Quantum numbers describe values of conserved quantities in the dynamics of the quantum system. Perhaps the most peculiar aspect of
quantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particlelike and wavelike behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
is the
quantizationIn physics, quantization is the process of explaining a classical understanding of physical phenomena in terms of a newer understanding known as "quantum mechanics". It is a procedure for constructing a quantum field theory starting from a classical field theory. This is a generalization of the...
of observable quantities. This is distinguished from
classical mechanicsIn physics, classical mechanics is one of the two major subfields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
where the values can range continuously. They often describe specifically the energies of
electronThe electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s in
atomThe atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...
s, but other possibilities include
angular momentumIn physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
,
spinIn quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
etc. Any quantum system can have one or more quantum numbers; it is thus rigorous 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. A quantized system requires at least one quantum number. The dynamics of any quantum system are described by a quantum
HamiltonianIn quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
,
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
HO = OH). 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.
To completely describe an electron in an atom, four quantum numbers are needed: energy, angular momentum, magnetic moment and spin.
Traditional nomenclature
Many different models have been proposed throughout the
history of quantum mechanicsThe history of quantum mechanics, as it interlaces with the history of quantum chemistry, began essentially with a number of different scientific discoveries: the 1838 discovery of cathode rays by Michael Faraday; the 18591860 winter statement of the black body radiation problem by Gustav...
, but the most prominent system of nomenclature spawned from the HundMulliken
molecular orbitalIn chemistry, a molecular orbital is a mathematical function describing the wavelike behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of finding an electron in any specific region. The term "orbital" was first...
theory of
Friedrich HundFriedrich Hermann Hund was a German physicist from Karlsruhe known for his work on atoms and molecules.Hund worked at the Universities of Rostock, Leipzig, Jena, Frankfurt am Main, and Göttingen....
,
Robert S. MullikenRobert Sanderson Mulliken was an American physicist and chemist, primarily responsible for the early development of molecular orbital theory, i.e. the elaboration of the molecular orbital method of computing the structure of molecules. Dr. Mulliken received the Nobel Prize for chemistry in 1966...
, and contributions from
SchrödingerErwin Rudolf Josef Alexander Schrödinger was an Austrian physicist and theoretical biologist who was one of the fathers of quantum mechanics, and is famed for a number of important contributions to physics, especially the Schrödinger equation, for which he received the Nobel Prize in Physics in 1933...
,
SlaterJohn Clarke Slater was a noted American physicist who made major contributions to the theory of the electronic structure of atoms, molecules and solids. This work is of ongoing importance in chemistry, as well as in many areas of physics. He also made major contributions to microwave electronics....
and
John LennardJones Sir John Edward LennardJones KBE, FRS was a mathematician who was a professor of theoretical physics at Bristol University, and then of theoretical science at Cambridge University...
. This system of nomenclature incorporated
BohrNiels Henrik David Bohr was a Danish physicist who made foundational contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in Physics in 1922. Bohr mentored and collaborated with many of the top physicists of the century at his institute in...
energy levels, HundMulliken orbital theory, and observations on electron spin based on
spectroscopySpectroscopy is the study of the interaction between matter and radiated energy. Historically, spectroscopy originated through the study of visible light dispersed according to its wavelength, e.g., by a prism. Later the concept was expanded greatly to comprise any interaction with radiative...
and Hund's rules.
This model describes electrons using four quantum numbers,
n,
,
m_{},
m_{s}. It is also the common nomenclature in the classical description of nuclear particle states (e.g. protons and neutrons).
 The first, n, describes the electron shell
An electron shell may be thought of as an orbit followed by electrons around an atom's nucleus. The closest shell to the nucleus is called the "1 shell" , followed by the "2 shell" , then the "3 shell" , and so on further and further from the nucleus. The shell letters K,L,M,.....
, or energy level.
 The value of n ranges from 1 to "n", where "n" is the shell containing the outermost electron of that atom. For example, in caesium (Cs)
Caesium or cesium is the chemical element with the symbol Cs and atomic number 55. It is a soft, silverygold alkali metal with a melting point of 28 °C , which makes it one of only five elemental metals that are liquid at room temperature...
, the outermost valenceIn chemistry, valence, also known as valency or valence number, is a measure of the number of bonds formed by an atom of a given element. "Valence" can be defined as the number of valence bonds...
electron is in the shell with energy level 6, so an electron in caesium can have an n value from 1 to 6. This is known as the principal quantum number.
 The second, , describes the subshell (0 = s orbital, 1 = p orbital, 2 = d orbital, 3 = f orbital, etc.).
 The value of ranges from 0 to n − 1. This is because the first p orbital ( = 1) appears in the second electron shell (n = 2), the first d orbital ( = 2) appears in the third shell (n = 3), and so on. A quantum number beginning in 3, 0, … describes an electron in the s orbital of the third electron shell of an atom.
 The third, m_{}, describes the specific orbital
An electron orbital may refer to:* An atomic orbital, describing the behaviour of an electron in an atom* A molecular orbital, describing the behaviour of an electron in a molecule See also :...
(or "cloud") within that subshell.*
 The values of m_{} range from − to . The s subshell ( = 0) contains only one orbital, and therefore the m_{} of an electron in an s subshell will always be 0. The p subshell ( = 1) contains three orbitals (in some systems, depicted as three "dumbbellshaped" clouds), so the m_{} of an electron in a p subshell will be −1, 0, or 1. The d subshell ( = 2) contains five orbitals, with m_{} values of −2, −1, 0, 1, and 2.
 The fourth, m_{s}, describes the spin of the electron within that orbital.*
 An electron can have a spin of ±½, m_{s} will be either, corresponding with "spin" and "opposite spin." Each electron in any individual orbital must have different spins, therefore, an orbital never contains more than two electrons.
* Note that, since atoms and electrons are in a state of constant motion, there is no universal fixed value for m
_{} and m
_{s} values. Therefore, the m
_{} and m
_{s} values are defined somewhat arbitrarily. The only requirement is that the naming schematic used within a particular set of calculations or descriptions must be consistent (e.g. the orbital occupied by the first electron in a p subshell could be described as m
_{} = −1 or m
_{} = 0, or m
_{} = 1, but the m
_{} value of the other electron in that orbital must be the same, and the m
_{} assigned to electrons in other orbitals must be different).
These rules are summarized as follows:
name  symbol  orbital meaning  range of values  value example 

principal quantum number In atomic physics, the principal quantum symbolized as n is the firstof a set of quantum numbers of an atomic orbital. The principal quantum number can only have positive integer values... 
n 
shell 
1 ≤ n 
n = 1, 2, 3, … 
azimuthal quantum numberThe azimuthal quantum number is a quantum number for an atomic orbital that determines its orbital angular momentum and describes the shape of the orbital... (angular momentumIn physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system... ) 

subshell (s orbital is listed as 0, p orbital as 1 etc.) 
0 ≤ ≤ n − 1 
for n = 3: = 0, 1, 2 (s, p, d) 
magnetic quantum number In atomic physics, the magnetic quantum number is the third of a set of quantum numbers which describe the unique quantum state of an electron and is designated by the letter m... , (projection of angular momentumIn physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system... ) 
m_{} 
energy shift (orientation of the subshell's shape) 
− ≤ m_{} ≤ 
for = 2: m_{} = −2, −1, 0, 1, 2 
spin projection quantum number In atomic physics, the spin quantum number is a quantum number that parameterizes the intrinsic angular momentum of a given particle... 
m_{s} 
spin of the electron (−½ = counterclockwise, ½ = clockwise) 
−½, ½ 
for an electron, either: −½, ½ 
Example: The quantum numbers used to refer to the outermost valenceIn chemistry, valence, also known as valency or valence number, is a measure of the number of bonds formed by an atom of a given element. "Valence" can be defined as the number of valence bonds...
electronThe electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s of the CarbonCarbon is the chemical element with symbol C and atomic number 6. As a member of group 14 on the periodic table, it is nonmetallic and tetravalent—making four electrons available to form covalent chemical bonds...
(C) atomThe atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...
, which are located in the 2p atomic orbitalAn atomic orbital is a mathematical function that describes the wavelike behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus...
, are; n = 2 (2nd electron shell), = 1 (p orbital subshell), m_{} = 1, 0 or −1, m_{s} = ½ (parallel spins).
As applied to the Hamiltonian and Schrödinger equation
 The principal quantum number
In atomic physics, the principal quantum symbolized as n is the firstof a set of quantum numbers of an atomic orbital. The principal quantum number can only have positive integer values...
(n = 1, 2, 3, 4, …) denotes the eigenvalue of H with the J^{2} part removed. This number therefore has a dependence only on the distance between the electron and the nucleus (i.e., 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
The azimuthal quantum number is a quantum number for an atomic orbital that determines its orbital angular momentum and describes the shape of the orbital...
( = 0, 1, …, n − 1) (also known as the angular quantum number or orbital quantum number) gives the orbital angular momentumIn physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
through the relation . In chemistry, this quantum number is very important, since it specifies the shape of an atomic orbitalAn atomic orbital is a mathematical function that describes the wavelike behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus...
and strongly influences chemical bondA chemical bond is an attraction between atoms that allows the formation of chemical substances that contain two or more atoms. The bond is caused by the electromagnetic force attraction between opposite charges, either between electrons and nuclei, or as the result of a dipole attraction...
s and bond angles. In some contexts, " = 0" is called an s orbital, " = 1" a p orbital, " = 2" a d orbital, and " = 3" an f orbital.
 The magnetic quantum number
In atomic physics, the magnetic quantum number is the third of a set of quantum numbers which describe the unique quantum state of an electron and is designated by the letter m...
(m_{} = −, − + 1, …, 0, …, − 1, ) yields the projection of the orbital angular momentumIn physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
along a specified axis. L_{z} = m_{}ħ.
 The spin projection quantum number
In atomic physics, the spin quantum number is a quantum number that parameterizes the intrinsic angular momentum of a given particle...
(m_{s} = ±½), is the intrinsic angular momentumIn physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
of the electron or nucleon. This is the projection of the spinIn quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
s = ½ along the specified axis.
 Results from spectroscopy
Spectroscopy is the study of the interaction between matter and radiated energy. Historically, spectroscopy originated through the study of visible light dispersed according to its wavelength, e.g., by a prism. Later the concept was expanded greatly to comprise any interaction with radiative...
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 Rules, which addresses the Pauli exclusion principleThe Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is antisymmetric with respect to exchange of the particles...
. 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 renowned SternGerlach experiment.
Molecular orbitals require different quantum numbers, because the HamiltonianIn quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
and its symmetries are quite different.
Quantum numbers with spinorbit interaction
When one takes the spinorbit interactionIn quantum physics, the spinorbit interaction is any interaction of a particle's spin with its motion. The first and best known example of this is that spinorbit interaction causes shifts in an electron's atomic energy levels due to electromagnetic interaction between the electron's spin and...
into consideration, the , m and soperators no longer commuteIn mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...
with the HamiltonianIn quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
, and their eigenvalues therefore change over time. Thus another set of quantum numbers should be used. This set includes
 The total angular momentum quantum number  ± s gives the total angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
through the relation .
 The projection of the total angular momentum along a specified axis (m_{j} = −j, −j + 1, …, j), which is analogous to m, and satisfies m_{j} = m_{} + m_{s} where m_{} + m_{s} < j.
 Parity
In physics, a parity transformation is the flip in the sign of one spatial coordinate. In three dimensions, it is also commonly described by the simultaneous flip in the sign of all three spatial coordinates:...
. This is the eigenvalue under reflection, and is positive (i.e. +1) for states which came from even and negative (i.e. −1) for states which came from odd . 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:

n 

m_{} 
m_{s} 

+ s 
 s 
m_{l} + m_{s} 

#1. 
2 
1 
1 
+1/2 
3/2 
1/2 
3/2 
#2. 
2 
1 
1 
1/2 
3/2 
1/2 
1/2 
#3. 
2 
1 
0 
+1/2 
3/2 
1/2 
1/2 
#4. 
2 
1 
0 
1/2 
3/2 
1/2 
1/2 
#5. 
2 
1 
1 
+1/2 
3/2 
1/2 
1/2 
#6. 
2 
1 
1 
1/2 
3/2 
1/2 
3/2 
#7. 
2 
0 
0 
+1/2 
1/2 
1/2 
1/2 
#8. 
2 
0 
0 
1/2 
1/2 
1/2 
1/2 
The quantum states in the system can be described as linear combination of these eight states. However, in the presence of spinorbit interactionIn quantum physics, the spinorbit interaction is any interaction of a particle's spin with its motion. The first and best known example of this is that spinorbit interaction causes shifts in an electron's atomic energy levels due to electromagnetic interaction between the electron's spin and...
, if one wants to describe the same system by eight states which are eigenvectors of the HamiltonianIn quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
(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) above) 
j = 1/2, 
m_{j} = 
1/2, 
odd parity 
(coming from states (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 states of the standard modelThe Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...
of particle physicsParticle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...
, and hence the quantum numbers of these particles bear the same relation to the HamiltonianIn quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
of this model as the quantum numbers of the Bohr atom does to its HamiltonianIn quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
. In other words, each quantum number denotes a symmetry of the problem. It is more useful in quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and manybody systems. It is the natural and quantitative language of particle physics and...
to distinguish between spacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being threedimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
and internal symmetries.
Typical quantum numbers related to spacetime symmetriesSpacetime symmetries are features of spacetime that can be described as exhibiting some form of symmetry. The role of symmetry in physics is important in simplifying solutions to many problems, spacetime symmetries finding ample application in the study of exact solutions of Einstein's field...
are spinIn quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
(related to rotational symmetry), the parityIn physics, a parity transformation is the flip in the sign of one spatial coordinate. In three dimensions, it is also commonly described by the simultaneous flip in the sign of all three spatial coordinates:...
, Cparity and Tparity (related to the Poincare symmetry of spacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being threedimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
). Typical internal symmetries are lepton numberIn particle physics, the lepton number is the number of leptons minus the number of antileptons.In equation form,so all leptons have assigned a value of +1, antileptons −1, and nonleptonic particles 0...
and baryon number or the electric chargeElectric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
. (For a full list of quantum numbers of this kind see the article on flavourIn particle physics, flavour or flavor is a quantum number of elementary particles. In quantum chromodynamics, flavour is a global symmetry...
.)
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; i.e., 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 groupIn mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
called Z_{2}.
Atomic physics
Particle physics