Quantum number

A quantum number is any one of a set of numbers used to specify the full quantum state of any system in quantum mechanics. Each quantum number specifies the value of a conserved quantity in the dynamics of the quantum system. Since any quantum system can have one or more quantum numbers, it is a futile job to list all possible quantum numbers. This article therefore illustrates the concepts by choosing two well-known examples, after a brief introduction to the general concept of quantum numbers.

How many quantum numbers?

How many quantum numbers are needed to describe any given system? There is 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 is described by a quantum Hamiltonian, H. There is one quantum number of the system corresponding to the energy, ie, 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. In various fields of study, there may be slightly different conventions for writing the quantum numbers, but they can all be related to the definition given here.

A necessary condition to completely define the state of the system is that at least as many quantum numbers as degrees of freedom are required. (Note: In this context the term degrees of freedom can be used as a synonym for generalized coordinates, as opposed to the way it is used in thermodynamics such as when one determines the heat capacity.)

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, Schroedinger equation and the Dirac equation.

The most widely studied set of quantum numbers is that for a single electron in an atom: not only because it is useful in chemistry, being the basic notion behind the periodic table, valence (chemistry) 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 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 nucleus, and the remainder. Since the potential is spherically symmetric, the full Hamiltonian commutes with J². J² itself commutes with any one of the components of the angular momentum vector, conventionally taken to be J_z. 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,...) denotes the eigenvalue of H with the J² part removed. This number therefore has a dependence only 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 angular momentum through the relation J² = l(l+1) h/2π, where h is the universal constant known as the Planck's constant. 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, J_z=m_lh/2π.
• The spin quantum number (m_s = −1/2 or +1/2) was found experimentally from spectroscopy.

To summarize, the quantum state of an electron is determined by the quantum numbers:

name	symbol	orbital meaning	range of values	value example
principal quantum number	$n\,\!$	shell	$1\leq n\,\!$	$n=1,2,3...\,\!$
azimuthal quantum number	$\ell \,\!$	subshell	$0\leq \ell \leq n-1\,\!$	for $n=3\,\!$: $\ell =0,1,2\,(s,p,d)\,\!$
magnetic quantum number	$m_{\ell }\,\!$	energy shift	$-\ell \leq m_{\ell }\leq \ell \,\!$	for $\ell =2\,\!$: $m_{\ell }=-2,-1,0,1,2\,\!$
spin quantum number	$m_{s}\,\!$	spin	$-{\begin{matrix}{\frac {1}{2}}\end{matrix}},{\begin{matrix}{\frac {1}{2}}\end{matrix}}\,\!$	always only: $-{\begin{matrix}{\frac {1}{2}}\end{matrix}},{\begin{matrix}{\frac {1}{2}}\end{matrix}}\,\!$

Example: The quantum numbers used to refer to the outer most valence electron of the Fluorine (F) atom, which is located in the 2p atomic orbital, are; n = 2, l = 1, m_l = 1, m_s = −1/2.

Note that molecular orbitals require totally different quantum numbers, because the Hamiltonian and its symmetries are quite different.

Elementary particles

For a more complete description of the quantum states of elementary particles see the articles on the 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; 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 called Z₂.

See also

• Principal quantum number
• Azimuthal quantum number
• Magnetic quantum number
• Spin quantum number
• Quantum mechanics and quantum state
• The Bohr atom and particle physics
• Symmetries, conservation laws and Noether's theorem

References and external links

General principles

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Atomic physics

• Quantum Numbers and Electron Configurations
• Quantum numbers for the hydrogen atom
• Lecture notes on quantum numbers

Particle physics

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• The particle data group