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.
Definition
It is given by:[1]
It acts on the wave function (the probability amplitude for different configurations of the system)
Application
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.
Schrödinger equation
Using the energy operator to the Schrödinger equation:
where i is the imaginary unit, ħ is the reduced Planck constant, and is the Hamiltonian operator.
Constant energy
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,[2]
Klein–Gordon equation
The relativistic mass-energy relation:
Derivation
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 is the operator. Summarizing these results:
For a 3-d plane wave
See also
- Time translation symmetry
- Planck constant
- Schrödinger equation
- Momentum operator
- Hamiltonian (quantum mechanics)
- Conservation of energy
- Complex number
- Stationary state
References
- ^ Quantum Mechanics Demystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN 0-07-145546-9
- ^ Young, Hugh D. (2020). Sears and Zemansky's university physics with modern physics. Roger A. Freedman, A. Lewis Ford, Hugh D. Young (Fifteenth edition, extended edtion ed.). [Hoboken, N.J.] ISBN 978-0-13-515955-2. OCLC 1057733965.
- ^ 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