# 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:

References in periodicals archive ?
When a quantum well is in the presence of a monochromatic laser beam--linearly polarized in the axial direction (z axis)--we can study the electronic states of the confinement potential and the electromagnetic field, including the term representing the external field in the kinetic part of the Hamiltonian operator.
The full Hamiltonian operator we consider for the compound LiNiP04 is the one proposed in [3,9]:
In the case where quantum mechanical effects are present this Hamiltonian is transformed into the related Hamiltonian operator and Schrrdinger's, or Dirac's equations should be employed.
Keywords: Self adjoiness, Hamiltonian operator, potential, Hilbert space, Norm, Laplace Operator.
2] = 1/2m which gives the standard Hamiltonian operator (2C) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
At this point the wave functions of the Hilbert space basis as well as the Hamiltonian operator depend on the radial and the angular coordinates of single particle functions.
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To find the Hamiltonian operator H of the system, we use the formalism of Dirac [10]:
The Hamiltonian operator is Hermitian, this equation is linear and clearly is homogeneous of degree one under the substitution [psi] [right arrow] [lambda][psi].

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