- Quantum number
**Quantum numbers**describe values of conserved numbers in the dynamics of thequantum system . They often describe specifically theenergies ofelectron s inatom s, but other possibilities includeangular 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.**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**. There is one quantum number of the system corresponding to the energy, i.e., theeigenvalue 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.(Editing needed here) Symmetric, the full Hamiltonian commutes with

**J**.^{2}**J**itself commutes with any one of the components of the^{2}angular momentum vector, conventionally taken to be**J**. These are the only mutually commuting operators in this problem; hence, there are three quantum numbers._{z}These are conventionally known as

*The

principal quantum number ("n" = 1, 2, 3, 4 ...) denotes the eigenvalue of**H**with the**J**part removed. This number therefore has a dependence only on the distance between the electron and the nucleus (ie, the radial coordinate,^{2}**r**). The average distance increases with**n**, and hence quantum states with different principal quantum numbers are said to belong to different shells.

*Theazimuthal quantum number ("l" = 0, 1 ... "n"−1) (also known as the**angular quantum number**or**orbital quantum number**) gives the orbitalangular 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 anatomic orbital and strongly influenceschemical bond s andbond angle s. 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.

*Themagnetic quantum number ("m_{l}" = −"l", −"l"+1 ... 0 ... "l"−1, "l") is theeigenvalue , $L\_z\; =\; m\_l\; hbar$. This is the projection of the orbitalangular 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 nor the same set of quantum numbers according toHund's Rule s, which addresses thePauli exclusion principle . 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 renownedStern-Gerlach experiment .* The spin projection quantum number ("m

_{s}" = −1/2 or +1/2), the intrinsicangular 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 its quantum numbers:

Example: The quantum numbers used to refer to the outermost valence

electron of theFluorine (F)atom , which is located in the 2patomic orbital , are; "n" = 2, "l" = 1, "m_{l}" = 1, or 0, or −1, "m_{s}" = −1/2 or 1/2.Note that

molecular orbitals require totally different quantum numbers, because the Hamiltonian and its symmetries are quite different.**Quantum numbers with spin-orbit interaction**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

* The ("j" = 1/2,3/2 ... "n"−1/2) gives the totalangular momentum through the relation $J^2\; =\; hbar^2\; j(j+1)$.

* The ("m_{j}" = -j,-j+1... "j"), which is analogous to m, and satisfies $m\_j\; =\; m\_l\; +\; m\_s$.

* Parity. This is theeigenvalue under reflection, and is positive (i.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:

* (2) "l" = 1, "m_{l}" = 0, "m_{s}" = +1/2

* (3) "l" = 1, "m_{l}" = 0, "m_{s}" = -1/2

* (4) "l" = 1, "m_{l}" = -1, "m_{s}" = +1/2

* (5) "l" = 1, "m_{l}" = -1, "m_{s}" = -1/2

* (6) "l" = 0, "m_{l}" = 0, "m_{s}" = +1/2

* (7) "l" = 0, "m_{l}" = 0, "m_{s}" = -1/2The

quantum state s in the system can be described as linear combination of these eight states. However, in the presence ofspin-orbit interaction , if one wants to describe the same system by eight states which areeigenvector s of the Hamiltonian (i.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**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 state s of thestandard model ofparticle physics , and hence the quantum numbers of these particles bear the same relation to the Hamiltonian of this model as the quantum numbers of theBohr atom does to its Hamiltonian. In other words, each quantum number denotes a symmetry of the problem. It is more useful infield theory to distinguish betweenspacetime andinternal symmetries.Typical quantum numbers related to

spacetime symmetries are spin (related to rotational symmetry), the parity,C-parity andT-parity (related to thePoincare symmetry ofspacetime ). Typical**internal symmetries**arelepton number andbaryon number or theelectric charge . (For a full list of quantum numbers of this kind see the article on flavour.)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

**Z**._{2}**References and external links****General principles***cite book | author=Dirac, Paul A.M. | title=Principles of quantum mechanics | publisher=Oxford University Press |year=1982 |id=ISBN 0-19-852011-5

**Atomic physics*** [

*http://www.perfectperiodictable.com/novelty#102 Pictorial representation of the Quantum Numbers "n, l and m*]_{l}"

* [*http://hyperphysics.phy-astr.gsu.edu/hbase/qunoh.html Quantum numbers for the hydrogen atom*]

* [*http://pprc.qmul.ac.uk/~lloyd/epp/lectures/Quantum_Numbers.pdf Lecture notes on quantum numbers*]**Particle physics***cite book | author=Griffiths, David J.|title=Introduction to Quantum Mechanics (2nd ed.) | publisher=Prentice Hall |year=2004 |id=ISBN 0-13-805326-X

*cite book | author=Halzen, Francis and Martin, Alan D. | title=QUARKS AND LEPTONS: An Introductory Course in Modern Particle Physics | publisher=John Wiley & Sons |year=1984 |id=ISBN 0-471-88741-2* [

*http://pdg.lbl.gov/ The particle data group*]**See also***

Quantum

*Quantum mechanics

*Quantum field theory

*Many-worlds interpretation

*Interpretation of quantum mechanics

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