In
quantum physics and
chemistry
Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, proper ...
, quantum numbers describe values of
conserved quantities in the dynamics of a
quantum system. Quantum numbers correspond to
eigenvalues of
operators that commute with the
Hamiltonian—quantities that can be known with precision at the same time as the system's energy
[specifically, observables that ]commute
Commute, commutation or commutative may refer to:
* Commuting, the process of travelling between a place of residence and a place of work
Mathematics
* Commutative property, a property of a mathematical operation whose result is insensitive to th ...
with the Hamiltonian are simultaneously diagonalizable
In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique. ...
with it and so the eigenvalues and the energy (eigenvalues of the Hamiltonian) are not limited by an uncertainty relation arising from non-commutativity.—and their corresponding eigenspaces. Together, a specification of all of the quantum numbers of a quantum system fully characterize a
basis state of the system, and can in principle be
measured together.
An important aspect of quantum mechanics is the
quantization of many observable quantities of interest.
[Many observables have discrete spectra (sets of eigenvalues) in quantum mechanics, so the quantities can only be measured in discrete (often integer) values.] In particular, this leads to quantum numbers that take values in
discrete sets of integers or
half-integers; although they could approach
infinity
Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol .
Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions am ...
in some cases. This distinguishes quantum mechanics from
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
where the values that characterize the system such as mass, charge, or momentum, all range continuously. Quantum numbers often describe specifically the
energy levels of electrons in atoms, but other possibilities include
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
,
spin, etc. An important family is
flavour quantum numbers
In particle physics, flavour or flavor refers to the ''species'' of an elementary particle. The Standard Model counts six flavours of quarks and six flavours of leptons. They are conventionally parameterized with ''flavour quantum numbers'' th ...
–
internal quantum numbers which determine the type of a particle and its interactions with other particles through the
fundamental forces. Any quantum system can have one or more quantum numbers; it is thus difficult to list all possible quantum numbers.
Quantum numbers needed for a given system
The tally of quantum numbers varies from system to system and has no universal answer. Hence these parameters must be found for each system to be analyzed. A quantized system requires at least one quantum number. The dynamics (i.e. time evolution) of any quantum system are described by a
quantum operator
In physics, an operator is a function over a space of physical states onto another space of physical states. The simplest example of the utility of operators is the study of symmetry (which makes the concept of a group useful in this context). Beca ...
in the form of a
Hamiltonian, . There is one quantum number of the system corresponding to the system's energy; i.e., one of the
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s of the Hamiltonian. There is also one quantum number for each
linearly independent operator that
commutes with the Hamiltonian. A
complete set of commuting observables In quantum mechanics, a complete set of commuting observables (CSCO) is a set of commuting operators whose common eigenvectors can be used as a basis to express any quantum state. In the case of operators with discrete spectra, a CSCO is a set of co ...
(CSCO) that commute with the Hamiltonian characterizes the system with all its quantum numbers. There is a one-to-one relationship between the quantum numbers and the operators of the CSCO, with each quantum number taking one of the eigenvalues of its corresponding operator. As a result of the different
basis that may be arbitrarily chosen to form a complete set of commuting operators, different sets of quantum numbers may be used for the description of the same system in different situations.
Electron in an atom
Four quantum numbers can describe an electron in an atom completely:
*
Principal quantum number ()
*
Azimuthal quantum number ()
*
Magnetic quantum number ()
*
Spin quantum number ()
The
spin–orbital interaction, however, relates these numbers. Thus, a complete description of the system can be given with fewer quantum numbers, if orthogonal choices are made for these basis vectors.
Specificity
Different electrons in a system will have different quantum numbers. For example, the highest occupied orbital electron, the actual differentiating electron (i.e. the electron that differentiates an element from the previous one); , r the differentiating electron according to the
''aufbau'' approximation. In
lanthanum
Lanthanum is a chemical element with the symbol La and atomic number 57. It is a soft, ductile, silvery-white metal that tarnishes slowly when exposed to air. It is the eponym of the lanthanide series, a group of 15 similar elements between l ...
, as a further illustration, the electrons involved are in the 6s; 5d; and 4f orbitals, respectively. In this case the principal quantum numbers are 6, 5, and 4.
Common terminology
The model used here describes electrons using four quantum numbers, , , , , given below. It is also the common nomenclature in the classical description of nuclear particle states (e.g. protons and neutrons). A quantum description of
molecular orbitals require other quantum numbers, because the
Hamiltonian and its symmetries are different.
Principal quantum number
The principal quantum number describes the
electron shell, or energy level, of an electron. The value of ranges from 1 to the shell containing the outermost electron of that atom, that is
:
For example, in
caesium
Caesium (IUPAC spelling) (or cesium in American English) is a chemical element with the symbol Cs and atomic number 55. It is a soft, silvery-golden alkali metal with a melting point of , which makes it one of only five elemental metals that a ...
(Cs), the outermost
valence electron is in the shell with energy level 6, so an electron in caesium can have an value from 1 to 6.
For particles in a time-independent potential (see
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
), it also labels the th eigenvalue of Hamiltonian (), that is, the energy , with the contribution due to angular momentum (the term involving ) left out. So this number depends only on the distance between the electron and the nucleus (that is, the radial coordinate ). The average distance increases with . Hence quantum states with different principal quantum numbers are said to belong to different shells.
Azimuthal quantum number
The azimuthal quantum number, also known as the (angular momentum quantum number or orbital quantum number), describes the
subshell, and gives the magnitude of the orbital
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
through the relation.
:
In chemistry and spectroscopy, is called s orbital, , p orbital, , d orbital, and , f orbital.
The value of ranges from 0 to , so the first p orbital () appears in the second electron shell (), the first d orbital () appears in the third shell (), 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. In chemistry, this quantum number is very important, since it specifies the shape of an
atomic orbital
In atomic theory and quantum mechanics, an atomic orbital is a function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in any ...
and strongly influences
chemical bonds and
bond angles. The azimuthal quantum number can also denote the number of angular nodes present in an orbital. For example, for p orbitals, and thus the amount of angular nodes in a p orbital is 1.
Shape of orbital is also given by azimuthal quantum number.
Magnetic quantum number
The magnetic quantum number describes the specific
orbital (or "cloud") within that subshell, and yields the ''projection'' of the orbital
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
''along a specified axis'':
:
The values of range from to , with integer intervals.
The s subshell () contains only one orbital, and therefore the of an electron in an s orbital will always be 0. The p subshell () contains three orbitals (in some systems, depicted as three "dumbbell-shaped" clouds), so the of an electron in a p orbital will be −1, 0, or 1. The d subshell () contains five orbitals, with values of −2, −1, 0, 1, and 2.
Spin quantum number
The spin quantum number describes the intrinsic spin
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
of the electron within each orbital and gives the projection of the
spin angular momentum along the specified axis:
:.
In general, the values of range from to , where is the
spin quantum number, associated with the particle's intrinsic spin angular momentum:
:.
An electron has spin number , consequently will be ±, referring to "spin up" and "spin down" states. Each electron in any individual orbital must have different quantum numbers because of the
Pauli exclusion principle
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formula ...
, therefore an orbital never contains more than two electrons.
Rules
There are no universal fixed values for and . Rather, the and values are
arbitrary. The only restrictions on the choices for these constants 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 orbital could be described as or or , but the value of the next unpaired electron in that orbital must be different; yet, the assigned to electrons in other orbitals again can be or or ).
These rules are summarized as follows:
:
Example: The quantum numbers used to refer to the outermost
valence electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have n ...
s of a
carbon
Carbon () is a chemical element with the symbol C and atomic number 6. It is nonmetallic and tetravalent—its atom making four electrons available to form covalent chemical bonds. It belongs to group 14 of the periodic table. Carbon ma ...
(C)
atom
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, a ...
, which are located in the 2p
atomic orbital
In atomic theory and quantum mechanics, an atomic orbital is a function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in any ...
, are; (2nd electron shell), (p orbital
subshell), , (parallel spins).
Results from
spectroscopy
Spectroscopy is the field of study that measures and interprets the electromagnetic spectra that result from the interaction between electromagnetic radiation and matter as a function of the wavelength or frequency of the radiation. Matter ...
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
In atomic physics, Hund's rules refers to a set of rules that German physicist Friedrich Hund formulated around 1927, which are used to determine the term symbol that corresponds to the ground state of a multi- electron atom. The first rule is ...
, which addresses the
Pauli exclusion principle
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formula ...
. A fourth quantum number, which represented spin with two possible values, was added as an ''
ad hoc
Ad hoc is a Latin phrase meaning literally 'to this'. In English, it typically signifies a solution for a specific purpose, problem, or task rather than a generalized solution adaptable to collateral instances. (Compare with ''a priori''.)
Com ...
'' assumption to resolve the conflict; this supposition would later be explained in detail by relativistic quantum mechanics and from the results of the renowned
Stern–Gerlach experiment.
Background
Many different models have been proposed throughout the
history of quantum mechanics, but the most prominent system of nomenclature spawned from the Hund-Mulliken
molecular orbital
In chemistry, a molecular orbital is a mathematical function describing the location and wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of find ...
theory of
Friedrich Hund,
Robert S. Mulliken, and contributions from
Schrödinger,
Slater
A slater, or slate mason, is a tradesperson who covers buildings with slate.
Tools of the trade
The various tools of the slater's trade are all drop-forged
Forging is a manufacturing process involving the shaping of metal using localiz ...
and
John Lennard-Jones. This system of nomenclature incorporated
Bohr
Niels Henrik David Bohr (; 7 October 1885 – 18 November 1962) was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922. B ...
energy levels, Hund-Mulliken orbital theory, and observations on electron spin based on
spectroscopy
Spectroscopy is the field of study that measures and interprets the electromagnetic spectra that result from the interaction between electromagnetic radiation and matter as a function of the wavelength or frequency of the radiation. Matter ...
and
Hund's rules
In atomic physics, Hund's rules refers to a set of rules that German physicist Friedrich Hund formulated around 1927, which are used to determine the term symbol that corresponds to the ground state of a multi- electron atom. The first rule is ...
.
Total angular momenta numbers
Total angular momentum of a particle
When one takes the
spin–orbit interaction
In quantum physics, the spin–orbit interaction (also called spin–orbit effect or spin–orbit coupling) is a relativistic interaction of a particle's spin with its motion inside a potential. A key example of this phenomenon is the spin–orb ...
into consideration, the and operators no longer
commute
Commute, commutation or commutative may refer to:
* Commuting, the process of travelling between a place of residence and a place of work
Mathematics
* Commutative property, a property of a mathematical operation whose result is insensitive to th ...
with the
Hamiltonian, and their eigenvalues therefore change over time. Thus another set of quantum numbers should be used. This set includes
For example, consider the following 8 states, defined by their quantum numbers:
:
The
quantum state
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
s in the system can be described as linear combination of these 8 states. However, in the presence of
spin–orbit interaction
In quantum physics, the spin–orbit interaction (also called spin–orbit effect or spin–orbit coupling) is a relativistic interaction of a particle's spin with its motion inside a potential. A key example of this phenomenon is the spin–orb ...
, if one wants to describe the same system by 8 states that are
eigenvectors of the
Hamiltonian (i.e. each represents a state that does not mix with others over time), we should consider the following 8 states:
:
Nuclear angular momentum quantum numbers
In
nuclei, the entire assembly of
protons and
neutron
The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the atomic nucleus, nuclei of atoms. Since protons and ...
s (
nucleons) has a resultant
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
due to the angular momenta of each nucleon, usually denoted . If the total angular momentum of a neutron is and for a proton is (where for protons and neutrons happens to be again (''see note'')), then the nuclear angular momentum quantum numbers are given by:
:
''Note: ''The orbital angular momenta of the nuclear (and atomic) states are all integer multiples of ħ while the intrinsic angular momentum of the neutron and proton are half-integer multiples. It should be immediately apparent that the combination of the intrinsic spins of the nucleons with their orbital motion will always give half-integer values for the total spin, , of any odd-A nucleus and integer values for any even-A nucleus.
Parity with the number is used to label nuclear angular momentum states, examples for some isotopes of
hydrogen
Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-to ...
(H),
carbon
Carbon () is a chemical element with the symbol C and atomic number 6. It is nonmetallic and tetravalent—its atom making four electrons available to form covalent chemical bonds. It belongs to group 14 of the periodic table. Carbon ma ...
(C), and
sodium
Sodium is a chemical element with the symbol Na (from Latin ''natrium'') and atomic number 11. It is a soft, silvery-white, highly reactive metal. Sodium is an alkali metal, being in group 1 of the periodic table. Its only stable ...
(Na) are;
:
The reason for the unusual fluctuations in , even by differences of just one nucleon, are due to the odd and even numbers of protons and neutrons – pairs of nucleons have a total angular momentum of zero (just like electrons in orbitals), leaving an odd or even number of unpaired nucleons. The property of nuclear spin is an important factor for the operation of
NMR spectroscopy in
organic chemistry
Organic chemistry is a subdiscipline within chemistry involving the scientific study of the structure, properties, and reactions of organic compounds and organic materials, i.e., matter in its various forms that contain carbon atoms.Clayden, J ...
,
and
MRI in
nuclear medicine
Nuclear medicine or nucleology is a medical specialty involving the application of radioactive substances in the diagnosis and treatment of disease. Nuclear imaging, in a sense, is " radiology done inside out" because it records radiation emi ...
,
due to the
nuclear magnetic moment interacting with an external
magnetic field.
Elementary particles
Elementary particle
In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions ( quarks, leptons, ...
s 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
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
s of the
standard model of
particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
, 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
quantum field theory to distinguish between
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
and
internal symmetries.
Typical quantum numbers related to
spacetime symmetries are
spin (related to rotational symmetry), the
parity
Parity may refer to:
* Parity (computing)
** Parity bit in computing, sets the parity of data for the purpose of error detection
** Parity flag in computing, indicates if the number of set bits is odd or even in the binary representation of the ...
,
C-parity
In physics, the C parity or charge parity is a multiplicative quantum number of some particles that describes their behavior under the symmetry operation of charge conjugation.
Charge conjugation changes the sign of all quantum charges (that is, ...
and
T-parity (related to the
Poincaré symmetry of
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
). Typical internal symmetries are
lepton number and
baryon number or the
electric charge
Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons res ...
. (For a full list of quantum numbers of this kind see the article on
flavour.)
Multiplicative quantum numbers
Most conserved quantum numbers are additive, so 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
Parity may refer to:
* Parity (computing)
** Parity bit in computing, sets the parity of data for the purpose of error detection
** Parity flag in computing, indicates if the number of set bits is odd or even in the binary representation of the ...
'', 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 (
involution).
See also
*
Electron configuration
*
Multiplicative quantum number
Notes
References
Further reading
*
*
*
*
External links
Quantum numbers for the hydrogen atomLecture notes on quantum numbersThe particle data group
{{Authority control
Physical quantities
Quantum numbers
Dimensionless numbers