In this paper we will examine the quantum postulates in a finite dimensional complex Hilbert space \(\mathcal {H}\) from the classical perspectives initiated by R.W. Hamilton in 1835. In particular we show that the space of density operators can be given the structure of a homogeneous Riemannian manifold that can be naturally identified with certain co-adjoint orbits in the semi-direct product associated with the action of the unitary group on the space of Hermitian matrices. This identification then provides density operators with canonical symplectic structure inherited from the orbit and sheds new light on the associated Schrödinger equation. In the special case that the densities are pure states (projectors) this identification reduces to a representation on the cotangent bundle of the sphere \(S^{2n-1}\) .