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Concept
Energy operator
Summary
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 energy operator to the Schrödinger equation:
can be obtained:
where i is the imaginary unit, ħ is the reduced Planck constant, and is the Hamiltonian operator.
Working from the definition, a partial solution for a wavefunction of a particle with a constant energy can be constructed. If the wavefunction is assumed to be separable, then the time dependence can be stated as , where E is the constant energy. In full,
where is the partial solution of the wavefunction dependent on position. Applying the energy operator, we have
This is also known as the stationary state, and can be used to analyse the time-independent Schrödinger equation:
where E is an eigenvalue of energy.
The relativistic mass-energy relation:
where again E = total energy, p = total 3-momentum of the particle, m = invariant mass, and c = speed of light, can similarly yield the Klein–Gordon equation:
where is the momentum operator. That is:
The energy operator is easily derived from using the free particle wave function (plane wave solution to Schrödinger's equation). Starting in one dimension the wave function is
The time derivative of Ψ is
By the De Broglie relation:
we have
Re-arranging the equation leads to
where the energy factor E is a scalar value, the energy the particle has and the value that is measured.
Official source
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