ⓘ 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.
1. Definition
It is given by:
E ^ = i ℏ ∂ ∂ t {\displaystyle {\hat {E}}=i\hbar {\frac {\partial }{\partial t}}\,\!}It acts on the wave function the probability amplitude for different configurations of the system
Ψ r, t {\displaystyle \Psi \left\mathbf {r},t\right\,\!}2. Application
The energy operator corresponds to the full energy of a system. The Schrodinger equation describes the space and timedependence of the slow changing nonrelativistic 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.
2.1. Application Schrodinger equation
Using the energy operator to the Schrodinger equation:
i ℏ ∂ ∂ t Ψ r, t = H ^ Ψ r, t {\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi \mathbf {r},\,t={\hat {H}}\Psi \mathbf {r},t\,\!}can be obtained:
E ^ Ψ r, t = H ^ Ψ r, t {\displaystyle {\begin{aligned}&{\hat {E}}\Psi \mathbf {r},\,t={\hat {H}}\Psi \mathbf {r},\,t\\\end{aligned}}\,\!}where i is the imaginary unit, ħ is the reduced Planck constant, and H ^ {\displaystyle {\hat {H}}} is the Hamiltonian operator.
In a stationary state additionally occurs the timeindependent Schrodinger equation:
E Ψ r, t = H ^ Ψ r, t {\displaystyle {\begin{aligned}&E\Psi \mathbf {r},\,t={\hat {H}}\Psi \mathbf {r},\,t\\\end{aligned}}\,\!}where E is an eigenvalue of energy.
2.2. Application Klein–Gordon equation
The relativistic massenergy relation:
E 2 = p c 2 + m c 2 {\displaystyle E^{2}=pc^{2}+mc^{2}^{2}\,\!}where again E = total energy, p = total 3momentum of the particle, m = invariant mass, and c = speed of light, can similarly yield the Klein–Gordon equation:
E ^ 2 = c 2 p ^ 2 + m c 2 E ^ 2 Ψ = c 2 p ^ 2 Ψ + m c 2 Ψ {\displaystyle {\begin{aligned}&{\hat {E}}^{2}=c^{2}{\hat {p}}^{2}+mc^{2}^{2}\\&{\hat {E}}^{2}\Psi =c^{2}{\hat {p}}^{2}\Psi +mc^{2}^{2}\Psi \\\end{aligned}}\,\!}that is:
∂ 2 Ψ ∂ t 2 = c 2 ∇ 2 Ψ − m c 2 ℏ 2 Ψ {\displaystyle {\frac {\partial ^{2}\Psi }{\partial t^{2}}}=c^{2}\nabla ^{2}\Psi \left{\frac {mc^{2}}{\hbar }}\right^{2}\Psi \,\!}3. Derivation
The energy operator is easily derived from using the free particle wave function plane wave solution to Schrodingers equation. Starting in one dimension the wave function is
Ψ = e i k x − ω t {\displaystyle \Psi =e^{ikx\omega t}\,\!}The time derivative of Ψ is
∂ Ψ ∂ t = − i ω e i k x − ω t = − i ω Ψ {\displaystyle {\frac {\partial \Psi }{\partial t}}=i\omega e^{ikx\omega t}=i\omega \Psi \,\!}.By the De Broglie relation:
E = ℏ ω {\displaystyle E=\hbar \omega \,\!},we have
∂ Ψ ∂ t = − i E ℏ Ψ {\displaystyle {\frac {\partial \Psi }{\partial t}}=i{\frac {E}{\hbar }}\Psi \,\!}.Rearranging the equation leads to
E Ψ = i ℏ ∂ Ψ ∂ t {\displaystyle E\Psi =i\hbar {\frac {\partial \Psi }{\partial t}}\,\!},where the energy factor E is a scalar value, the energy the particle has and the value that is measured. The partial derivative is a linear operator so this expression is the operator for energy:
E ^ = i ℏ ∂ ∂ t {\displaystyle {\hat {E}}=i\hbar {\frac {\partial }{\partial t}}\,\!}.It can be concluded that the scalar E is the eigenvalue of the operator, while E ^ {\displaystyle {\hat {E}}\,\!} is the operator. Summarizing these results:
E ^ Ψ = i ℏ ∂ ∂ t Ψ = E Ψ {\displaystyle {\hat {E}}\Psi =i\hbar {\frac {\partial }{\partial t}}\Psi =E\Psi \,\!}For a 3d plane wave
Ψ = e i k ⋅ r − ω t {\displaystyle \Psi =e^{i\mathbf {k} \cdot \mathbf {r} \omega t}\,\!}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.
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