An occupation number basis, often used in the context of quantum mechanics, is a way of describing the quantum state of a system of identical particles. The occupation number for each quantum state indicates how many particles occupy that state.
For bosons, which obey Bose-Einstein statistics, the occupation number is a sequence of non-negative integers, because there is no restriction on the number of bosons that can occupy the same quantum state. This sequence of integers must have a finite square sum, meaning the sum of the squares of the occupation numbers is finite. This ensures that the overall state is normalizable, which is a requirement for a physical quantum state.
Fermions, on the other hand, obey the Pauli exclusion principle and Fermi-Dirac statistics, which means that no two fermions can occupy the same quantum state. This is reflected in their occupation number sequences, which consist of binary values (0's and 1's), where 0 indicates the state is unoccupied and 1 indicates the state is occupied by one fermion. The square sum of this sequence is automatically finite, as the occupation numbers are either 0 or 1.
An occupation number basis, often used in the context of quantum mechanics, is a way of describing the quantum state of a system of identical particles. The occupation number for each quantum state indicates how many particles occupy that state.
For bosons, which obey Bose-Einstein statistics, the occupation number is a sequence of non-negative integers, because there is no restriction on the number of bosons that can occupy the same quantum state. This sequence of integers must have a finite square sum, meaning the sum of the squares of the occupation numbers is finite. This ensures that the overall state is normalizable, which is a requirement for a physical quantum state.
Fermions, on the other hand, obey the Pauli exclusion principle and Fermi-Dirac statistics, which means that no two fermions can occupy the same quantum state. This is reflected in their occupation number sequences, which consist of binary values (0's and 1's), where 0 indicates the state is unoccupied and 1 indicates the state is occupied by one fermion. The square sum of this sequence is automatically finite, as the occupation numbers are either 0 or 1.