Magnetic quantum number

The four quantum numbers conventionally used to describe the quantum state of an electron in an atom are the principal quantum number n, the azimuthal (orbital) quantum number ${\displaystyle \ell }$, and the magnetic quantum numbers m_l and m_s. Electrons in a given subshell of an atom (such as s, p, d, or f) are defined by values of ${\displaystyle \ell }$ (0, 1, 2, or 3). The orbital magnetic quantum number takes integer values in the range from ${\displaystyle -\ell }$ to ${\displaystyle +\ell }$, including zero. Thus the s, p, d, and f subshells contain 1, 3, 5, and 7 orbitals each. Each of these orbitals can accommodate up to two electrons (with opposite spins), forming the basis of the periodic table.

Other magnetic quantum numbers are similarly defined, such as m_j for the z-axis component the total electronic angular momentum j, and m_I for the nuclear spin I. Magnetic quantum numbers are capitalized to indicate totals for a system of particles, such as M_L or m_L for the total z-axis orbital angular momentum of all the electrons in an atom.

Derivation

There is a set of quantum numbers associated with the energy states of the atom. The four quantum numbers ${\displaystyle n}$, ${\displaystyle \ell }$, ${\displaystyle m_l}$, and ${\displaystyle m_s}$ specify the complete quantum state of a single electron in an atom called its wavefunction or orbital. The Schrödinger equation for the wavefunction of an atom with one electron is a separable partial differential equation. (This is not the case for the neutral helium atom or other atoms with mutually interacting electrons, which require more sophisticated methods for solution) This means that the wavefunction as expressed in spherical coordinates can be broken down into the product of three functions of the radius, colatitude (or polar) angle, and azimuth:

${\displaystyle \psi (r,\theta ,\varphi )=R(r)P(\theta )F(\varphi )}$

The differential equation for ${\displaystyle F}$ can be solved in the form ${\displaystyle F(\varphi )=A{e}^{\lambda \varphi }}$. Because values of the azimuth angle ${\displaystyle \varphi }$ differing by 2${\displaystyle \pi }$ radians (360 degrees) represent the same position in space, and the overall magnitude of ${\displaystyle F}$ does not grow with arbitrarily large ${\displaystyle \varphi }$ as it would for a real exponent, the coefficient ${\displaystyle \lambda }$ must be quantized to integer multiples of ${\displaystyle i}$, producing an imaginary exponent: ${\displaystyle \lambda =i{m}_l}$. These integers are the magnetic quantum numbers. The same constant appears in the colatitude equation, where larger values of ${\displaystyle {m_l}^2}$ tend to decrease the magnitude of ${\displaystyle P(\theta ),}$ and values of ${\displaystyle m_l}$ greater than the azimuthal quantum number ${\displaystyle \ell }$ do not permit any solution for ${\displaystyle P(\theta ).}$