# Hamiltonian Operator

## Hamiltonian operator

[‚ham·əl′tō·nē·ən ¦äp·ə‚rād·ər] (quantum mechanics)

## Hamiltonian Operator

(also del, or ▽-operator), a differential operator of the form

where i, j, and k are coordinate unit vectors. It was introduced by Sir W. R. Hamilton in 1853. If the Hamiltonian operator is applied to a scalar function φ (*x, y, z*) and ▽φ is understood to be the product of a vector and a scalar, the gradient of the function is produced:

If the operator is applied to a vector function **r**(*x, y, z*), when ▽ r is understood to be the scalar product of vectors, the divergence of the vector **r** is produced:

(*u, v*, and *w* are the coordinates of the vector **r**). The scalar product of the Hamiltonian operator and itself gives the Laplacian operator: