Why can't two electrons have the same set of quantum numbers?

Two electrons cannot have the same set of quantum numbers due to the Pauli Exclusion Principle.

The Pauli Exclusion Principle is a fundamental principle in quantum mechanics that was formulated by Austrian physicist Wolfgang Pauli in 1925. It states that no two electrons in an atom can have the same set of four quantum numbers. These quantum numbers are used to describe the properties and behaviour of an electron in an atom. They include the principal quantum number (n), the azimuthal quantum number (l), the magnetic quantum number (m), and the spin quantum number (s).

The principal quantum number (n) describes the energy level of the electron and its distance from the nucleus. The azimuthal quantum number (l) defines the shape of the orbital, and the magnetic quantum number (m) describes the orientation of the orbital in space. The spin quantum number (s) indicates the direction of the electron's spin - either up or down.

The Pauli Exclusion Principle is crucial in determining the electronic structure of atoms and the way electrons fill the atomic orbitals. It explains why electrons fill lower energy levels before moving to higher ones, a concept known as the Aufbau Principle. It also accounts for the structure of the periodic table, with each period corresponding to the filling of a different principal energy level.

In essence, the Pauli Exclusion Principle is a consequence of the laws of quantum mechanics, specifically the property of electron wavefunctions. Electrons are fermions, particles that have half-integer spin, and their wavefunctions are antisymmetric. This means that if you swap two electrons, the wavefunction changes its sign. If two electrons were to have the same set of quantum numbers, their wavefunction would be identical, and swapping them would not change the wavefunction, violating the antisymmetry property. Therefore, the Pauli Exclusion Principle is a fundamental aspect of the quantum mechanical description of matter.