# 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 ?
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.
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.
These expressions determine the reduced form of the Hamiltonian operator Hl by the form:
In fact, the Hamiltonian of system (31) describes an ideal Bose gas consisting of phonons with spin 1 at a small wave number k [much less than] 2m[upsilon]/h but at k [much greater than] 2m[upsilon]/h the Hamiltonian operator describes an ideal gas of sound particles.
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].
Self-adjoint, globally defined Hamiltonian operators for systems with boundaries.
Specific topics include phase-parameter relation and sharp statistical properties for general families of unimodal maps, Hamiltonian operators in a noncommutative world, the deformation of space as a complete affine structure on the 2-torus smooth, parabolic foliations, the H-principle and pseudoconcave CF manifolds, wild knots as limit sets of Kleinian groups, an extension of the Burau representation to a mapping class group associated to Thompson's group T, Paonleve's Theorems I and II, and the growth rate of contractible closed geodesics on reducible manifolds.
In order to calculate the energy shifts due to the hyperfine interaction and to an external magnetic field B [equivalent] BZ, we define effective Hamiltonian operators [H'.

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