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 acts on the wave function (the probability amplitude for different configurations of the 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 the Schrödinger 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 in the Schrödinger equation:
where i is the imaginary unit, ħ is the reduced Planck constant, and
is the Hamiltonian operator expressed as:
From the equation, the equality can be made:
It can be shown that the expectation value of energy will always be greater than or equal to the minimum potential of the system.
Consider computing the expectation value of kinetic energy:
Hence the expectation value of kinetic energy is always non-negative.
This result can be used with the linearity condition to calculate the expectation value of the total energy which is given for a normalized wavefunction as:
Similarly, the same condition can be generalized to any higher dimensions.
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
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:
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:
The energy operator is easily derived from using the free particle wave function (plane wave solution to Schrödinger's equation).
[3] Starting in one dimension the wave function is
The time derivative of Ψ is
By the De Broglie relation:
Re-arranging the equation leads to
It can be concluded that the scalar E is the eigenvalue of the operator, while
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.