Hcdot

Hcdot, usually written as H⋅, is a notation used primarily in theoretical physics and applied mathematics to denote a specific type of inner product that involves a Hamiltonian operator. The symbol Hcdot represents the action of the Hamiltonian H on a state vector or function, followed by a dot or scalar product with another vector or function, resulting in a scalar quantity that often corresponds to an observable or energy expectation value. The notation is concise and distinguishes itself from other forms of the dot product that involve unrelated vectors or operators.

In quantum mechanics, Hcdot plays a pivotal role in the time‑evolution equation for wavefunctions. For a wavefunction ψ, the expression H⋅ψ is interpreted as the energy operator acting on ψ, and the result can be inserted into the Schrödinger equation or used to compute probability densities. The dot emphasises that the operation is a contraction over all internal degrees of freedom, such as spin or spatial indices.

Beyond quantum theory, Hcdot occasionally appears in the study of lattice models in statistical mechanics, where it denotes the interaction energy between neighboring sites represented by the Hamiltonian matrix. In computational chemistry, an Hcdot term is sometimes employed to simplify matrix representations of complex systems, allowing efficient diagonalisation protocols.

Although the notation is not universally standard—some authors replace Hcdot with Hψ or 𝐻ψ to signify the same operation—its widespread use in Hamiltonian dynamics makes it a familiar shorthand among researchers.