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. Any quantum system can have one or more quantum numbers; it is thus rigorous to list all possible quantum numbers.
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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. A quantized system requires at least one quantum number. 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: energy, angular momentum, magnetic moment and spin.
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 Schrödinger, 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, ℓ, mℓ, ms. It is also the common nomenclature in the classical description of nuclear particle states (e.g. protons and neutrons).
* Note that, since atoms and electrons are in a state of constant motion, there is no universal fixed value for mℓ and ms values. Therefore, the mℓ and ms 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 mℓ = −1 or mℓ = 0, or mℓ = 1, but the mℓ value of the other electron in that orbital must be the same, and the mℓ 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 ≤ n | n = 1, 2, 3, … |
| azimuthal quantum number (angular momentum) | ℓ | subshell (s orbital is listed as 0, p orbital as 1 etc.) | 0 ≤ ℓ ≤ n − 1 | for n = 3: ℓ = 0, 1, 2 (s, p, d) |
| magnetic quantum number, (projection of angular momentum) | mℓ | energy shift (orientation of the subshell's shape) | −ℓ ≤ mℓ ≤ ℓ | for ℓ = 2: mℓ = −2, −1, 0, 1, 2 |
| spin projection quantum number | ms | spin of the electron (−½ = counter-clockwise, ½ = clockwise) | −½, ½ | for an electron, either: −½, ½ |
Example: The quantum numbers used to refer to the outermost valence electrons of the Carbon (C) atom, which are located in the 2p atomic orbital, are; n = 2 (2nd electron shell), ℓ = 1 (p orbital subshell), mℓ = 1, 0 or −1, ms = ½ (parallel spins).
Molecular orbitals require different quantum numbers, because the Hamiltonian and its symmetries are quite different.
When one takes the spin-orbit interaction into consideration, the ℓ-, m- and s-operators no longer commute 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 eight states, defined by their quantum numbers:
| n | ℓ | mℓ | ms | ℓ + s | ℓ - s | ml + ms | ||
|---|---|---|---|---|---|---|---|---|
| #1. | 2 | 1 | 1 | +1/2 | 3/2 | 3/2 | ||
| #2. | 2 | 1 | 1 | -1/2 | 3/2 | 1/2 | 1/2 | |
| #3. | 2 | 1 | 0 | +1/2 | 3/2 | 1/2 | 1/2 | |
| #4. | 2 | 1 | 0 | -1/2 | 3/2 | 1/2 | -1/2 | |
| #5. | 2 | 1 | -1 | +1/2 | 3/2 | 1/2 | -1/2 | |
| #6. | 2 | 1 | -1 | -1/2 | 3/2 | -3/2 | ||
| #7. | 2 | 0 | 0 | +1/2 | 1/2 | -1/2 | 1/2 | |
| #8. | 2 | 0 | 0 | -1/2 | 1/2 | -1/2 | -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, | 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) above) |
| j = 1/2, | mj = | 1/2, | odd parity | (coming from states (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 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 quantum 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.