# Principal quantum number

The **principal quantum number**, usually designated by *n*, appears in the description of the electronic structure of atoms. The quantum number first arose in the Bohr-Sommerfeld theory of the hydrogen atom, but it is also part of the solution of the Schrödinger equation for hydrogen-like atoms. It is a positive integer (*n* = 1, 2, 3, …) that indexes atomic shells. Historically, atomic shells were indicated by the capital letters K, L, M, … for *n=1,2,3,*…, respectively, but this usage is dying out.

In the Bohr-Sommerfeld ("old") quantum theory, the electron in a hydrogen-like (one-electron) atom moves in elliptic orbits. The principal quantum number appears in this theory at two places: in the energy *E*_{n} of the electron and in the length *a*_{n} of the semi-major axis of the *n*th orbit,

where Ry is the Rydberg energy for infinite nuclear mass (= 13.605 6923 eV). Further, *m*_{e} is the mass of the electron, −*e* is the charge of the electron, *Ze* is the charge of the nucleus, ε_{0} is the electric constant, and is Planck's reduced constant.

In the "new" quantum mechanics (of Heisenberg, Schrödinger, and others) the energy *E*_{n} of a bound electron in a hydrogen-like atom satisfies the exact same equation, but the electron *orbit* is replaced by an electron *orbital*; the latter has no radius. However, in the new quantum theory the same expression for *a*_{n} appears in the form of the expectation value of *r* (the length of the position vector of the electron) with respect to a state with principal quantum number *n*. That is, quantum mechanics gives the same measure for the "size" of a one-electron atom (in state *n*) as the old quantum theory.

Strictly speaking, the principal quantum number is not defined for many-electron atoms. However, in a fairly good approximate description (central field plus independent-particle model) of the many-electron atom, the principal quantum number does appear and hence *n* is a label that is often applied to many-electron atoms as well.

## Azimuthal and magnetic quantum numbers

An atomic shell consists of atomic subshells that are labeled by the *azimuthal quantum number*, commonly denoted by *ℓ*. The azimuthal quantum number is more often referred to as *angular momentum quantum number*, because the eigenvalues of the squared orbital angular momentum operator are equal to *ℓ*(*ℓ*+1) ℏ².

For a given atomic shell of principal quantum number *n*, *ℓ* runs from 0 to *n*−1, as follows from the solution of the Schrödinger equation. In total, an atomic shell with quantum number *n* consists of *n* subshells and has

spatial (i.e., function of the position vector of the electron) atomic orbitals.
An atomic subshell consists of 2*ℓ*+1 atomic orbitals labeled by the *orbital magnetic quantum number*, almost invariably denoted by *m*. For given *ℓ*, *m* runs over 2*ℓ*+1 values: *m* = −*ℓ*, −*ℓ*+1, …, *ℓ*−1, *ℓ*. The number *m* is proportional to the eigenvalues of the *z*-component of the orbital angular momentum operator that has eigenvalues *m*ℏ.

For historical reasons the orbitals of certain azimuthal quantum number *ℓ* are denoted by letters:
*s*, *p*, *d*, *f*, *g*, for *ℓ* = 0, 1, 2, 3, 4, respectively. For instance an atomic orbital with *n* = 4 and *ℓ* = 2 is written as 4*d*. If the *n* = 4, *ℓ* = 2 subshell is occupied *k* times (there are *k* electrons in the 4*d* orbital), we indicate this by writing (4*d*)^{k}.

Any spatial atomic orbital can be occupied at most twice, which is because the *spin magnetic quantum number* *m*_{s} (proportional to the eigenvalues of the *z*-component of the spin angular momentum operator ) can have only two values: +½ (spin up) and −½ (spin down). In addition, the Pauli exclusion principle states that no two electrons with the same four quantum numbers *n*, *ℓ*, *m*, and *m*_{s} can occupy the same atomic orbital. As a consequence, the spatial orbital (*n*,*ℓ*,*m*) can be occupied at most by two electrons with spin *m*_{s} = ±½.
Hence a subshell can accommodate at most 2(2*l*+1) electrons. If a subshell accommodates the maximum number of electrons, it is called *closed*. If the atomic shell *n* contains the maximum of 2*n*^{2} electrons, it is also called *closed*. For instance the noble gas neon in its lowest energy state has the electron configuration (1*s*)^{2}(2*s*)^{2}(2*p*)^{6}, that is, all its shells and subshells are closed.