Hamiltonian operator

In quantum mechanics, the Hamiltonian is the operator corresponding to the total energy of the system in most of the cases. It is usually denoted by H, also Ȟ or Ĥ. Its spectrum is the set of possible outcomes when one measures the total energy of a system. Because of its close relation to the time-evolution of a system, it is of fundamental importance in most formulations of quantum theory.

The Hamiltonian is named after William Rowan Hamilton, who also created a revolutionary reformation of Newtonian mechanics, now called Hamiltonian mechanics, that is important in quantum physics.

The Hamiltonian is the sum of the kinetic energies of all the particles, plus the potential energy of the particles associated with the system. For different situations or number of particles, the Hamiltonian is different since it includes the sum of kinetic energies of the particles, and the potential energy function corresponding to the situation.

By analogy with classical mechanics, the Hamiltonian is commonly expressed as the sum of operators corresponding to the kinetic and potential energies of a system in the form

where

is the potential energy operator and

is the kinetic energy operator in which m is the mass of the particle, the dot denotes the dot product of vectors, and

is the momentum operator wherein ∇ is the del operator. The dot product of ∇ with itself is the Laplacian ∇². In three dimensions using Cartesian coordinates the Laplace operator is

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