Wavefunctions are analogous to vectors in 3D space. The unit vectors of our vector space are eigenstates.
In normal 3D space, we represent a vector by its components.
Eigenfunctions - the unit vectors of our space - are orthonormal.
In normal 3D space, we can compute the
dot product between two vectors using the components.
In our vector space, we define the dot product to be
From a more mathematical point of view, the square integrable functions form a (vector) Hilbert Space.
The scalar product is defined as above.
Linear operators take vectors in the space into other vectors.
Jim Branson 2013-04-22