quantum number

# tag · @ author · ! scientist

Electrons in an atom are like birds in a tree. Each bird occupies its own branch, at a certain height and with a certain flight direction. Quantum numbers are a precise description of the place: which branch (energy), from which side (orbital shape), in which nest (magnetic number), and which way it's facing (spin). No two addresses are the same.

History

The principal quantum number n was introduced by Niels Bohr in 1913 to explain hydrogen spectra. In 1916, Arnold Sommerfeld added the orbital and magnetic numbers. The concept took its final form in 1925, when Wolfgang Pauli discovered spin and formulated the exclusion principle.

How it works

An electron in an atom can only be in strictly defined states. The four numbers are like the coordinates of a parking space in a multistory garage: floor (n), row (l), spot (mₗ), and direction of the car (spin). No two cars can occupy the same position — that's how orbitals are filled.

💡 The spin quantum number of an electron takes only two values: +1/2 or -1/2, like a coin that always lands heads or tails, but never on its edge. It is precisely thanks to the Pauli principle, which forbids identical sets of numbers, that atoms do not collapse and form all the variety of elements.

In quantum mechanics, the state of a particle is fully described by a wave function, and quantum numbers arise as eigenvalues of operators of observable quantities. For an electron in a hydrogen atom, from the Schrödinger equation, we get the principal n (energy), orbital l (angular momentum), and magnetic mₗ (projection). The fourth — spin s and its projection mₛ — follows from the Dirac equation and describes the intrinsic angular momentum. The Pauli principle forbids two fermions from having the same set of these numbers, determining the structure of atoms and the periodic law.

History

The idea of quantum numbers originated in Bohr's atomic model (1913) with the introduction of the principal number n. Arnold Sommerfeld generalized the model in 1916, adding orbital and magnetic numbers to explain fine structure. A decisive contribution was made by Wolfgang Pauli, who proposed spin and the exclusion principle in 1925. Erwin Schrödinger created wave mechanics in 1926, where the numbers n, l, mₗ received a rigorous interpretation as eigenvalues of operators, and the Dirac equation (1928) gave spin a relativistic foundation.

How it works

The wave function of an electron is determined from the Schrödinger equation. When separating variables in a central potential, quantum conditions arise: the radial equation gives n = 1, 2, 3…, the angular one gives l = 0, …, n−1 and mₗ = −l, …, l. Spin is introduced via the Dirac equation and for the electron takes values s=1/2, mₛ = ±1/2. The Pauli principle forbids two electrons from occupying the same quantum state, which explains the sequential filling of shells and the periodicity of chemical properties. Thus, quantum numbers are a direct consequence of the Schrödinger equation and the Pauli principle.

E_n = -\frac{m_e e^4}{8\epsilon_0^2 h^2}\frac{1}{n^2}

E_n is the energy of level n, m_e is the electron mass (≈9.11×10⁻³¹ kg), e is the electron charge (≈1.602×10⁻¹⁹ C), ε₀ is the electric constant (≈8.854×10⁻¹² F/m), h is Planck's constant (≈6.626×10⁻³⁴ J·s), n is the principal quantum number (n=1,2,3...)

L = \hbar \sqrt{l(l+1)}

L is the magnitude of the orbital angular momentum, l is the orbital quantum number (l=0,1,...,n-1), ℏ is the reduced Planck constant (h/2π ≈ 1.0546×10⁻³⁴ J·s)

Open problems & fun facts

Why do such specific quantum numbers and symmetries exist?; Measurement problem: Are quantum numbers real before the act of observation?; How will quantum numbers change in the regime of quantum gravity, at Planck scales?

💡 The electron's spin has no classical analogue: it's not the rotation of a tiny ball, but an intrinsic property manifested in magnetic moments. Sometimes spin is compared to rotation, but it's actually a quantum 'left-right' that cannot be seen directly.

No articles yet

quantum number

History

How it works

History

How it works

History

How it works

Open problems & fun facts

Related articles