Hamiltoniansform

Hamiltoniansform is a term used in some mathematical and physical discussions to denote the canonical or standardized representation of a Hamiltonian after applying a structure-preserving transformation. The idea is to express the Hamiltonian in a form that reveals essential structure such as separability, symmetries, or normal modes, while preserving the Hamiltonian nature of the equations of motion.

In classical mechanics, the transformation is typically a canonical transformation in phase space, mapping coordinates and momenta (q, p) to (Q, P) in a way that preserves Hamilton’s equations. The resulting Hamiltonian H′(Q, P, t) is considered the Hamiltoniansform when it exposes features like integrals of motion, reduced degrees of freedom, or a diagonal (or block-diagonal) kinetic–potential structure. Techniques such as generating functions, action-angle variables, or normal form transformations are often employed to obtain such representations.

In quantum mechanics, an analogous idea applies under unitary transformations. If U is a unitary operator, the transformed Hamiltonian H′ = UHU† is the quantum counterpart of a Hamiltoniansform, chosen to simplify spectra, dynamics, or perturbative calculations while preserving eigenvalues and time evolution up to the transformed basis.

The concept is not universally standardized and is used primarily as a descriptive label in contexts emphasizing simplification, diagonalization, or symmetry-adapted forms of a Hamiltonian. Related ideas include canonical transformations, unitary transformations, and normal form theory.