Energy operator

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Main article: operator (physics)

In quantum mechanics, energy is defined in terms of the energy operator, acting on the wave function of the system.


It is given by:[1]

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 slow changing (non-relativistic) wave function of quantum systems. The solution of this equation for bound system is discrete (a set of permitted states, each characterized by an energy level) which results in the concept of quanta.

Schrödinger equation[edit]

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.

Klein–Gordon equation[edit]

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:

that is:


The energy operator is easily derived from using the free particle wave function (plane wave solution to Schrödinger's equation).[2] 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. Cancellation of Ψ leads to

The partial derivative is a linear operator so this expression is the operator for energy:


It can be concluded that the scalar E is the eigenvalue of the operator, while is the operator. Summarizing these results:

For a 3-d plane wave

the derivation is exactly identical, as no change is made to the term including time and therefore the time derivative. Since the operator is linear, they are valid for any linear combination of plane waves, and so they can act on any wave function without affecting the properties of the wave function or operators. Hence this must be true for any wave function. It turns out to work even in relativistic quantum mechanics, such as the Klein–Gordon equation above.

See also[edit]


  1. ^ Quantum Mechanics Demystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN 0-07-145546-9
  2. ^ Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd Edition), R. Resnick, R. Eisberg, John Wiley & Sons, 1985, ISBN 978-0-471-87373-0