Hamiltonian Operator

Hamiltonian operator

[‚ham·əl′tō·nē·ən ¦äp·ə‚rād·ər]
(quantum mechanics)
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

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:

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
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Keywords: Self adjoiness, Hamiltonian operator, potential, Hilbert space, Norm, Laplace Operator.