Editing Adiabatic theorem (section)

== Diabatic vs. adiabatic processes ==
{| class="wikitable"
|+ Comparison
|-
! style="width: 50%" | Diabatic
! style="width: 50%" | Adiabatic
|-
| Rapidly changing conditions prevent the system from adapting its configuration during the process, hence the spatial probability density remains unchanged. Typically there is no eigenstate of the final Hamiltonian with the same functional form as the initial state. The system ends in a linear combination of states that sum to reproduce the initial probability density.
| Gradually changing conditions allow the system to adapt its configuration, hence the probability density is modified by the process. If the system starts in an eigenstate of the initial Hamiltonian, it will end in the ''corresponding'' eigenstate of the final Hamiltonian.<ref name="Kato">{{cite journal |author=Kato |first=T. |year=1950 |title=On the Adiabatic Theorem of Quantum Mechanics |journal=Journal of the Physical Society of Japan |volume=5 |issue=6 |pages=435–439 |bibcode=1950JPSJ....5..435K |doi=10.1143/JPSJ.5.435}}</ref>
|}

At some initial time <math>t_0</math> a quantum-mechanical system has an energy given by the Hamiltonian <math>\hat{H}(t_0)</math>; the system is in an eigenstate of <math>\hat{H}(t_0)</math> labelled <math>\psi(x,t_0)</math>. Changing conditions modify the Hamiltonian in a continuous manner, resulting in a final Hamiltonian <math>\hat{H}(t_1)</math> at some later time <math>t_1</math>. The system will evolve according to the time-dependent [[Schrödinger equation]], to reach a final state <math>\psi(x,t_1)</math>. The adiabatic theorem states that the modification to the system depends critically on the time <math>\tau = t_1 - t_0</math> during which the modification takes place.

For a truly adiabatic process we require <math>\tau \to \infty</math>; in this case the final state <math>\psi(x,t_1)</math> will be an eigenstate of the final Hamiltonian <math>\hat{H}(t_1) </math>, with a modified configuration:

:<math>|\psi(x,t_1)|^2 \neq |\psi(x,t_0)|^2 .</math>

The degree to which a given change approximates an adiabatic process depends on both the energy separation between <math>\psi(x,t_0)</math> and adjacent states, and the ratio of the interval <math>\tau</math> to the characteristic timescale of the evolution of <math>\psi(x,t_0)</math> for a time-independent Hamiltonian, <math>\tau_\text{int} = 2\pi\hbar/E_0</math>, where <math>E_0</math> is the energy of <math>\psi(x,t_0)</math>.

Conversely, in the limit <math>\tau \to 0</math> we have infinitely rapid, or diabatic passage; the configuration of the state remains unchanged:

:<math>|\psi(x,t_1)|^2 = |\psi(x,t_0)|^2 .</math>

The so-called "gap condition" included in Born and Fock's original definition given above refers to a requirement that the [[Spectrum of an operator|spectrum]] of <math>\hat{H}</math> is [[Discrete mathematics|discrete]] and [[Degenerate energy level|nondegenerate]], such that there is no ambiguity in the ordering of the states (one can easily establish which eigenstate of <math>\hat{H}(t_1)</math> ''corresponds'' to <math>\psi(t_0)</math>). In 1999 J. E. Avron and A. Elgart reformulated the adiabatic theorem to adapt it to situations without a gap.<ref name="Avron-Elgart">{{cite journal |author=Avron |first=J. E. |last2=Elgart |first2=A. |name-list-style=and |year=1999 |title=Adiabatic Theorem without a Gap Condition |journal=Communications in Mathematical Physics |volume=203 |issue=2 |pages=445–463 |arxiv=math-ph/9805022 |bibcode=1999CMaPh.203..445A |doi=10.1007/s002200050620 |s2cid=14294926}}</ref>

=== Comparison with the adiabatic concept in thermodynamics ===
The term "adiabatic" is traditionally used in [[thermodynamics]] to describe processes without the exchange of heat between system and environment (see [[adiabatic process]]), more precisely these processes are usually faster than the timescale of heat exchange. (For example, a pressure wave is adiabatic with respect to a heat wave, which is not adiabatic.) Adiabatic in the context of thermodynamics is often used as a synonym for fast process.

The [[Classical mechanics|classical]] and [[Quantum mechanics|quantum]] mechanics definition<ref name=Griffiths>{{cite book |last=Griffiths |first=David J. |title=Introduction to Quantum Mechanics |year=2005 |publisher=Pearson Prentice Hall |isbn=0-13-111892-7 |chapter=10 }}</ref> is instead closer to the thermodynamical concept of a [[quasistatic process]], which are processes that are almost always at equilibrium (i.e. that are slower than the internal energy exchange interactions time scales, namely a "normal" atmospheric heat wave is quasi-static, and a pressure wave is not). Adiabatic in the context of mechanics is often used as a synonym for slow process.

In the quantum world adiabatic means for example that the time scale of electrons and photon interactions is much faster or almost instantaneous with respect to the average time scale of electrons and photon propagation. Therefore, we can model the interactions as a piece of continuous propagation of electrons and photons (i.e. states at equilibrium) plus a quantum jump between states (i.e. instantaneous).

The adiabatic theorem in this heuristic context tells essentially that quantum jumps are preferably avoided, and the system tries to conserve the state and the quantum numbers.<ref name=":1">{{cite web |author=Zwiebach |first=Barton |date=Spring 2018 |title=L15.2 Classical adiabatic invariant |url=https://www.youtube.com/watch?v=qxBhW2DRnPg&t=254s?t=03m00s |url-status=live |archive-url=https://ghostarchive.org/varchive/youtube/20211221/qxBhW2DRnPg |archive-date=2021-12-21 |publisher=MIT 8.06 Quantum Physics III}}{{cbignore}}</ref>

The quantum mechanical concept of adiabatic is related to [[adiabatic invariant]], it is often used in the [[old quantum theory]] and has no direct relation with heat exchange.