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.

1 How many quantum numbers?
- 1.1 Quantum numbers with spin-orbit interaction
2 Elementary particles
3 References and external links
4 See also

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 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.

To completely describe an electron in an atom, four quantum numbers are needed, these are conventionally known as:

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.

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.

The spin projection quantum number (m_s = −1/2 or +1/2), is the intrinsic angular momentum of the electron. This is the projection of the spin 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	$n \$	shell	$1 \le n \,\!$	$n=1,2,3...\,\!$
azimuthal quantum number (angular momentum)	$\ell \$	subshell (s orbital is listed as 0, p orbital as 1 etc.)	$(0 \le \ell \le 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 \le m_\ell \le \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.

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

Quantum numbers with spin-orbit interaction

For more details on this topic, see 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:

n = 2, l = 1, m_l = 1, m_s = +1/2
n = 2, l = 1, m_l = 1, m_s = -1/2
n = 2, l = 1, m_l = 0, m_s = +1/2
n = 2, l = 1, m_l = 0, m_s = -1/2
n = 2, l = 1, m_l = -1, m_s = +1/2
n = 2, l = 1, m_l = -1, m_s = -1/2
n = 2, l = 0, m_l = 0, m_s = +1/2
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

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.

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

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.
The particle data group
Lecture notes on quantum numbers

Quantum number

Contents

How many quantum numbers?

Quantum numbers with spin-orbit interaction

Elementary particles

References and external links

General principles

Atomic physics

Particle physics

See also

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