Quantum number

Quantum numbers describe values of conserved numbers in the dynamics of the quantum system. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons They often describe specifically the energies of electrons in atoms, but other possibilities include angular momentum, spin etc. In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin Since any quantum system can have one or more quantum numbers, it is a futile job to list all possible quantum numbers.

1 How Many Quantum Numbers?
2 Single electron in an atom
- 2.1 Quantum numbers with spin-orbit interaction
3 Elementary particles
4 References and external links
5 See also

How Many Quantum Numbers?

The question of how many quantum numbers are needed to describe any given system has no universal answer, although for each system one must find the answer for a full analysis of the system. The dynamics of any quantum system are described by a quantum Hamiltonian, H. In Quantum mechanics, the Hamiltonian H is the Observable corresponding to the Total energy of the system There is one quantum number of the system corresponding to the energy, i. e. , the eigenvalue of the Hamiltonian. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes There is also one quantum number for each operator O that commutes with the Hamiltonian (i. e. satisfies the relation OH = HO). These are all the quantum numbers that the system can have. Note that the operators O defining the quantum numbers should be independent of each other. Often there is more than one way to choose a set of independent operators. Consequently, in different situations different sets of quantum numbers may be used for the description of the same system.

Single electron in an atom

This section is not meant to be a full description of this problem. For that, see the article on the Hydrogen-like atom, Bohr atom, Schrödinger equation and the Dirac equation. A hydrogen-like atom is an Atom with one Electron and thus is Isoelectronic with Hydrogen. In Atomic physics, the Bohr model created by Niels Bohr depicts the Atom as a small positively charged nucleus surrounded by Electrons In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system In Physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 and provides

The most widely studied set of quantum numbers is that for a single electron in an atom: because it is not only useful in chemistry, being the basic notion behind the periodic table, valence and a host of other properties, but also because it is a solvable and realistic problem, and, as such, finds widespread use in textbooks. The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny Chemistry (from Egyptian kēme (chem meaning "earth") is the Science concerned with the composition structure and properties The periodic table of the chemical elements is a tabular method of displaying the Chemical elements Although precursors to this table exist its invention is In Chemistry, valence, also known as valency or valency number, is a measure of the number of Chemical bonds formed by the Atoms

In non-relativistic quantum mechanics the Hamiltonian of this system consists of the kinetic energy of the electron and the potential energy due to the Coulomb force between the nucleus and the electron. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons The kinetic energy of an object is the extra Energy which it possesses due to its motion Potential energy can be thought of as Energy stored within a physical system ---- Bold text Coulomb's law', developed in the 1780s by French physicist Charles Augustin de Coulomb, may be stated in scalar form The kinetic energy can be separated into a piece which is due to angular momentum, J, of the electron around the nucleus, and the remainder. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position The nucleus of an Atom is the very dense region consisting of Nucleons ( Protons and Neutrons, at the center of an atom Since the potential is spherically symmetric, the full Hamiltonian commutes with J². J² itself commutes with any one of the components of the angular momentum vector, conventionally taken to be J_z. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position These are the only mutually commuting operators in this problem; hence, there are three quantum numbers.

These are conventionally known as

The principal quantum number (n = 1, 2, 3, 4 . In Atomic physics, the principal quantum number symbolized as n is the first of a set of Quantum numbers (which includes the principal quantum . . ) denotes the eigenvalue of H with the J² part removed. This number therefore has a dependence only on the distance between the electron and the nucleus (ie, the radial coordinate, r). The average distance increases with n, and hence quantum states with different principal quantum numbers are said to belong to different shells.
The azimuthal quantum number (l = 0, 1 . The Azimuthal quantum number (or orbital angular momentum quantum number, second quantum number) symbolized as l (lower-case L is a Quantum number . . n−1) (also known as the angular quantum number or orbital quantum number) gives the orbital angular momentum through the relation $L^2 = \hbar^2 l(l+1)$ . In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position In chemistry, this quantum number is very important, since it specifies the shape of an atomic orbital and strongly influences chemical bonds and bond angles. An atomic orbital is a Mathematical function that describes the wave-like behavior of an electron in an atom A chemical bond is the physical process responsible for the attractive interactions between Atoms and Molecules and which confers stability to diatomic and polyatomic Molecular geometry or molecular structure is the three- Dimensional arrangement of the Atoms that constitute a Molecule. In some contexts, l=0 is called an s orbital, l=1, a p orbital, l=2, a d orbital and l=3, an f orbital.
The magnetic quantum number (m_l = −l, −l+1 . In Atomic physics, the magnetic quantum number is the third of a set of Quantum numbers (the Principal quantum number, the Azimuthal quantum number . . 0 . . . l−1, l) is the eigenvalue, $L_z = m_l \hbar$ . In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes This is the projection of the orbital angular momentum along a specified axis. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position

Results from spectroscopy indicated that up to two electrons can occupy a single orbital. Spectroscopy was originally the study of the interaction between Radiation and Matter as a function of Wavelength (λ However two electrons can never have the same exact quantum state nor the same set of quantum numbers according to Hund's Rules, which addresses the Pauli exclusion principle. In Atomic physics, Hund's rules refer to a simple set of rules used to determine which is the Term symbol that corresponds to the ground state of a multi- Electron The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925 A fourth quantum number with two possible values was added as an ad hoc assumption to resolve the conflict; this supposition could later be explained in detail by relativistic quantum mechanics and from the results of the renown Stern-Gerlach experiment. In Quantum mechanics, the Stern–Gerlach experiment, named after Otto Stern and Walther Gerlach, is an important 1922 experiment on the Deflection

The spin projection quantum number (m_s = −1/2 or +1/2), the intrinsic angular momentum of the electron. In Atomic physics, the spin quantum number is a Quantum number that parameterizes the intrinsic Angular momentum (or spin angular momentum or simply In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position This is the projection of the spin s=1/2 along the specified axis. In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin

To summarize, the quantum state of an electron is determined by its quantum numbers:

name	symbol	orbital meaning	range of values	value example
principal quantum number	$n \$	shell	$1 \le n \,\!$	$n=1,2,3...\,\!$
azimuthal quantum number (angular momentum)	$\ell \$	subshell	( $0 \le \ell \le n-1) \$	for $n=3\,\!$ : $\ell=0,1,2\,(s, p, d) \$
magnetic quantum number, (projection of angular momentum)	$m_\ell \$	energy shift	$-\ell \le m_\ell \le \ell \$	for $\ell=2 \$ : $m_\ell=-2,-1,0,1,2\,\!$
spin projection quantum number	$m_s\,\!$	spin	$- \begin{matrix} \frac{1}{2} \end{matrix} , \begin{matrix} \frac{1}{2} \end{matrix} \$	for an electron, either: $- \begin{matrix} \frac{1}{2} \end{matrix} , \begin{matrix} \frac{1}{2} \end{matrix} \$

Example: The quantum numbers used to refer to the outermost valence electron of the Fluorine (F) atom, which is located in the 2p atomic orbital, are; n = 2, l = 1, m_l = 1, or 0, or −1, m_s = −1/2 or 1/2. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position In Chemistry, valence, also known as valency or valency number, is a measure of the number of Chemical bonds formed by the Atoms The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J Fluorine, fluorum meaning "to flow" is the Chemical element with the symbol F and Atomic number 9 History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny An atomic orbital is a Mathematical function that describes the wave-like behavior of an electron in an atom

Note that molecular orbitals require totally different quantum numbers, because the Hamiltonian and its symmetries are quite different. In Chemistry, a molecular orbital (or MO) is a region in which an Electron may be found in a Molecule.

Quantum numbers with spin-orbit interaction

For more details on this topic, see Clebsch-Gordan coefficients. In Physics, the Clebsch-Gordan coefficients are sets of numbers that arise in Angular momentum coupling under the laws of Quantum mechanics.

When one takes the spin-orbit interaction into consideration, l, m and s no longer commute with the Hamiltonian, and their value therefore changes over time. In Quantum physics, the spin-orbit interaction (also called spin-orbit effect or spin-orbit coupling) is any interaction of a particle's spin In Mathematics, commutativity is the ability to change the order of something without changing the end result Thus another set of quantum numbers should be used. This set includes

The total angular momentum quantum number (j = 1/2,3/2 . The Azimuthal quantum number (or orbital angular momentum quantum number, second quantum number) symbolized as l (lower-case L is a Quantum number . . n−1/2) gives the total angular momentum through the relation $J^2 = \hbar^2 j(j+1)$ . In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position
The projection of the total angular momentum along a specified axis (m_j = -j,-j+1. The Azimuthal quantum number (or orbital angular momentum quantum number, second quantum number) symbolized as l (lower-case L is a Quantum number . . j), which is analogous to m, and satisfies $m j = m l + m s$ .
Parity. In Physics, a parity transformation (also called parity inversion) is the flip in the sign of one Spatial Coordinate. This is the eigenvalue under reflection, and is positive (i. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes e. +1) for states which came from even l and negative (i. e. -1) for states which came from odd l. The former is also known as even parity and the latter as odd parity

For example, consider the following eight states, defined by their quantum numbers:

(1) l = 1, m_l = 1, m_s = +1/2
(2) l = 1, m_l = 1, m_s = -1/2
(3) l = 1, m_l = 0, m_s = +1/2
(4) l = 1, m_l = 0, m_s = -1/2
(5) l = 1, m_l = -1, m_s = +1/2
(6) l = 1, m_l = -1, m_s = -1/2
(7) l = 0, m_l = 0, m_s = +1/2
(8) l = 0, m_l = 0, m_s = -1/2

The quantum states in the system can be described as linear combination of these eight states. In Quantum physics, a quantum state is a mathematical object that fully describes a quantum system. However, in the presence of spin-orbit interaction, if one wants to describe the same system by eight states which are eigenvectors of the Hamiltonian (i. In Quantum physics, the spin-orbit interaction (also called spin-orbit effect or spin-orbit coupling) is any interaction of a particle's spin In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes e. each represents a state which does not mix with others over time), we should consider the following eight states:

j = 3/2, m_j = 3/2, odd parity (coming from state (1) above)
j = 3/2, m_j = 1/2, odd parity (coming from states (2) and (3) above)
j = 3/2, m_j = -1/2, odd parity (coming from states (4) and (5) above)
j = 3/2, m_j = -3/2, odd parity (coming from state (6))
j = 1/2, m_j = 1/2, odd parity (coming from state (2) and (3) above)
j = 1/2, m_j = -1/2, odd parity (coming from states (4) and (5) above)
j = 1/2, m_j = 1/2, even parity (coming from state (7) above)
j = 1/2, m_j = -1/2, even parity (coming from state (8) above)

Elementary particles

For a more complete description of the quantum states of elementary particles see the articles on the standard model and flavour (particle physics). The Standard Model of Particle physics is a theory that describes three of the four known Fundamental interactions together with the Elementary particles In Particle physics, flavour or flavor (see spelling differences) is a Quantum number of Elementary particles related to their

Elementary particles contain many quantum numbers which are usually said to be intrinsic to them. However, it should be understood that the elementary particles are quantum states of the standard model of particle physics, and hence the quantum numbers of these particles bear the same relation to the Hamiltonian of this model as the quantum numbers of the Bohr atom does to its Hamiltonian. In Quantum physics, a quantum state is a mathematical object that fully describes a quantum system. The Standard Model of Particle physics is a theory that describes three of the four known Fundamental interactions together with the Elementary particles Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them In Atomic physics, the Bohr model created by Niels Bohr depicts the Atom as a small positively charged nucleus surrounded by Electrons In other words, each quantum number denotes a symmetry of the problem. It is more useful in field theory to distinguish between spacetime and internal symmetries. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS

Typical quantum numbers related to spacetime symmetries are spin (related to rotational symmetry), the parity, C-parity and T-parity (related to the Poincare symmetry of spacetime). Spacetime symmetries refers to aspects of Spacetime that can be described as exhibiting some form of Symmetry. In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin In Physics, a parity transformation (also called parity inversion) is the flip in the sign of one Spatial Coordinate. In Physics, C-symmetry means the symmetry of physical laws under a charge -conjugation transformation. T Symmetry is the symmetry of physical laws under a Time reversal transformation &mdash T t \mapsto -t In Physics and Mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS Typical internal symmetries are lepton number and baryon number or the electric charge. In High energy physics, the lepton number is the number of Leptons minus the number of antileptons In Particle physics, the baryon number is an approximate conserved Quantum number of a system Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction. For a full list of quantum numbers of this kind see the article on flavour. In Particle physics, flavour or flavor (see spelling differences) is a Quantum number of Elementary particles related to their

It is worth mentioning here a minor but often confusing point. Most conserved quantum numbers are additive. Thus, in an elementary particle reaction, the sum of the quantum numbers should be the same before and after the reaction. However, some, usually called a parity, are multiplicative; ie, their product is conserved. All multiplicative quantum numbers belong to a symmetry (like parity) in which applying the symmetry transformation twice is equivalent to doing nothing. These are all examples of an abstract group called Z₂. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element

References and external links

General principles

Dirac, Paul A. M. (1982). Principles of quantum mechanics. Oxford University Press. ISBN 0-19-852011-5.

Atomic physics

Particle physics

Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed. ). Prentice Hall. ISBN 0-13-805326-X.
Halzen, Francis and Martin, Alan D. (1984). QUARKS AND LEPTONS: An Introductory Course in Modern Particle Physics. John Wiley & Sons. ISBN 0-471-88741-2.

The particle data group

Dictionary

-noun

(quantum mechanics) One of certain integers or half-integers that specify the state of a quantum mechanical system (such as an electron in an atom).

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