All Topics  
Quantum number

 

 
   Email Print
   Bookmark   Link
 

 

Quantum number
 
 
  Topics Quantum number Topic Home

Quantum numbers describe values of conserved numbers in the dynamics of the quantum system. They often describe specifically the energies of electron
Electron

The electron is a subatomic particle that carries a negative electric charge. It has elementary particle and is believed to be a point particle....
s in atom
Atom

|-! bgcolor=gray | Properties|-||}The atom is a basic unit of matter consisting of a dense, central atomic nucleus surrounded by a electron cloud of electric charge electrons....
s, but other possibilities include angular momentum
Angular momentum

In physics, the angular momentum of a particle about an origin is a vector quantity related to rotation, equal to the mass of the particle multiplied by the cross product of the position vector of the particle with its velocity vector....
, spin
Spin (physics)

In quantum mechanics, spin is a fundamental property of atomic nucleus, hadrons, and elementary particles. For particles with non-zero spin, spin direction is an important intrinsic degrees of freedom ....
 etc. Since any quantum system can have one or more quantum numbers, it is a rigorous 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.






Discussion
Ask a question about 'Quantum number'
Start a new discussion about 'Quantum number'
Answer questions from other users
Full Discussion Forum



Encyclopedia


Quantum numbers describe values of conserved numbers in the dynamics of the quantum system. They often describe specifically the energies of electron
Electron

The electron is a subatomic particle that carries a negative electric charge. It has elementary particle and is believed to be a point particle....
s in atom
Atom

|-! bgcolor=gray | Properties|-||}The atom is a basic unit of matter consisting of a dense, central atomic nucleus surrounded by a electron cloud of electric charge electrons....
s, but other possibilities include angular momentum
Angular momentum

In physics, the angular momentum of a particle about an origin is a vector quantity related to rotation, equal to the mass of the particle multiplied by the cross product of the position vector of the particle with its velocity vector....
, spin
Spin (physics)

In quantum mechanics, spin is a fundamental property of atomic nucleus, hadrons, and elementary particles. For particles with non-zero spin, spin direction is an important intrinsic degrees of freedom ....
 etc. Since any quantum system can have one or more quantum numbers, it is a rigorous 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
Hamiltonian (quantum mechanics)

In quantum mechanics, the Hamiltonian H is the observable corresponding to the total energy of the system. As with all observables, the Spectrum of the Hamiltonian 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 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.

Since it is symmetric, the full Hamiltonian
Hamiltonian (quantum mechanics)

In quantum mechanics, the Hamiltonian H is the observable corresponding to the total energy of the system. As with all observables, the Spectrum of the Hamiltonian is the set of possible outcomes when one measures the total energy of a system....
 commutes with J2, which itself commutes with any one of the components of the angular momentum
Angular momentum

In physics, the angular momentum of a particle about an origin is a vector quantity related to rotation, equal to the mass of the particle multiplied by the cross product of the position vector of the particle with its velocity vector....
 vector, conventionally taken to be Jz. 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
    Principal quantum number

    In atomic physics, the principal quantum number symbolized as n is the firstof a set of quantum numbers of an atomic orbital. The quantum number n labels the energy levels of hydrogenic atoms....
     (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
    Azimuthal quantum number

    The Azimuthal quantum number symbolized as l is a quantum number for an atomic orbital that determines its orbital angular momentum. The azimuthal quantum number is the second of a set of quantum numbers which describe the unique quantum state of an electron and is designated by the letter l....
     (l = 0, 1 ... n−1) (also known as the angular quantum number or orbital quantum number) gives the orbital angular momentum
    Angular momentum

    In physics, the angular momentum of a particle about an origin is a vector quantity related to rotation, equal to the mass of the particle multiplied by the cross product of the position vector of the particle with its velocity vector....
     through the relation . In chemistry, this quantum number is very important, since it specifies the shape of an atomic orbital
    Atomic orbital

    An atomic orbital is a mathematical function that describes the 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 specific region around the atom's nucleus....
     and strongly influences chemical bond
    Chemical bond

    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....
    s 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
    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....
     (ml = −l, −l+1 ... 0 ... l−1, l) is the eigenvalue, . This is the projection of the orbital angular momentum
    Angular momentum

    In physics, the angular momentum of a particle about an origin is a vector quantity related to rotation, equal to the mass of the particle multiplied by the cross product of the position vector of the particle with its velocity vector....
     along a specified axis.


Results from spectroscopy
Spectroscopy

Spectroscopy was originally the study of the interaction between radiation and matter as a function of wavelength . In fact, historically, spectroscopy referred to the use of visible light dispersed according to its wavelength, e.g....
 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 principle
Pauli exclusion principle

The Pauli exclusion principle is a quantum mechanics principle formulated by Wolfgang Pauli in 1925. It states that no two identical particles fermions may occupy the same quantum state simultaneously....
. 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 Stern-Gerlach experiment.

  • The spin projection quantum number
    Spin quantum number

    In atomic physics, the spin quantum number is a quantum number that parameterizes the intrinsic angular momentum of a given Elementary particle....
     (ms = −1/2 or +1/2), the intrinsic angular momentum
    Angular momentum

    In physics, the angular momentum of a particle about an origin is a vector quantity related to rotation, equal to the mass of the particle multiplied by the cross product of the position vector of the particle with its velocity vector....
     of the electron. This is the projection of the spin
    Spin (physics)

    In quantum mechanics, spin is a fundamental property of atomic nucleus, hadrons, and elementary particles. For particles with non-zero spin, spin direction is an important intrinsic degrees of freedom ....
     s=1/2 along the specified axis.


To summarize, the quantum state of an electron is determined by its quantum numbers:
name symbol orbital meaning range of values value example
principal quantum number shell
azimuthal quantum number (angular momentum
Angular momentum

In physics, the angular momentum of a particle about an origin is a vector quantity related to rotation, equal to the mass of the particle multiplied by the cross product of the position vector of the particle with its velocity vector....
)		 subshell for :
magnetic quantum number, (projection of angular momentum
Angular momentum

In physics, the angular momentum of a particle about an origin is a vector quantity related to rotation, equal to the mass of the particle multiplied by the cross product of the position vector of the particle with its velocity vector....
)		 energy shift for :
spin projection quantum number spin for an electron, either:


Example: The quantum numbers used to refer to the outermost valence
Valence (chemistry)

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 chemical element....
 electron
Electron

The electron is a subatomic particle that carries a negative electric charge. It has elementary particle and is believed to be a point particle....
 of the Fluorine
Fluorine

Fluorine is the chemical element with the symbol F and atomic number 9. Fluorine forms a single bond with itself in elemental form, resulting in the diatomic F2 molecule....
 (F) atom
Atom

|-! bgcolor=gray | Properties|-||}The atom is a basic unit of matter consisting of a dense, central atomic nucleus surrounded by a electron cloud of electric charge electrons....
, which is located in the 2p atomic orbital
Atomic orbital

An atomic orbital is a mathematical function that describes the 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 specific region around the atom's nucleus....
, are; n = 2, l = 1 or 0, ml = 1, or 0, or −1, ms = −1/2 or 1/2.

Note that molecular orbitals require totally different quantum numbers, because the Hamiltonian
Hamiltonian (quantum mechanics)

In quantum mechanics, the Hamiltonian H is the observable corresponding to the total energy of the system. As with all observables, the Spectrum of the Hamiltonian is the set of possible outcomes when one measures the total energy of a system....
 and its symmetries are quite different.

Quantum numbers with spin-orbit interaction

When one takes the spin-orbit interaction
Spin-orbit interaction

In quantum physics, the spin-orbit interaction is any interaction of a particle's spin with its motion. The first and best known example of this is that spin-orbit interaction causes shifts in an electron's energy level , due to electromagnetic interaction between the electron's spin and the nucleus's electric field, through which it moves...
 into consideration, l, m and s no longer commute
Commutativity

In mathematics, commutativity is the process to change the order of something without changing the end result. It is a fundamental property of many binary operations throughout mathematics, and many Mathematical proof depend on it....
 with the Hamiltonian
Hamiltonian (quantum mechanics)

In quantum mechanics, the Hamiltonian H is the observable corresponding to the total energy of the system. As with all observables, the Spectrum of the Hamiltonian is the set of possible outcomes when one measures the total energy of a system....
, and their value therefore changes over time. Thus another set of quantum numbers should be used. This set includes
  • The total angular momentum quantum number
    Azimuthal quantum number

    The Azimuthal quantum number symbolized as l is a quantum number for an atomic orbital that determines its orbital angular momentum. The azimuthal quantum number is the second of a set of quantum numbers which describe the unique quantum state of an electron and is designated by the letter l....
     (j = 1/2,3/2 ... n−1/2) gives the total angular momentum
    Angular momentum

    In physics, the angular momentum of a particle about an origin is a vector quantity related to rotation, equal to the mass of the particle multiplied by the cross product of the position vector of the particle with its velocity vector....
     through the relation .
  • The projection of the total angular momentum along a specified axis
    Azimuthal quantum number

    The Azimuthal quantum number symbolized as l is a quantum number for an atomic orbital that determines its orbital angular momentum. The azimuthal quantum number is the second of a set of quantum numbers which describe the unique quantum state of an electron and is designated by the letter l....
     (mj = -j,-j+1... j), which is analogous to m, and satisfies .
  • Parity
    Parity (physics)

    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 spatial coordinates:...
    . 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. n = 2 l = 1, ml = 1, ms = +1/2
  2. n = 2 l = 1, ml = 1, ms = -1/2
  3. n = 2 l = 1, ml = 0, ms = +1/2
  4. n = 2 l = 1, ml = 0, ms = -1/2
  5. n = 2 l = 1, ml = -1, ms = +1/2
  6. n = 2 l = 1, ml = -1, ms = -1/2
  7. n = 2 l = 0, ml = 0, ms = +1/2
  8. n = 2 l = 0, ml = 0, ms = -1/2


The quantum state
Quantum state

In quantum physics, a quantum State is a mathematical object that fully describes a Quantum system. One typically imagines some experimental apparatus and procedure which "prepares" this quantum state; the mathematical object then reflects the setup of the apparatus....
s in the system can be described as linear combination of these eight states. However, in the presence of spin-orbit interaction
Spin-orbit interaction

In quantum physics, the spin-orbit interaction is any interaction of a particle's spin with its motion. The first and best known example of this is that spin-orbit interaction causes shifts in an electron's energy level , due to electromagnetic interaction between the electron's spin and the nucleus's electric field, through which it moves...
, if one wants to describe the same system by eight states which are eigenvectors of the Hamiltonian
Hamiltonian (quantum mechanics)

In quantum mechanics, the Hamiltonian H is the observable corresponding to the total energy of the system. As with all observables, the Spectrum of the Hamiltonian 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, mj = 3/2, odd parity (coming from state (1) above)
  • j = 3/2, mj = 1/2, odd parity (coming from states (2) and (3) above)
  • j = 3/2, mj = -1/2, odd parity (coming from states (4) and (5) above)
  • j = 3/2, mj = -3/2, odd parity (coming from state (6))
  • j = 1/2, mj = 1/2, odd parity (coming from state (2) and (3) above)
  • j = 1/2, mj = -1/2, odd parity (coming from states (4) and (5) above)
  • j = 1/2, mj = 1/2, even parity (coming from state (7) above)
  • j = 1/2, mj = -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
Quantum state

In quantum physics, a quantum State is a mathematical object that fully describes a Quantum system. One typically imagines some experimental apparatus and procedure which "prepares" this quantum state; the mathematical object then reflects the setup of the apparatus....
s of the standard model
Standard Model

The Standard Model of particle physics is a theory of three of the four known fundamental interactions and the elementary particles that take part in these interactions....
 of particle physics
Particle physics

Particle physics is a branch of physics that studies the elementary particle constituents of matter and radiation, and the interactions between them....
, and hence the quantum numbers of these particles bear the same relation to the Hamiltonian
Hamiltonian (quantum mechanics)

In quantum mechanics, the Hamiltonian H is the observable corresponding to the total energy of the system. As with all observables, the Spectrum of the Hamiltonian 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 Hamiltonian
Hamiltonian (quantum mechanics)

In quantum mechanics, the Hamiltonian H is the observable corresponding to the total energy of the system. As with all observables, the Spectrum of the Hamiltonian 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 field theory
Field theory

Field theory may refer to:*Field theory , the theory of the algebraic concept of field*Field theory , a physical theory which employs fields in the physical sense...
 to distinguish between spacetime
Spacetime

In physics, spacetime is any mathematical model that combines space and Time in physics into a single continuum . Spacetime is usually interpreted with space being Three-dimensional space and time playing the role of a fourth dimension that is of a different sort than the spatial dimensions....
 and internal symmetries.

Typical quantum numbers related to spacetime symmetries
Spacetime symmetries

Spacetime symmetries refers to aspects of spacetime that can be described as exhibiting some form of symmetry. The role of symmetry in physics is important, for example, in simplifying solutions to many problems....
 are spin
Spin (physics)

In quantum mechanics, spin is a fundamental property of atomic nucleus, hadrons, and elementary particles. For particles with non-zero spin, spin direction is an important intrinsic degrees of freedom ....
 (related to rotational symmetry), the parity
Parity (physics)

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 spatial coordinates:...
, C-parity and T-parity (related to the Poincare symmetry of spacetime
Spacetime

In physics, spacetime is any mathematical model that combines space and Time in physics into a single continuum . Spacetime is usually interpreted with space being Three-dimensional space and time playing the role of a fourth dimension that is of a different sort than the spatial dimensions....
). Typical internal symmetries are lepton number
Lepton number

In high energy 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 non-leptonic particles 0....
 and baryon number
Baryon number

In particle physics, the baryon number is an conservation laws quantum number of a system. It is defined as:whereWhy one third? According to the laws of strong interaction there cannot be any bare color charge, i.e....
 or the electric charge
Electric charge

Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. Electrically charged matter is influenced by, and produces, electromagnetic fields....
. (For a full list of quantum numbers of this kind see the article on flavour
Flavour (particle physics)

In 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; 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
Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
 called Z2.

General principles


Atomic physics



Particle physics





See also

  • Quantum
    Quantum

    In physics, a quantum is an indivisible entity of a quantity that has the same units as the Planck constant and is related to both energy and momentum of elementary particles of matter and of photons and other bosons....
  • Quantum mechanics
    Quantum mechanics

    Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
  • Quantum field theory
    Quantum field theory

    Quantum field theory or QFT provides a theoretical framework for constructing quantum mechanics models of systems classically described by field or of Many-body problem....
  • Many-worlds interpretation
    Many-worlds interpretation

    The many-worlds interpretation is an interpretation of quantum mechanics.It is also known as MWI, the relative state formulation, theory of the universal wavefunction, parallel universes, many-universes interpretation or just many worlds....
  • Interpretation of quantum mechanics
    Interpretation of quantum mechanics

    An interpretation of quantum mechanics is a statement which attempts to explain how quantum mechanics informs our understanding of nature. Although quantum mechanics has received thorough experimental testing, many of these experiments are open to different interpretations....