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Encyclopedia > Quantum number

Quantum numbers describe values of conserved quantity in the dynamics of the quantum system. They often describe specifically the energies of electrons in atoms, but other possibilities include angular momentum, spin etc. Since any quantum system can have one or more quantum numbers, it is a futile job to list all possible quantum numbers. In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ... Fig. ... e- redirects here. ... Properties In chemistry and physics, an atom (Greek á¼„Ï„Î¿Î¼Î¿Ï‚ or Ã¡tomos meaning indivisible) is the smallest particle still characterizing a chemical element. ... This gyroscope remains upright while spinning due to its angular momentum. ... The terms spin and SPIN have several meanings, including those primarily discussed as spinning: For spin in sub-atomic physics, see spin (physics) For the stalled aircraft maneuver or any of several forms of loss of control in aircraft, see spin (flight) For the periodical, see Spin Magazine For the...

## How many quantum numbers? GA_googleFillSlot("encyclopedia_square");

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. There is one quantum number of the system corresponding to the energy, i.e., the eigenvalue of the Hamiltonian. 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. The quantum Hamiltonian is the physical state of a system, which may be characterized as a ray in an abstract Hilbert space (or, in the case of ensembles, as a trace class operator with trace 1). ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...

## 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.

The most widely studied set of quantum numbers is that for a single electron in an atom: not only because it is 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. Hydrogen-like atoms are atoms with one single electron. ... The Bohr model of the atom The Bohr Model is a physical model that depicts the atom as a small positively charged nucleus with electrons in orbit at different levels, similar in structure to the solar system. ... For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ... In physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 and provides a description of elementary spin-Â½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity. ... e- redirects here. ... Properties In chemistry and physics, an atom (Greek á¼„Ï„Î¿Î¼Î¿Ï‚ or Ã¡tomos meaning indivisible) is the smallest particle still characterizing a chemical element. ... Chemistry - the study of atoms, made of nuclei (conglomeration of center particles) and electrons (outer particles), and the structures they form. ... The periodic table of the chemical elements is a tabular method of displaying the chemical elements, first devised by English analytical chemist John Newlands in 1863. ... In chemistry, valence, also known as valency or valency number, is a measure of the number of chemical bonds formed by the atoms of a given element. ...

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. 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. 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 Jz. These are the only mutually commuting operators in this problem; hence, there are three quantum numbers. Fig. ... The kinetic energy of an object is the extra energy which it possesses due to its motion. ... Potential energy is the energy available within a physical system due to an objects position in conjunction with a conservative force which acts upon it (such as the gravitational force or Coulomb force). ... In physics, Coulombs law is an inverse-square law indicating the magnitude and direction of electrical force that one stationary, electrically charged substance of small volume (ideally, a point source) exerts on another. ... The nucleus of an atom is the very small dense region, of positive charge, in its centre consisting of nucleons (protons and neutrons). ...

These are conventionally known as

• The principal quantum number (n = 1, 2, 3, 4 ...) 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 ... 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 chemistry, this quantum number is very important, since it specifies the shape of an atomic orbital and strongly influences chemical bonds and bond angles. 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 (ml = −l, −l+1 ... 0 ... l−1, l) is the eigenvalue, $L_z = m_l hbar$. This is the projection of the orbital angular momentum along a specified axis.

Results from spectroscopy indicated that up to two electrons can occupy a single orbital. However two electrons can never have the same exact quantum state. A fourth quantum number with two possible values was added to as an ad hoc assumption to resolve the conflict. It could later be explained in detail by relativistic quantum mechanics. In atomic physics, the principal quantum number symbolized as n is the first quantum number of an atomic orbital. ... The Azimuthal quantum number (or orbital angular momentum quantum number) symbolized as l (lower-case L) is a quantum number for an atomic orbital which determines its orbital angular momentum. ... This gyroscope remains upright while spinning due to its angular momentum. ... In chemistry, an atomic orbital is the region in which an electron may be found around a single atom. ... A chemical bond is the physical process responsible for the attractive interactions between atoms and molecules, and that which confers stability to diatomic and polyatomic chemical compounds. ... Geometry of the water molecule Molecules have fixed equilibrium geometries--bond lengths and angles--that are dictated by the laws of quantum mechanics. ... By virtue of its charge and spin motion, an electron develops a magnetic field. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... This gyroscope remains upright while spinning due to its angular momentum. ... Extremely high resolution spectrogram of the Sun showing thousands of elemental absorption lines (fraunhofer lines) Spectroscopy is the study of the interaction between radiation (electromagnetic radiation, or light, as well as particle radiation) and matter. ...

• The spin quantum number (ms = −1/2 or +1/2), the intrinsic angular momentum of the electron. This is the projection of the spin s=1/2 along the specified axis.

To summarize, the quantum state of an electron is determined by the quantum numbers: In atomic physics, the spin quantum number is a quantum number that parametrizes the intrinsic angular momentum (or spin angular momentum, or simply spin) of a given particle. ... The terms spin and SPIN have several meanings, including those primarily discussed as spinning: For spin in sub-atomic physics, see spin (physics) For the stalled aircraft maneuver or any of several forms of loss of control in aircraft, see spin (flight) For the periodical, see Spin Magazine For the...

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 quantum number $m_s,!$ spin $- begin{matrix} frac{1}{2} end{matrix} , begin{matrix} frac{1}{2} end{matrix}$ always only: $- begin{matrix} frac{1}{2} end{matrix} , begin{matrix} frac{1}{2} end{matrix}$

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, ml = 1, or 0, or −1, ms = −1/2 or 1/2. This gyroscope remains upright while spinning due to its angular momentum. ... This gyroscope remains upright while spinning due to its angular momentum. ... In chemistry, valence, also known as valency or valency number, is a measure of the number of chemical bonds formed by the atoms of a given element. ... e- redirects here. ... General Name, Symbol, Number fluorine, F, 9 Chemical series halogens Group, Period, Block 17, 2, p Appearance Yellowish brown gas Atomic mass 18. ... Properties In chemistry and physics, an atom (Greek á¼„Ï„Î¿Î¼Î¿Ï‚ or Ã¡tomos meaning indivisible) is the smallest particle still characterizing a chemical element. ... In chemistry, an atomic orbital is the region in which an electron may be found around a single atom. ...

Note that molecular orbitals require totally different quantum numbers, because the Hamiltonian and its symmetries are quite different. In quantum chemistry, molecular orbitals are the statistical states electrons can have within molecules. ... In physics, Hamiltonian has distinct but closely related meanings. ...

### Quantum numbers with spin-orbit interaction

For more details on this topic, see Clebsch-Gordan coefficients.

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. Thus another set of quantum numbers should be used. This set includes This article may be too technical for most readers to understand. ... Spin-orbit interaction, in quantum mechanics, is a shift in energy levels due to the potential energy of the spin magnetic moment of the electron in the magnetic field it feels as it moves through the electric field of the nucleus. ... A map or binary operation from a set to a set is said to be commutative if, (A common example in school-math is the + function: , thus the + function is commutative) Otherwise, the operation is noncommutative. ... In physics, Hamiltonian has distinct but closely related meanings. ...

For example, consider the following eight states, defined by their quantum numbers: The Azimuthal quantum number (or orbital angular momentum quantum number) symbolized as l (lower-case L) is a quantum number for an atomic orbital which determines its orbital angular momentum. ... This gyroscope remains upright while spinning due to its angular momentum. ... The Azimuthal quantum number (or orbital angular momentum quantum number) symbolized as l (lower-case L) is a quantum number for an atomic orbital which determines its orbital angular momentum. ... In physics, a parity transformation (also called parity inversion) is the simultaneous flip in the sign of all spatial coordinates: A 3Ã—3 matrix representation of P would have determinant equal to â€“1, and hence cannot reduce to a rotation. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...

• (1) l = 1, ml = 1, ms = +1/2
• (2) l = 1, ml = 1, ms = -1/2
• (3) l = 1, ml = 0, ms = +1/2
• (4) l = 1, ml = 0, ms = -1/2
• (5) l = 1, ml = -1, ms = +1/2
• (6) l = 1, ml = -1, ms = -1/2
• (7) l = 0, ml = 0, ms = +1/2
• (8) l = 0, ml = 0, ms = -1/2

The quantum states in the system can be described as linear combination of these eight states. 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.e. each represents a state which does not mix with others over time), we should consider the following eight states: A quantum state is any possible state in which a quantum mechanical system can be. ... Spin-orbit interaction, in quantum mechanics, is a shift in energy levels due to the potential energy of the spin magnetic moment of the electron in the magnetic field it feels as it moves through the electric field of the nucleus. ... In linear algebra, the eigenvectors (from the German eigen meaning own) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... In physics, Hamiltonian has distinct but closely related meanings. ...

• j = 3/2, mj = 3/2, even parity (coming from state (1) above)
• j = 3/2, mj = 1/2, even parity (coming from states (2) and (3) above)
• j = 3/2, mj = -1/2, even parity (coming from states (4) and (5) above)
• j = 3/2, mj = -3/2, even parity (coming from state (6))
• j = 1/2, mj = 1/2, even parity (coming from state (2) and (3) above)
• j = 1/2, mj = -1/2, even parity (coming from states (4) and (5) above)
• j = 1/2, mj = 1/2, odd parity (coming from state (7) above)
• j = 1/2, mj = -1/2, odd 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 Fundamental Particles and Interactions For the Standard Model in Cryptography, see Standard Model (cryptography). ... Flavour (or flavor) is a quantum number of elementary particles related to their weak interactions. ...

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 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. A quantum state is any possible state in which a quantum mechanical system can be. ... The Standard Model of Fundamental Particles and Interactions For the Standard Model in Cryptography, see Standard Model (cryptography). ... Thousands of particles explode from the collision point of two relativistic (100 GeV per ion) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... The Bohr model of the atom The Bohr Model is a physical model that depicts the atom as a small positively charged nucleus with electrons in orbit at different levels, similar in structure to the solar system. ... Field theory (mathematics), the theory of the algebraic concept of field. ... In physics, spacetime is any mathematical model that combines space and time into a single construct called the space-time continuum. ... ...

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). Typical internal symmetries are lepton number and baryon number or the electric charge. For a full list of quantum numbers of this kind see the article on flavour. The role of symmetry in physics is important, for example, in simplifying solutions to many problems. ... In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. ... In physics, a parity transformation (also called parity inversion) is the simultaneous flip in the sign of all spatial coordinates: A 3Ã—3 matrix representation of P would have determinant equal to â€“1, and hence cannot reduce to a rotation. ... C parity or charge parity is a multiplicative quantum number of some particles that describes its behavior under a symmetry operation of charge conjugation (see C-symmetry). ... T-symmetry is the symmetry of physical laws under a time-reversal transformationâ€” The universe is not symmetric under time reversal, although in restricted contexts one may find this symmetry. ... Poincare symmetry is the full symmetry of special relativity and includes translations (ie, displacements) in time and space (these form the Abelian Lie group of translations on space-time) rotations in space (this forms the non-Abelian Lie group of 3-dimensional rotations) boosts, ie, transformations connecting two uniformly moving... In physics, spacetime is any mathematical model that combines space and time into a single construct called the space-time continuum. ... 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. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ... Flavour (or flavor) is a quantum number of elementary particles related to their weak interactions. ...

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. This picture illustrates how the hours in a clock form a group. ...

### General principles

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

### 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.

Results from FactBites:

 ScienceDaily: Quantum number (1536 words) Each quantum number specifies the value of a conserved quantity in the dynamics of the quantum system. Quantum number -- A quantum number describes the energies of electrons in atoms. Specifically, atomic orbitals are the quantum states of the individual electrons in the...
 Quantum number Summary (1755 words) Quantum numbers are the four numbers used to describe not only the distribution of electrons in atoms and molecular systems but also the allowable values of certain physical quantities of an electron's behavior. It is an intrinsic quantum number that is unrelated to the s-shaped orbital. 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.
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