Quantum number

Quantum numbers describe values of conserved quantities in the dynamics of the quantum system. Perhaps the most peculiar aspect of quantum mechanics is the quantization of observable quantities. This is distinguished from classical mechanics where the values can range continuously. They often describe specifically the energies of electrons in atoms, but other possibilities include angular momentum, spin 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, 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.

Traditional nomenclature

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 theory of Friedrich Hund, Robert S. Mulliken, and contributions from Schrodinger, Slater and John Lennard-Jones. This system of nomenclature incorporated Bohr energy levels, Hund-Mulliken orbital theory, and observations on electron spin based on spectroscopy and Hund's rules.

This model describes electrons using four quantum numbers, $n\$, $\ell \$, $m_{\ell }\$, and $m_{s}\,\!$.

• The first, $n\$, describes the electron shell, 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 cesium (Cs), the outermost valence electron is in the shell with energy level 6, so an electron in cesium can have an $n\$ value from 1 to 6.
• The second, $\ell \$, describes the subshell (0 = s orbital, 1 = p orbital, 2 = d orbital, 3 = f orbital, etc.).
  • The value of $\ell \$ ranges from $0$ to $\ n-1$. This is because the first p orbital (l=1) appears in the second electron shell (n=2), the first d orbital (l=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_{\ell }\$, describes the specific orbital (or "cloud") within that subshell.*
  • The values of $m_{\ell }\$ range from $-\ell \$ to $\ell \$. The s subshell (l=0) contains only one orbital, and therefore the m_l of an electron in an s subshell will always be 0. The p subshell (l=1) contains three orbitals (in some systems, depicted as three "dumbbell-shaped" clouds), so the m_l of an electron in a p subshell will be -1, 0, or 1. The d subshell (l=2) contains five orbitals, with m_l values of -2,-1,0,1, and 2.
• The fourth, $m_{s}\,\!$, describes the spin of the electron within that orbital.*
  • Because an orbital never contains more than two electrons, $m_{s}\,\!$ will be either ${\begin{matrix}{\frac {1}{2}}\end{matrix}}$ or $-{\begin{matrix}{\frac {1}{2}}\end{matrix}}$, corresponding with "spin" and "opposite spin".

* Note that, since atoms and electrons are in a state of constant motion, there is no universal fixed value for m_l and m_s values. Therefore, the m_l 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_l=-1 or m_l=0, or m_l=1, but the m_l value of the other electron in that orbital must be the same, and the m_l 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	$n\$	shell	$1\leq n\,\!$	$n=1,2,3...\,\!$
azimuthal quantum number (angular momentum)	$\ell \$	subshell (s orbital is listed as 0, p orbital as 1 etc.)	$(0\leq \ell \leq n-1)\$	for $n=3\,\!$: $\ell =0,1,2\,(s,p,d)\$
magnetic quantum number, (projection of angular momentum)	$m_{\ell }\$	energy shift (orientation of the subshell's shape)	$-\ell \leq m_{\ell }\leq \ell \$	for $\ell =2\$: $m_{\ell }=-2,-1,0,1,2\,\!$
spin projection quantum number	$m_{s}\,\!$	spin of the electron (-1/2 = counter-clockwise, 1/2 = clockwise)	$-{\begin{matrix}{\frac {1}{2}}\end{matrix}},{\begin{matrix}{\frac {1}{2}}\end{matrix}}\$	for an electron, either: $-{\begin{matrix}{\frac {1}{2}}\end{matrix}},{\begin{matrix}{\frac {1}{2}}\end{matrix}}\$

Example: The quantum numbers used to refer to the outermost valence electron of the Carbon (C) atom, which is located in the 2p atomic orbital, are; n = 2 (group 2), l = 1 or 0, m_l = 1, or 0, or −1, m_s = −1/2 or 1/2.

As applied to the Hamiltonian and Schrodinger equation

• The principal quantum number (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 (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 (l = 0, 1 ... n−1) (also known as the angular quantum number or orbital quantum number) gives the orbital angular 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 an atomic orbital and strongly influences chemical bonds 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 (m_l = −l, −l+1 ... 0 ... l−1, l) is the eigenvalue, $L_{z}=m_{\ell }\hbar$. This is the projection of the orbital angular momentum along a specified axis.
• The spin projection quantum number (m_s = −1/2 or +1/2), is the intrinsic angular momentum of the electron or nucleon. This is the projection of the spin s=1/2 along the 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 Rules, 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 renowned Stern-Gerlach experiment.

Molecular orbitals require different quantum numbers, because the Hamiltonian and its symmetries are quite different.

Quantum numbers with spin-orbit interaction

Further information: Clebsch-Gordan coefficients

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 total angular momentum quantum number (j = 1/2,3/2 ... n−1/2) gives the total angular momentum through the relation $J^{2}=\hbar ^{2}j(j+1)$.
• The projection of the total angular momentum along a specified axis (m_j = -j,-j+1... j), which is analogous to m, and satisfies $m_{j}=m_{l}+m_{s}$.
• 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. n = 2, l = 1, m_l = 1, m_s = +1/2
2. n = 2, l = 1, m_l = 1, m_s = -1/2
3. n = 2, l = 1, m_l = 0, m_s = +1/2
4. n = 2, l = 1, m_l = 0, m_s = -1/2
5. n = 2, l = 1, m_l = -1, m_s = +1/2
6. n = 2, l = 1, m_l = -1, m_s = -1/2
7. n = 2, l = 0, m_l = 0, m_s = +1/2
8. n = 2, l = 0, m_l = 0, m_s = -1/2

The quantum states 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 eigenvectors 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 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

For a more complete description of the quantum states of elementary particles, see Standard model and Flavour (particle physics).

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 model of particle physics, 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 field theory to distinguish between spacetime and internal symmetries.

Typical quantum numbers related to spacetime symmetries are spin (related to rotational symmetry), the 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.)

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 group called Z₂.

See also

• Quantum mechanics
• Electronic configuration

References and external links

General principles

• Dirac, Paul A.M. (1982). Principles of quantum mechanics. Oxford University Press. ISBN 0-19-852011-5.

Atomic physics

• Quantum numbers for the hydrogen atom

Particle physics

• Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 0-13-805326-X.
• Halzen, Francis and Martin, Alan D. (1984). QUARKS AND LEPTONS: An Introductory Course in Modern Particle Physics. John Wiley & Sons. ISBN 0-471-88741-2.{{cite book}}: CS1 maint: multiple names: authors list (link)
• The particle data group
• Lecture notes on quantum numbers