Quantum number

Different electrons in a system will have different quantum numbers. For example, the highest occupied orbital electron, the actual differentiating electron (i.e. the electron that differentiates an element from the previous one); , r the differentiating electron according to the aufbau approximation. In lanthanum, as a further illustration, the electrons involved are in the 6s; 5d; and 4f orbitals, respectively. In this case the principal quantum numbers are 6, 5, and 4.

The model used here describes electrons using four quantum numbers, n, ℓ, m_ℓ, m_s, given below. It is also the common nomenclature in the classical description of nuclear particle states (e.g. protons and neutrons). A quantum description of molecular orbitals require other quantum numbers, because the Hamiltonian and its symmetries are different.

The principal quantum number describes the electron shell, or energy level, of an electron. The value of n ranges from 1 to the shell containing the outermost electron of that atom, that is[1]

n = 1, 2, ...

For example, in caesium (Cs), the outermost valence electron is in the shell with energy level 6, so an electron in caesium can have an n value from 1 to 6.

For particles in a time-independent potential (see Schrödinger equation), it also labels the nth eigenvalue of Hamiltonian (H), that is, the energy E, with the contribution due to angular momentum (the term involving J²) left out. So this number depends only on the distance between the electron and the nucleus (that is, the radial coordinate r). The average distance increases with n. Hence quantum states with different principal quantum numbers are said to belong to different shells.

The azimuthal quantum number, also known as the (angular momentum quantum number or orbital quantum number), describes the subshell, and gives the magnitude of the orbital angular momentum through the relation.

L² = ħ² ℓ (ℓ + 1)

In chemistry and spectroscopy, ℓ = 0 is called s orbital, ℓ = 1, p orbital, ℓ = 2, d orbital, and ℓ = 3, f orbital.

The value of ℓ ranges from 0 to n − 1, so 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:[2]

ℓ = 0, 1, 2,..., n − 1

A quantum number beginning in n = 3,ℓ = 0, describes an electron in the s orbital of the third electron shell of an atom. 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. The azimuthal quantum number can also denote the number of angular nodes present in an orbital. For example, for p orbitals, ℓ = 1 and thus the amount of angular nodes in a p orbital is 1.

Shape of orbital is also given by azimuthal quantum number.

The magnetic quantum number describes the specific orbital (or "cloud") within that subshell, and yields the projection of the orbital angular momentum along a specified axis:

L_z = m_ℓ ħ

The values of m_ℓ range from −ℓ to ℓ, with integer intervals.[3]

The s subshell (ℓ = 0) contains only one orbital, and therefore the m_ℓ of an electron in an s orbital 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 orbital 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 spin quantum number describes the intrinsic spin angular momentum of the electron within each orbital and gives the projection of the spin angular momentum S along the specified axis:

S_z = m_s ħ.

In general, the values of m_s range from −s to s, where s is the spin quantum number, associated with the particle's intrinsic spin angular momentum:[4]

m_s = −s, −s + 1, −s + 2, ..., s − 2, s − 1, s.

An electron has spin number s = 1/2, consequently m_s will be ±1/2, referring to "spin up" and "spin down" states. Each electron in any individual orbital must have different quantum numbers because of the Pauli exclusion principle, therefore an orbital never contains more than two electrons.

There are no universal fixed values for m_ℓ and m_s. Rather, the m_ℓ and m_s values are arbitrary. The only restrictions on the choices for these constants 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 orbital could be described as m_ℓ = −1 or m_ℓ = 0 or m_ℓ = 1, but the m_ℓ value of the next unpaired electron in that orbital must be different; yet, the m_ℓ assigned to electrons in other orbitals again can be m_ℓ = −1 or m_ℓ = 0 or m_ℓ = 1).

These rules are summarized as follows:

Name	Symbol	Meaning	Range of values	Value examples
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	Orbital (orientation of the orbital)	−ℓ ≤ mℓ ≤ ℓ	for ℓ = 2: mℓ = −2, −1, 0, 1, 2
Spin quantum number	ms	spin of the electron (−1/2 = "spin down", 1/2 = "spin up")	−s ≤ ms ≤ s	for an electron s = 1/2, so ms = −1/2, +1/2

Example: The quantum numbers used to refer to the outermost valence electrons of a carbon (C) atom, which are located in the 2p atomic orbital, are; n = 2 (2nd electron shell), ℓ = 1 (p orbital subshell), m_ℓ = 1, 0, −1, m_s = 1/2 (parallel spins).

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, which represented spin with two possible values, was added as an ad hoc assumption to resolve the conflict; this supposition would later be explained in detail by relativistic quantum mechanics and from the results of the renowned Stern–Gerlach experiment.

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.[5]

When one takes the spin–orbit interaction into consideration, the L 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[6][7]

For example, consider the following 8 states, defined by their quantum numbers:

	n	ℓ	mℓ	ms		ℓ + s	ℓ − s	mℓ + ms
(1)	2	1	1	+1/2		3/2	1/2	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	1/2	−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 8 states. However, in the presence of spin–orbit interaction, if one wants to describe the same system by 8 states that are eigenvectors of the Hamiltonian (i.e. each represents a state that does not mix with others over time), we should consider the following 8 states:

j	mj	parity
3/2	3/2	odd	coming from state (1) above
3/2	1/2	odd	coming from states (2) and (3) above
3/2	−1/2	odd	coming from states (4) and (5) above
3/2	−3/2	odd	coming from state (6) above
1/2	1/2	odd	coming from states (2) and (3) above
1/2	−1/2	odd	coming from states (4) and (5) above
1/2	1/2	even	coming from state (7) above
1/2	−1/2	even	coming from state (8) above

In nuclei, the entire assembly of protons and neutrons (nucleons) has a resultant angular momentum due to the angular momenta of each nucleon, usually denoted I. If the total angular momentum of a neutron is j_n = ℓ + s and for a proton is j_p = ℓ + s (where s for protons and neutrons happens to be 1/2 again (see note)), then the nuclear angular momentum quantum numbers I are given by:

I = |j_n − j_p|, |j_n − j_p| + 1, |j_n − j_p| + 2, ..., (j_n + j_p) − 2, (j_n + j_p) − 1, (j_n + j_p)

Note: The orbital angular momenta of the nuclear (and atomic) states are all integer multiples of ħ while the intrinsic angular momentum of the neutron and proton are half-integer multiples. It should be immediately apparent that the combination of the intrinsic spins of the nucleons with their orbital motion will always give half-integer values for the total spin, I, of any odd-A nucleus and integer values for any even-A nucleus.

Parity with the number I is used to label nuclear angular momentum states, examples for some isotopes of hydrogen (H), carbon (C), and sodium (Na) are;[8]

1 1H	I = (1/2)+		9 6C	I = (3/2)−		20 11Na	I = 2+
2 1H	I = 1+		10 6C	I = 0+		21 11Na	I = (3/2)+
3 1H	I = (1/2)+		11 6C	I = (3/2)−		22 11Na	I = 3+
			12 6C	I = 0+		23 11Na	I = (3/2)+
			13 6C	I = (1/2)−		24 11Na	I = 4+
			14 6C	I = 0+		25 11Na	I = (5/2)+
			15 6C	I = (1/2)+		26 11Na	I = 3+

The reason for the unusual fluctuations in I, even by differences of just one nucleon, are due to the odd and even numbers of protons and neutrons – pairs of nucleons have a total angular momentum of zero (just like electrons in orbitals), leaving an odd or even number of unpaired nucleons. The property of nuclear spin is an important factor for the operation of NMR spectroscopy in organic chemistry,[7] and MRI in nuclear medicine,[8] due to the nuclear magnetic moment interacting with an external magnetic field.