The ground state of a quantum mechanical system is defined as the lowest energy state that the system can occupy. In many cases, the ground state of a quantum system is non-degenerate, meaning there is only one unique lowest energy state. This property is known as the non-degeneracy of the ground state.
The non-degeneracy of the ground state arises from a fundamental principle in quantum mechanics called the Pauli exclusion principle. According to this principle, no two identical fermions (particles with half-integer spin, such as electrons) can occupy the same quantum state simultaneously.
Consider a system of identical fermions, such as electrons in an atom. The electrons are subject to the laws of quantum mechanics, which means they must obey the Pauli exclusion principle. This principle implies that each electron in the system must occupy a distinct quantum state, characterized by a unique set of quantum numbers (such as energy, angular momentum, and spin).
As a result, when arranging the electrons in increasing energy levels, the lowest energy level can accommodate only one electron (with its specific set of quantum numbers) due to the exclusion principle. Since no other electron can occupy the same state, the ground state is non-degenerate.
On the other hand, systems of particles with different statistics, such as bosons (particles with integer spin), can occupy the same quantum state simultaneously. In such cases, the ground state may be degenerate, meaning there are multiple distinct states with the same lowest energy.
It's worth noting that there are exceptions to the non-degeneracy of the ground state, especially in the presence of external factors like symmetry breaking or interactions that can lead to degenerate ground states. However, in many systems, the non-degeneracy of the ground state is a consequence of the Pauli exclusion principle and the unique nature of identical fermions.