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Energy operator


In quantum mechanics, energy is defined in terms of the energy operator, acting on the wave function of the system as a consequence of time translation symmetry.

It is given by:

It acts on the wave function (the probability amplitude for different configurations of the system)

The energy operator corresponds to the full energy of a system. The Schrödinger equation describes the space- and time-dependence of the slow changing (non-relativistic) wave function of a quantum system. The solution of this equation for a bound system is discrete (a set of permitted states, each characterized by an energy level) which results in the concept of quanta.

Using the classical equation for conservation of energy of a particle:

where E = total energy, H = hamiltonian, T = kinetic energy and V = potential energy of the particle, substituting the energy and Hamiltonian operators and multiplying by the wave function obtains the Schrödinger equation

that is

where i is the imaginary unit, ħ is the reduced Planck constant, and is the Hamiltonian operator.


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