Quantum number
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2008-07-11T20:50:13Z
24.1.202.242
/* Elementary particles */
'''Quantum numbers''' describe values of conserved numbers in the dynamics of the [[quantum system]]. They often describe specifically the [[energies]] of [[electron]]s in [[atom]]s, but other possibilities include [[angular momentum]], [[Spin (physics)|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 (quantum mechanics)|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.
==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]]: because it is not only useful in [[chemistry]], being the basic notion behind the [[periodic table]], [[valence (chemistry)|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.
In non-relativistic [[quantum mechanics]] the [[Hamiltonian (quantum mechanics)|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 [[Atomic nucleus|nucleus]], and the remainder. Since the potential is spherically symmetric, the full [[Hamiltonian (quantum mechanics)|Hamiltonian]] commutes with '''J<sup>2</sup>'''. '''J<sup>2</sup>''' itself commutes with any one of the components of the [[angular momentum]] vector, conventionally taken to be '''J<sub>z</sub>'''. 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 ...) denotes the eigenvalue of '''H''' with the '''J<sup>2</sup>''' 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 <math>L^2 = \hbar^2 l(l+1)</math>. In chemistry, this quantum number is very important, since it specifies the shape of an [[atomic orbital]] and strongly influences [[chemical bond]]s and [[bond 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.
*The [[magnetic quantum number]] (''m<sub>l</sub>'' = −''l'', −''l''+1 ... 0 ... ''l''−1, ''l'') is the [[eigenvalue]], <math>L_z = m_l \hbar </math>. 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 nor the same set of quantum numbers according to [[Hund's Rule]]s, which addresses the [[Pauli 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 renown [[Stern-Gerlach experiment]].
* [[spin quantum number|The spin projection quantum number]] (''m<sub>s</sub>'' = −1/2 or +1/2), the intrinsic [[angular momentum]] of the electron. This is the projection of the [[Spin (physics)|spin]] ''s''=1/2 along the specified axis.
To summarize, the quantum state of an electron is determined by its quantum numbers:
{| class="wikitable"
! name !! symbol !! orbital meaning !! range of values !! value example
|-
| principal quantum number || <math>n \ </math> || shell || <math>1 \le n \,\!</math> || <math>n=1,2,3...\,\!</math>
|-
| azimuthal quantum number ([[angular momentum]])|| <math>\ell \ </math> || subshell || ( <math>0 \le \ell \le n-1) \ </math> || for <font style="vertical-align:+15%;"><math>n=3\,\!</math></font>: <br> <math>\ell=0,1,2\,(s, p, d) \ </math>
|-
| magnetic quantum number, (projection of [[angular momentum]])|| <math>m_\ell \ </math> || energy shift || <math>-\ell \le m_\ell \le \ell \ </math> || for <font style="vertical-align:+15%;"><math>\ell=2 \ </math></font>: <br> <math>m_\ell=-2,-1,0,1,2\,\!</math>
|-
| spin projection quantum number || <math>m_s\,\!</math> || spin || <math>- \begin{matrix} \frac{1}{2} \end{matrix} , \begin{matrix} \frac{1}{2} \end{matrix} \ </math> || for an electron, either: <math>- \begin{matrix} \frac{1}{2} \end{matrix} , \begin{matrix} \frac{1}{2} \end{matrix} \ </math>
|}
Example: The quantum numbers used to refer to the outermost [[valence (chemistry)|valence]] [[electron]] of the [[Fluorine]] (F) [[atom]], which is located in the 2p [[atomic orbital]], are; ''n'' = 2, ''l'' = 1, ''m<sub>l</sub>'' = 1, or 0, or −1, ''m<sub>s</sub>'' = −1/2 or 1/2.
Note that [[molecular orbitals]] require totally different quantum numbers, because the [[Hamiltonian (quantum mechanics)|Hamiltonian]] and its symmetries are quite different.
===Quantum numbers with spin-orbit interaction===
{{details|Clebsch-Gordan coefficients}}
When one takes the [[spin-orbit interaction]] into consideration, ''l'', ''m'' and ''s'' no longer [[Commutativity|commute]] with the [[Hamiltonian (quantum mechanics)|Hamiltonian]], and their value therefore changes over time. Thus another set of quantum numbers should be used. This set includes
* The [[Azimuthal_quantum_number#Example:_total_angular_momentum_in_the_atom|total angular momentum quantum number]] (''j'' = 1/2,3/2 ... ''n''−1/2) gives the total [[angular momentum]] through the relation <math>J^2 = \hbar^2 j(j+1)</math>.
* The [[Azimuthal_quantum_number#Example:_total_angular_momentum_in_the_atom|projection of the total angular momentum along a specified axis]] (''m<sub>j</sub>'' = -j,-j+1... ''j''), which is analogous to m, and satisfies <math>m_j = m_l + m_s</math>.
* [[Parity (physics)|Parity]]. This is the [[eigenvalue]] 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:
* (1) ''l'' = 1, ''m<sub>l</sub>'' = 1, ''m<sub>s</sub>'' = +1/2
* (2) ''l'' = 1, ''m<sub>l</sub>'' = 1, ''m<sub>s</sub>'' = -1/2
* (3) ''l'' = 1, ''m<sub>l</sub>'' = 0, ''m<sub>s</sub>'' = +1/2
* (4) ''l'' = 1, ''m<sub>l</sub>'' = 0, ''m<sub>s</sub>'' = -1/2
* (5) ''l'' = 1, ''m<sub>l</sub>'' = -1, ''m<sub>s</sub>'' = +1/2
* (6) ''l'' = 1, ''m<sub>l</sub>'' = -1, ''m<sub>s</sub>'' = -1/2
* (7) ''l'' = 0, ''m<sub>l</sub>'' = 0, ''m<sub>s</sub>'' = +1/2
* (8) ''l'' = 0, ''m<sub>l</sub>'' = 0, ''m<sub>s</sub>'' = -1/2
The [[quantum state]]s 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 [[eigenvector]]s of the [[Hamiltonian (quantum mechanics)|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<sub>j</sub>'' = 3/2, odd parity (coming from state (1) above)
* ''j'' = 3/2, ''m<sub>j</sub>'' = 1/2, odd parity (coming from states (2) and (3) above)
* ''j'' = 3/2, ''m<sub>j</sub>'' = -1/2, odd parity (coming from states (4) and (5) above)
* ''j'' = 3/2, ''m<sub>j</sub>'' = -3/2, odd parity (coming from state (6))
* ''j'' = 1/2, ''m<sub>j</sub>'' = 1/2, odd parity (coming from state (2) and (3) above)
* ''j'' = 1/2, ''m<sub>j</sub>'' = -1/2, odd parity (coming from states (4) and (5) above)
* ''j'' = 1/2, ''m<sub>j</sub>'' = 1/2, even parity (coming from state (7) above)
* ''j'' = 1/2, ''m<sub>j</sub>'' = -1/2, even parity (coming from state (8) above)
==Elementary particles==
<i>For a more complete description of the quantum states of elementary particles see the articles on the [[standard model]] and [[flavour (particle physics)]].</i>
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 the [[standard model]] of [[particle physics]], and hence the quantum numbers of these particles bear the same relation to the [[Hamiltonian (quantum mechanics)|Hamiltonian]] of this model as the quantum numbers of the [[Bohr atom]] does to its [[Hamiltonian (quantum mechanics)|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.
Typical quantum numbers related to [[spacetime symmetries]] are [[spin (physics)|spin]] (related to rotational symmetry), the [[parity (physics)|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 (particle physics)|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 (mathematics)|group]] called '''Z<sub>2</sub>'''.
==References and external links==
{{reflist}}
===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://chemed.chem.purdue.edu/genchem/topicreview/bp/ch6/quantum.html Quantum Numbers and Electron Configurations]
* [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 ==
{{portal|Physics}}
*[[Quantum]]
*[[Quantum mechanics]]
*[[Quantum field theory]]
*[[Many-worlds interpretation]]
*[[Interpretation of quantum mechanics]]
[[Category:Quantum mechanics]]
[[Category:Quantum measurement| ]]
[[Category:Fundamental physics concepts]]
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