A * projection operator* is a linear operator that transforms a vector in the direction of another vector, i.e. it projects one vector onto another.

In general,

where is a scalar.

It is useful in quantum mechanics to have a projection operator that maps a vector onto another vector, which is part of a complete set of orthonormal basis vectors in a Hilbert space. We define the operator as:

This allows us to project a vector onto the basis vector :

If is a wavefunction that is a linear combination of a complete set of orthonormal basis functions, i.e. ,

When we measure an observable of a system whose state is described by , we get an eigenvalue corresponding to an eigenfunction, which is one of the set of orthonormal basis functions. We say that the wavefunction is projected onto (or collapses into) the eigenfunction .