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:

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