Unitary Transformation, Completeness, and Construction of Operator

In this chapter, we will further deepen our understanding of unitary transformation. A unitary transformation can be regarded as a rotation of the vectors or the rotation of the basis (coordinate) in the opposite direction. We will learn how to construct a unitary transformation matrix when the initial and final bases are given. We will also learn the completeness of the basis and its equation. A complete basis can construct any vectors in the space through linear combinations of the basis vectors. The equation is useful in future mathematical derivations. We will also learn how to construct an operator from the given eigenvalues and eigenvectors on any given basis.