2, the action of external radiation with a frequency that satisfies hn = E2 - E1, then a forced (induced) to the ground state with the emission of one photon with the same energy hn = E2 - E1. With such transition is emitted by the atom in addition to the photon, the action of which there was a transition. Radiation, which occurs as a result of external exposure is called induced. Thus, in the process of stimulated emission of two photons are involved: the primary photon, causing the emission of radiation by an excited atom and the secondary photon emitted by the atom. Secondary photons are indistinguishable from virgin.

            Einstein and Dirac proved the identity of the stimulated emission radiation forcing: they have the same phase, frequency, polarization and direction of propagation. Stimulated emission strictly coherent with compelling light.

The emitted photons are moving in the same direction and by meeting other excited atoms, stimulating further induced transitions and the number of photons increases avalanche. However, along with the induced radiation absorption will occur. Therefore, to enhance the incident radiation is necessary that the number of photons in the stimulated emission (which is proportional to the population of excited states) exceeded the number of absorbed photons. In the system of atoms are in thermodynamic equilibrium, the absorption will dominate the stimulated emission, i.e. incident radiation passing through matter will be weakened.           

That the medium intensified radiation incident upon it to create a non-equilibrium state of the system in which the number of atoms in the excited state than in the core. These states are called states with population inversion. Process of creating a non-equilibrium state of matter is called pumping. Pumping can be achieved by optical, electrical, and other means.
In environments with inverted population stimulated emission may exceed the absorption, i.e. incident light passing through the medium will increase (these are called active environment). For these media in the Bouguer’s law I = I0e-
ax , the absorption coefficienta - negative.


§ 7 Lasers - optical quantum generators

In the early 60's was created quantum generator optical range - laser “Light light amplification by stimulated emission of radiation. The properties of laser radiation: high monochromaticity (extremely high frequency of the light), acute spatial orientation, a huge spectral brightness.

According to the laws of quantum mechanics, the energy of an electron in an atom is not arbitrary: it can only have a certain (discrete) set of value Е1, Е2, Е3... Еn, called energy levels. These values ??are different for different atoms. Set of allowed values ??of energy is called the energy spectrum of the atom. In normal conditions (in the absence of external influences), most of the electrons in the atoms resides at the lowest excited state Е1, i.e. atom has a minimum reserve of internal energy, and the remaining levels of Е2, Е3.....Еn corresponding to the higher the energy of the atom are called excited

            At transition of an electron from one energy level to another atom can emit or absorb electromagnetic waves with a frequency nmn = (Еm - Еn)h,

where h - Planck's constant (h = 6.62 · 10-34 J·s);

Еn - final, Еm - beginner level.

An excited atom can give my some excess energy received from an external source or acquired by him as a result of the thermal motion of electrons in two different ways.
Any excited state of the atom is unstable, and there is always a probability of spontaneous transition to a lower energy state and emit a photon of electromagnetic radiation. Such a transition is called spontaneous. He is irregular, chaotic. All the usual sources produce light by spontaneous emission.

This is the first mechanism of emission (electromagnetic radiation). In the considered two-level system of light emission is no amplification of the radiation can not be achieved. The absorbed energy hn is released as a photon with the same energy hn can speak of thermodynamic equilibrium: the processes of excitation of the atoms in the gas is always balanced by the reverse process of emission.

§8 A three-level scheme

In the atoms of matter in thermodynamic equilibrium at each successive level of the excited electron is less than the previous. If an effect on the system of the exciting radiation with a frequency that is in resonance with the transition between levels 1 and 3 (schematically 13), the atoms will absorb this radiation and go from level 1 to level 3. If the radiation intensity is high enough, the number of atoms that have fallen to the level of 3, can be very significant and we are in violation of the equilibrium distribution of the level populations, increase the population of level 3 and reduce, therefore, the population of level 1.

                        From the top of the third level, transitions 31 and 32. It turned out that transition 31 leads to the emission of energy Е31=hn3-1, a transition 32 is not a radiative: it leads to the settlement of the "top" of the intermediate level 2 (This second level is called metastable, and in the end it will be more atoms than on the first. Since the atoms in the layer 2 comes from the ground level 1 through the upper state 3, and back to the ground level back with "great delay", the level 1 "depleted."

            As a result, there is an inversion, i.e. inverse inverted population distribution levels. Population inversion of the energy levels generated intense secondary radiation, called the pump radiation and ultimately leads to induced (forced) the photon propagation in population inversion.

            As in any generator in the laser for lasing need feedback. In a laser, the feedback is implemented using mirrors. Amplifying (active) medium placed between two mirrors - flat or more concave. One mirror is solid, the other partially transparent.

"Seed" for the generation process is the spontaneous emission of a photon. As a result of the movement of the photon in the environment, it creates an avalanche of photons moving in the same direction. After reaching the semi-transparent mirror, avalanche partly reflected and partly pass through the mirror out. After reflection from the right mirror wave goes back, continuing to grow. After going the distance l, it reaches the left mirror, reflected again rushes to the right mirror.

                        Such conditions are created for axial waves. Quanta of other directions are not able to pick up a significant portion of the stored energy in the active medium.
Coming out of the laser wave is nearly flat front, a high degree of spatial and temporal coherence of the entire cross section of the beam.

In lasers as the active medium using various gases and gas mixtures (gas lasers), crystals and glasses doped with certain ions (solid state lasers), semiconductors (semiconductor lasers).
Excitation method (in the pump) are dependent on the type of the active medium. This is either a way to excitation energy transfer in a collision of particles in a gas-discharge plasma (gas lasers) or energy transfer radiation-active centers incoherent light from special sources (optical pumping of solid-state lasers), or injection of nonequilibrium carriers in p-n - the transition or excitation by an electron beam or optical pumping (semiconductor lasers).

                        There are now very many different lasers, which give radiation over a wide wavelength range (200÷2·104 nm). Lasers operate with a very short duration of the light pulse t 1·10-12 s, can give and continuous radiation. Power density of the laser radiation is of the order 1010 W/sm2 (the intensity of the sun is only 7·103 W/sm2).


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