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#1 2022-02-16 22:24:25

Jai Ganesh
Registered: 2005-06-28
Posts: 47,159

Quantum Numbers (Detailed)

Quantum Numbers

In chemistry and quantum physics, quantum numbers describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can be known with precision at the same time as the system's energy - and their corresponding eigenspaces. Together, a specification of all of the quantum numbers of a quantum system fully characterize a basis state of the system, and can in principle be measured together.

An important aspect of quantum mechanics is the quantization of many observable quantities of interest. In particular, this leads to quantum numbers that take values in discrete sets of integers or half-integers; although they could approach infinity in some cases. This distinguishes quantum mechanics from classical mechanics where the values that characterize the system such as mass, charge, or momentum, all range continuously. Quantum numbers often describe specifically the energy levels of electrons in atoms, but other possibilities include angular momentum, spin, etc. An important family is flavour quantum numbers – internal quantum numbers which determine the type of a particle and its interactions with other particles through the fundamental forces. Any quantum system can have one or more quantum numbers; it is thus difficult to list all possible quantum numbers.

Quantum numbers needed for a given system

The tally of quantum numbers varies from system to system and has no universal answer. Hence these parameters must be found for each system to be analyzed. A quantized system requires at least one quantum number. The dynamics (i.e. time evolution) of any quantum system are described by a quantum operator in the form of a Hamiltonian, H. There is one quantum number of the system corresponding to the system's energy; i.e., one of the eigenvalues of the Hamiltonian. There is also one quantum number for each linearly independent operator O that commutes with the Hamiltonian. A complete set of commuting observables (CSCO) that commute with the Hamiltonian characterizes the system with all its quantum numbers. There is a one-to-one relationship between the quantum numbers and the operators of the CSCO, with each quantum number taking one of the eigenvalues of its corresponding operator. As a result of the different basis that may be arbitrarily chosen to form a complete set of commuting operators, different sets of quantum numbers may be used for the description of the same system in different situations.

Electron in an atom

Four quantum numbers can describe an electron in an atom completely:

* Principal quantum number (n)
* Azimuthal quantum number (ℓ)
* Magnetic quantum number (mℓ)
* Spin quantum number (ms)

The spin-orbital interaction, however, relates these numbers. Thus, a complete description of the system can be given with fewer quantum numbers, if orthogonal choices are made for these basis vectors.


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.

Common terminology

The model used here describes electrons using four quantum numbers, n, ℓ, mℓ, ms, 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.

Principal quantum number

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

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

) 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.

Azimuthal quantum number

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.

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:

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.

Magnetic 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:

The values of mℓ range from -ℓ to ℓ, with integer intervals.

The s subshell (ℓ = 0) contains only one orbital, and therefore the mℓ 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 mℓ of an electron in a p orbital will be -1, 0, or 1. The d subshell (ℓ = 2) contains five orbitals, with mℓ values of -2, -1, 0, 1, and 2.

Spin quantum number

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:


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

An electron has spin number s = 1/2
will be

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.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.


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