Editing Conservation of energy (section)

==Quantum theory==
In [[quantum mechanics]], the energy of a quantum system is described by a [[self-adjoint operator|self-adjoint]] (or Hermitian) operator called the [[Hamiltonian (quantum mechanics)|Hamiltonian]], which acts on the [[Hilbert space]] (or a space of [[wave function]]s) of the system. If the Hamiltonian is a time-independent operator, emergence probability of the measurement result does not change in time over the evolution of the system. Thus the expectation value of energy is also time independent. The local energy conservation in quantum field theory is ensured by the quantum [[Noether's theorem]] for the energy-momentum tensor operator. Thus energy is conserved by the normal unitary evolution of a quantum system.

However, when the non-unitary [[Born rule]] is applied, the system's energy is [[Measurement in quantum mechanics|measured]] with an energy that can be below or above the expectation value, if the system was not in an energy eigenstate. (For macroscopic systems, this effect is usually too small to measure.) The disposition of this energy gap is not well-understood; most physicists believe that the energy is transferred to or from the macroscopic environment in the course of the measurement process,<ref>{{Cite web |last=Carroll |first=Sean |date=Jan 28, 2021 |title=Energy Conservation and Non-Conservation in Quantum Mechanics |url=https://www.preposterousuniverse.com/blog/2021/01/28/energy-conservation-and-non-conservation-in-quantum-mechanics/ |access-date=May 26, 2024 |website=Preposterousuniverse}}</ref> while others believe that the observable energy is only conserved "on average".<ref name="quanta-mag">{{cite news |title=Puzzling Quantum Scenario Appears Not to Conserve Energy |url=https://www.quantamagazine.org/puzzling-quantum-scenario-appears-not-to-conserve-energy-20220516/ |access-date=25 September 2022 |work=[[Quanta Magazine]] |date=2022}}</ref><ref name="Energy Non-conservation in Quantum">{{cite journal |last1=Carroll |first1=Sean M. |last2=Lodman |first2=Jackie |title=Energy Non-conservation in Quantum Mechanics |journal=Foundations of Physics |date=August 2021 |volume=51 |issue=4 |pages=83 |doi=10.1007/s10701-021-00490-5|arxiv=2101.11052 |bibcode=2021FoPh...51...83C |s2cid=226664820 }}</ref><ref>{{Cite journal |last=Aharonov |first=Yakir |date=Oct 2023 |title=Conservation laws and the foundations of quantum mechanics |journal=Proceedings of the National Academy of Sciences |volume=41 |issue=120 |doi=10.1073/pnas.22208101201of9 |doi-broken-date=1 November 2024 |doi-access=free |arxiv=2401.14261 }}</ref> No experiment has been confirmed as definitive evidence of violations of the conservation of energy principle in quantum mechanics, but that does not rule out that some newer experiments, as proposed, may find evidence of violations of the conservation of energy principle in quantum mechanics.<ref name="Energy Non-conservation in Quantum"/>