Magnetic quantum number in the context of "Atomic orbital"

In quantum mechanics, the Pauli exclusion principle (German: Pauli-Ausschlussprinzip) states that two or more identical particles with half-integer spins (i.e. fermions) cannot simultaneously occupy the same quantum state within a system that obeys the laws of quantum mechanics. This principle was formulated by Austrian physicist Wolfgang Pauli in 1925 for electrons, and later extended to all fermions with his spin–statistics theorem of 1940.

In the case of electrons in atoms, the exclusion principle can be stated as follows: in a poly-electron atom it is impossible for any two electrons to have the same two values of all four of their quantum numbers, which are: n, the principal quantum number; ℓ, the azimuthal quantum number; m_ℓ, the magnetic quantum number; and m_s, the spin quantum number. For example, if two electrons reside in the same orbital, then their values of n, ℓ, and m_ℓ are equal. In that case, the two values of m_s (spin) pair must be different. Since the only two possible values for the spin projection m_s are +1/2 and −1/2, it follows that one electron must have m_s = +1/2 and one m_s = −1/2.

In quantum mechanics, the azimuthal quantum number ℓ is a quantum number for an atomic orbital that determines its orbital angular momentum and describes aspects of the angular shape of the orbital. The azimuthal quantum number is the second of a set of quantum numbers that describe the unique quantum state of an electron (the others being the principal quantum number n, the magnetic quantum number m_ℓ, and the spin quantum number m_s).

For a given value of the principal quantum number n (electron shell), the possible values of ℓ are the integers from 0 to n − 1. For instance, the n = 1 shell has only orbitals with ${\displaystyle \ell =0}$, and the n = 2 shell has only orbitals with ${\displaystyle \ell =0}$, and ${\displaystyle \ell =1}$.

In quantum physics and chemistry, quantum numbers are quantities that characterize the possible states of the system. To fully specify the state of the electron in a hydrogen atom, four quantum numbers are needed. The traditional set of quantum numbers includes the principal, azimuthal, magnetic, and spin quantum numbers. To describe other systems, different quantum numbers are required. For subatomic particles, one needs to introduce new quantum numbers, such as the flavour of quarks, which have no classical correspondence.

Quantum numbers are closely related to eigenvalues of observables. When the corresponding observable commutes with the Hamiltonian of the system, the quantum number is said to be "good", and acts as a constant of motion in the quantum dynamics.