# Quantum Numbers

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## Quantum numbers

The quantities, usually discrete with integer or half-integer values, which are needed to characterize a physical system of one or more atomic or subatomic particles. Specification of the set of quantum numbers serves to define such a system or, in other words, to label the possible states the system may have. In general, quantum numbers are obtained from conserved quantities determinable by performing symmetry transformations consisting of arbitrary variations of the system which leave the system unchanged. For example, since the behavior of a set of particles should be independent of the location of the origin in space and time (that is, the symmetry operation is translation in space-time), it follows that momentum and energy are rigorously conserved. *See* Symmetry laws (physics)

In general, each physical system must be studied individually to find the symmetry transformations, and thus the conserved quantities and possible quantum numbers. The quantum numbers themselves, that is, the actual state labels, are usually the eigenvalues of the physical operators corresponding to the conserved quantities for the system in question. *See* Eigenvalue (quantum mechanics), Elementary particle, Parity (quantum mechanics)

It is not necessary that the conserved quantity be “quantized” in order to be regarded as a quantum number; for example, a free particle possesses energy and momentum, both of which can have values from a continuum but which are used to specify the state of the particle.

## Quantum Numbers

integers (0, 1, 2, … ) or half-integers (1/2, 3/2, 5/2, …) that define the possible discrete values of the physical quantities characterizing quantum systems (atomic nucleus, atom, molecule) and individual elementary particles. The use of quantum numbers in quantum mechanics reflects features of the discreteness of processes occurring in the mi-croworld, and is closely connected with the existence of the quantum of action, or the Planck constant ℏ. Quantum numbers were first introduced in physics to describe the empirically determined regularities of atomic spectra, but the meaning of quantum numbers and the associated discreteness of certain quantities characterizing the dynamics of microparticles were revealed only by quantum mechanics.

A set of quantum numbers that thoroughly defines the state of a quantum system is said to be complete. The aggregate of the states that satisfy all possible values of the quantum numbers of a complete set forms the complete system of states. The state of an electron in the atom is determined by four quantum numbers corresponding to the four degrees of freedom of the electron (three degrees of freedom are connected with the three coordinates that define the electron’s spatial position, and the fourth, internal, degree of freedom is connected with its spin). For the hydrogen atom and hydrogen-like atoms these quantum numbers forming a complete set are given below.

(1)The principal quantum number *n* = 1, 2, 3, … defines the energy levels of the electron.

(2)The azimuthal (or orbital) quantum number *l* = 0, 1, 2, …, *n* – 1 gives the spectrum of the possible values of the square of the orbital angular momentum of an electron: *M*^{2}_{i} = ℏ^{2}*l*(*l* + 1).

(3)The magnetic quantum number m_{l} characterizes the possible values of a projection M_{lz} of the orbital angular momentum *M _{l}* on some arbitrarily selected direction (taken as the

*z*-axis): M

_{lz}=ℏ

*m*. This number may assume integer values in the interval from –

_{l}*l*to +

*l*(for a total of 2

*l*+ 1 values).

(4)The magnetic spin quantum number, or simply the spin quantum number, m_{s} characterizes the possible values of a projection of the electron’s spin and may assume two values: m_{s} = ±1/2.

Definition of a state of an electron using the quantum numbers *n, l, m _{l}*, and

*m*does not take into account the so-called fine structure of energy levels—the splitting of levels with a given

_{s}*n*(when

*n*≥ 2) as a result of the spin’s influence on the orbital motion of the electron. When this interaction is taken into account, the quantum numbers

*j*and

*m*are used to characterize the state of the electron instead of

_{j}*m*and m

_{l}_{s}.

(5)The quantum number *j* of the total angular momentum *M* of an electron (the orbital moment + the spin moment) determines the possible values of the square of the total momentum: *M*^{2} = ℏ^{2}*j*(*j* + 1) and, for a given *l*, may assume two values: *j* = *l* ± 1/2.

(6)The magnetic quantum number *m _{j}* of the total angular momentum determines the possible values of the projection of the total angular momentum on the

*z*-axis,

*M*= ℏ

_{z}*m*, it may assume

_{j}*2j*+ 1 values:

*m*= -

_{j}*j*, -

*j*+ 1, …, +

*j*.

The same quantum numbers approximately describe the states of individual electrons in complex (multielectron) atoms as well as the state of the individual nucleons—protons and neutrons— in atomic nuclei. In this case, *n* represents the successive (in order of increasing energy) energy levels with a given *l*. The state of a multielectron atom as a whole is defined by the following quantum numbers: (1) the quantum number of the total orbital angular momentum of an atom *L*, which is defined by the motion of all electrons, *L* = 0, 1, 2, … ; (2) the quantum number of the total angular momentum of the atom *j*, which may assume all values differing by units of 1 from *J* = | *L — S* | to *J = L* + *S*, where *S* is the the total spin of the atom (in ℏ units); and (3) the magnetic quantum number m_{j}, which defines the possible values of a projection of the atom’s total angular momentum on the *z*-axis, *M*_{z} = *m _{j}*ℏ, and which assumes 2

*J*+ 1 values.

Still another quantum number—the parity of the state *P* —is used to characterize the state of the atom and of the quantum System in general. It assumes the values +1 or −1 depending on whether the wave function defining the state of the system retains its sign on inversion of the coordinates r with respect to the origin of coordinates (that is, when r —?> — r is replaced) or changes its sign. The parity Pis equal to (-1)^{l} for the hydrogen atom and (—1)^{L} for multielectron atoms.

Quantum numbers also have proved convenient for formulating the selection rules that determine the possible types of quantum transitions.

A number of other quantum numbers are introduced in the physics of elementary particles and nuclear physics. The quantum numbers of elementary particles are internal characteristics of the particles that define their interactions and the regularities of mutual transformations. In addition to the spin *s*, which may be an integer or half-integer (in ℏ units), the following includes quantum numbers of elementary particles: (1) the electric charge *Q*, which in all known elementary particles is equal either to 0 or to a positive or negative integer (in units of the elementary electric charge *e);* (2) the baryon charge *B*, which is equal to 0 or 1 (for antiparticles, 0 or — 1); (3) lepton charges, or lepton numbers—the electric number L_{e} and the muons —which are equal to 0 or +1 (for antiparticles, 0 or —1);(4) the isotopic spin *T*, which is an integer or half-integer; (5) the strangeness *S* or the hypercharge *Y* (which is related to *S* by the equation *Y = S* + *B*) —all known elementary particles (or antiparticles) have *S*=0 or ±1, ±2, ±3; and (6) the intrinsic parity II, a quantum number that characterizes the properties of the symmetry of elementary particles with respect to inversion of the coordinates and may be equal to +1 (such particles are called particles of even parity) or —1 (particles of odd parity). These quantum numbers also are applied to systems consisting of several elementary particles and, in particular, to atomic nuclei. Here, the total values of the electric, baryon, and lepton charges and of the strangeness of a system of particles are equal to the algebraic sum of the corresponding quantum numbers of the individual particles; the total spin and isotopic spin are obtained on the basis of the quantum rules for the addition of angular momentums; and the intrinsic parities of the particles are multiplied.

Physical quantities defining the motion of a quantum mechanical particle (or system) that are conserved in the process of motion but that do not necessarily belong to the discrete spectrum of possible values are often said to be quantum numbers in the broad sense. For example, the energy of a free-moving electron (which has a continuous spectrum of values) may be considered as one of its quantum numbers.

D. V. GAL’TSOV