completeness relation
The completeness relation is the statement that a basis really does cover the whole space, leaving no state out. Written as a sum over a basis of projection operators, Σ |n⟩⟨n| = 1, it says that adding up the projectors onto every basis direction rebuilds the identity operator — the operation that leaves every state unchanged. In plain terms: any state is fully accounted for by its components along the basis.
Its everyday use is as a tool you can slip into any expression without changing it, since inserting the identity changes nothing. Drop Σ |n⟩⟨n| into the middle of an overlap or an operator and it quietly breaks the calculation into a sum over basis states, each piece a simple number you can handle. This 'inserting a complete set of states' trick is one of the most-used moves in all of quantum mechanics.
Completeness is the partner of orthonormality. Orthonormality says the basis states are clean and distinct; completeness says there are enough of them to span everything, with none missing. A set that is orthonormal but incomplete leaves gaps — states it cannot represent — so both properties together are what make a basis trustworthy. For continuous bases like position, the sum becomes an integral, but the idea is identical.
Summing the projectors onto every basis state rebuilds the identity, so the basis misses nothing.
The '1' on the right is the identity operator, not the number one. The relation says the projectors add up to 'do nothing', which is why you can insert it freely without altering an expression.