Editing Adiabatic theorem

{{Short description|Concept in quantum mechanics}}
{{about|the adiabatic theorem in quantum mechanics|adiabatic processes in thermodynamics|adiabatic process}}

The '''adiabatic theorem''' is a concept in [[quantum mechanics]]. Its original form, due to [[Max Born]] and [[Vladimir Fock]] (1928), was stated as follows:
:''A physical system remains in its instantaneous [[eigenstate]] if a given [[perturbation theory (quantum mechanics)|perturbation]] is acting on it slowly enough and if there is a gap between the [[eigenvalue]] and the rest of the [[Hamiltonian (quantum mechanics)|Hamiltonian]]'s [[Spectrum of an operator|spectrum]].''<ref name="Born-Fock">{{cite journal |author=Born |first=M. |last2=Fock |first2=V. A. |name-list-style=and |year=1928 |title=Beweis des Adiabatensatzes |journal=Zeitschrift für Physik A |volume=51 |issue=3–4 |pages=165–180 |bibcode=1928ZPhy...51..165B |doi=10.1007/BF01343193 |s2cid=122149514}}</ref>
In simpler terms, a quantum mechanical system subjected to gradually changing external conditions adapts its functional form, but when subjected to rapidly varying conditions there is insufficient time for the functional form to adapt, so the spatial probability density remains unchanged.

== Adiabatic pendulum ==
{{See also|Rayleigh–Lorentz pendulum}}
At the 1911 Solvay conference, Einstein gave a lecture on the quantum hypothesis, which states that <math>E = nh \nu</math> for atomic oscillators. After Einstein's lecture, [[Hendrik Lorentz]] commented that, classically, if a simple pendulum is shortened by holding the wire between two fingers and sliding down, it seems that its energy will change smoothly as the pendulum is shortened. This seems to show that the quantum hypothesis is invalid for macroscopic systems, and if macroscopic systems do not follow the quantum hypothesis, then as the macroscopic system becomes microscopic, it seems the quantum hypothesis would be invalidated. Einstein replied that although both the energy <math>E</math> and the frequency <math>\nu</math> would change, their ratio <math>\frac{E}{\nu}</math> would still be conserved, thus saving the quantum hypothesis.<ref>{{Cite book |last=Instituts Solvay |first=Brussels Institut international de physique Conseil de physique |url=https://archive.org/details/lathoriedurayo00inst/page/450/mode/2up |title=La théorie du rayonnement et les quanta : rapports et discussions de la réunion tenue à Bruxelles, du 30 octobre au 3 novembre 1911, sous les auspices de M.E. Solvay |last2=Solvay |first2=Ernest |last3=Langevin |first3=Paul |last4=Broglie |first4=Maurice de |last5=Einstein |first5=Albert |date=1912 |publisher=Paris, France: Gauthier-Villars |others=University of British Columbia Library |page=450}}</ref>

Before the conference, Einstein had just read a paper by [[Paul Ehrenfest]] on the adiabatic hypothesis.<ref>EHRENFEST, P. (1911): ``Welche Züge der Lichtquantenhypothese spielen in der Theorie der Wärmestrahlung eine wesentliche Rolle?<nowiki>''</nowiki> Annalen der Physik 36, pp. 91–118. Reprinted in KLEIN (1959), pp. 185–212.</ref> We know that he had read it because he mentioned it in a letter to Michele Besso written before the conference.<ref>{{Cite web |title=Letter to Michele Besso, 21 October 1911, translated in Volume 5: The Swiss Years: Correspondence, 1902-1914 (English translation supplement), page 215 |url=https://einsteinpapers.press.princeton.edu/vol5-trans/237 |access-date=2024-04-17 |website=einsteinpapers.press.princeton.edu}}</ref><ref>{{Cite journal |last=Laidler |first=Keith J. |date=1994-03-01 |title=The meaning of "adiabatic" |url=http://www.nrcresearchpress.com/doi/10.1139/v94-121 |journal=Canadian Journal of Chemistry |language=en |volume=72 |issue=3 |pages=936–938 |doi=10.1139/v94-121 |issn=0008-4042}}</ref>

== 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.

== Example systems ==

=== Simple pendulum ===
As an example, consider a [[pendulum]] oscillating in a vertical plane. If the support is moved, the mode of oscillation of the pendulum will change. If the support is moved ''sufficiently slowly'', the motion of the pendulum relative to the support will remain unchanged. A gradual change in external conditions allows the system to adapt, such that it retains its initial character. The detailed classical example is available in the [[Adiabatic invariant#Classical mechanics – action variables|Adiabatic invariant]] page and here.<ref name=":2">{{cite web |author=Zwiebach |first=Barton |date=Spring 2018 |title=Classical analog: oscillator with slowly varying frequency |url=https://www.youtube.com/watch?v=DYJM_P4sG-c |url-status=live |archive-url=https://ghostarchive.org/varchive/youtube/20211221/DYJM_P4sG-c |archive-date=2021-12-21 |publisher=MIT 8.06 Quantum Physics III}}{{cbignore}}</ref>

=== Quantum harmonic oscillator ===

[[Image:HO adiabatic process.gif|thumb|right|300px|'''Figure 1.''' Change in the probability density, <math>|\psi(t)|^2</math>, of a ground state quantum harmonic oscillator, due to an adiabatic increase in spring constant.]]

The [[Classical physics|classical]] nature of a pendulum precludes a full description of the effects of the adiabatic theorem. As a further example consider a [[quantum harmonic oscillator]] as the [[spring constant]] <math>k</math> is increased. Classically this is equivalent to increasing the stiffness of a spring; quantum-mechanically the effect is a narrowing of the [[potential energy]] curve in the system [[Hamiltonian (quantum mechanics)|Hamiltonian]].

If <math>k</math> is increased adiabatically <math display="inline">\left(\frac{dk}{dt} \to 0\right)</math> then the system at time <math>t</math> will be in an instantaneous eigenstate <math>\psi(t)</math> of the ''current'' Hamiltonian <math>\hat{H}(t)</math>, corresponding to the initial eigenstate of <math>\hat{H}(0)</math>. For the special case of a system like the quantum harmonic oscillator described by a single [[quantum number]], this means the quantum number will remain unchanged. '''Figure 1''' shows how a harmonic oscillator, initially in its ground state, <math>n = 0</math>, remains in the ground state as the potential energy curve is compressed; the functional form of the state adapting to the slowly varying conditions.

For a rapidly increased spring constant, the system undergoes a diabatic process <math display="inline">\left(\frac{dk}{dt} \to \infty\right)</math> in which the system has no time to adapt its functional form to the changing conditions. While the final state must look identical to the initial state <math>\left(|\psi(t)|^2 = |\psi(0)|^2\right)</math> for a process occurring over a vanishing time period, there is no eigenstate of the new Hamiltonian, <math>\hat{H}(t)</math>, that resembles the initial state. The final state is composed of a [[linear superposition]] of many different eigenstates of <math>\hat{H}(t)</math> which sum to reproduce the form of the initial state.

=== Avoided curve crossing ===
{{main|Avoided crossing}}
[[File:Avoided_crossing_in_linear_field.svg|thumb|right|300px|'''Figure 2.''' An avoided energy-level crossing in a two-level system subjected to an external magnetic field. Note the energies of the diabatic states, <math>|1\rangle</math> and <math>|2\rangle</math> and the [[eigenvalues]] of the Hamiltonian, giving the energies of the eigenstates <math>|\phi_1\rangle</math> and <math>|\phi_2\rangle</math> (the adiabatic states). (Actually, <math>|\phi_1\rangle</math> and <math>|\phi_2\rangle</math> should be switched in this picture.)]]

For a more widely applicable example, consider a 2-[[Energy level|level]] atom subjected to an external [[magnetic field]].<ref name="Stenholm">{{cite journal |author=Stenholm |first=Stig |author-link=Stig Stenholm |year=1994 |title=Quantum Dynamics of Simple Systems |journal=The 44th Scottish Universities Summer School in Physics |pages=267–313}}</ref> The states, labelled <math>|1\rangle</math> and <math>|2\rangle</math> using [[bra–ket notation]], can be thought of as atomic [[Azimuthal quantum number|angular-momentum states]], each with a particular geometry. For reasons that will become clear these states will henceforth be referred to as the diabatic states. The system wavefunction can be represented as a linear combination of the diabatic states:

:<math>|\Psi\rangle = c_1(t)|1\rangle + c_2(t)|2\rangle.</math>

With the field absent, the energetic separation of the diabatic states is equal to <math>\hbar\omega_0</math>; the energy of state <math>|1\rangle</math> increases with increasing magnetic field (a low-field-seeking state), while the energy of state <math>|2\rangle</math> decreases with increasing magnetic field (a high-field-seeking state). Assuming the magnetic-field dependence is linear, the [[Hamiltonian matrix]] for the system with the field applied can be written

:<math>\mathbf{H} = \begin{pmatrix}
\mu B(t)-\hbar\omega_0/2 & a \\
a^* & \hbar\omega_0/2-\mu B(t)
\end{pmatrix}</math>

where <math>\mu</math> is the [[magnetic moment]] of the atom, assumed to be the same for the two diabatic states, and <math>a</math> is some time-independent [[Angular momentum coupling|coupling]] between the two states. The diagonal elements are the energies of the diabatic states (<math>E_1(t)</math> and <math>E_2(t)</math>), however, as <math>\mathbf{H}</math> is not a [[diagonal matrix]], it is clear that these states are not eigenstates of <math>\mathbf{H}</math> due to the off-diagonal coupling constant.

The eigenvectors of the matrix <math>\mathbf{H}</math> are the eigenstates of the system, which we will label <math>|\phi_1(t)\rangle</math> and <math>|\phi_2(t)\rangle</math>, with corresponding eigenvalues
<math display="block">\begin{align}
\varepsilon_1(t) &= -\frac{1}{2}\sqrt{4a^2 + (\hbar\omega_0 - 2\mu B(t))^2} \\[4pt]
\varepsilon_2(t) &= +\frac{1}{2}\sqrt{4a^2 + (\hbar\omega_0 - 2\mu B(t))^2}.
\end{align}</math>

It is important to realise that the eigenvalues <math>\varepsilon_1(t)</math> and <math>\varepsilon_2(t)</math> are the only allowed outputs for any individual measurement of the system energy, whereas the diabatic energies <math>E_1(t)</math> and <math>E_2(t)</math> correspond to the [[expectation value]]s for the energy of the system in the diabatic states <math>|1\rangle</math> and <math>|2\rangle</math>.

'''Figure 2''' shows the dependence of the diabatic and adiabatic energies on the value of the magnetic field; note that for non-zero coupling the [[eigenvalues]] of the Hamiltonian cannot be [[Degenerate energy level|degenerate]], and thus we have an avoided crossing. If an atom is initially in state <math>|\phi_2(t_0)\rangle</math> in zero magnetic field (on the red curve, at the extreme left), an adiabatic increase in magnetic field <math display="inline">\left(\frac{dB}{dt} \to 0\right)</math> will ensure the system remains in an eigenstate of the Hamiltonian <math>|\phi_2(t)\rangle</math> throughout the process (follows the red curve). A diabatic increase in magnetic field <math display="inline">\left(\frac{dB}{dt}\to \infty\right)</math> will ensure the system follows the diabatic path (the dotted blue line), such that the system undergoes a transition to state <math>|\phi_1(t_1)\rangle</math>. For finite magnetic field slew rates <math display="inline">\left(0 < \frac{dB}{dt} < \infty\right)</math> there will be a finite probability of finding the system in either of the two eigenstates. See [[#Calculating adiabatic passage probabilities|below]] for approaches to calculating these probabilities.

These results are extremely important in [[Atomic physics|atomic]] and [[molecular physics]] for control of the energy-state distribution in a population of atoms or molecules.

== Mathematical statement ==
Under a slowly changing Hamiltonian <math>H(t)</math> with instantaneous eigenstates <math>| n(t) \rangle</math> and corresponding energies <math>E_n(t)</math>, a quantum system evolves from the initial state
<math display="block">| \psi(0) \rangle = \sum_n c_n(0) | n(0) \rangle</math>
to the final state
<math display="block">| \psi(t) \rangle = \sum_n c_n(t) | n(t) \rangle ,</math>
where the coefficients undergo the change of phase
<math display="block">c_n(t) = c_n(0) e^{i \theta_n(t)} e^{i \gamma_n(t)}</math>

with the '''dynamical phase'''
<math display="block">\theta_m(t) = -\frac{1}{\hbar} \int_0^t E_m(t') dt'</math>

and '''[[geometric phase]]'''
<math display="block">\gamma_m(t) = i \int_0^t \langle m(t') | \dot{m}(t') \rangle dt' .</math>

In particular, <math>|c_n(t)|^2 = |c_n(0)|^2</math>, so if the system begins in an eigenstate of <math>H(0)</math>, it remains in an eigenstate of <math>H(t)</math> during the evolution with a change of phase only.

=== Proofs ===
:{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
!Sakurai in ''Modern Quantum Mechanics''<ref name="Modern Quantum Mechanics">{{Cite book|last1=Sakurai|first1=J. J.| url=https://www.cambridge.org/highereducation/books/modern-quantum-mechanics/DF43277E8AEDF83CC12EA62887C277DC#contents |title=Modern Quantum Mechanics |last2=Napolitano|first2=Jim |date=2020-09-17 |publisher=Cambridge University Press| isbn=978-1-108-58728-0| edition=3 |doi=10.1017/9781108587280|bibcode=2020mqm..book.....S }}</ref>
|-
|
This proof is partly inspired by one given by Sakurai in ''Modern Quantum Mechanics''.<ref name="Modern Quantum Mechanics"/>
The instantaneous eigenstates <math>| n(t) \rangle</math> and energies <math>E_n(t)</math>, by assumption, satisfy the time-independent Schrödinger equation
<math display="block">H(t) | n(t) \rangle = E_n(t) | n(t) \rangle</math>
at all times <math>t</math>. Thus, they constitute a basis that can be used to expand the state
<math display="block">| \psi(t) \rangle = \sum_n c_n(t) | n(t) \rangle</math>
at any time <math>t</math>. The evolution of the system is governed by the time-dependent Schrödinger equation
<math display="block">i \hbar |\dot{\psi}(t) \rangle = H(t) | \psi(t) \rangle,</math>
where <math>\dot{} = d / dt</math> (see {{slink|Notation for differentiation|Newton's notation}}). Insert the expansion of <math>| \psi(t) \rangle</math>, use <math>H(t) | n(t) \rangle = E_n(t) | n(t) \rangle</math>, differentiate with the product rule, take the inner product with <math>| m(t) \rangle</math> and use orthonormality of the eigenstates to obtain
<math display="block">i \hbar \dot{c}_m(t) + i \hbar \sum_n c_n(t) \langle m(t) | \dot{n}(t) \rangle = c_m(t) E_m(t) .</math>

This coupled first-order differential equation is exact and expresses the time-evolution of the coefficients in terms of inner products <math>\langle m(t) | \dot{n} (t) \rangle</math> between the eigenstates and the time-differentiated eigenstates. But it is possible to re-express the inner products for <math>m \neq n</math> in terms of matrix elements of the time-differentiated Hamiltonian <math>\dot{H}(t)</math>. To do so, differentiate both sides of the time-independent Schrödinger equation with respect to time using the product rule to get
<math display="block">\dot{H}(t)|n(t)\rangle + H(t)|\dot{n}(t)\rangle = \dot{E}_n(t) |n(t)\rangle + E_n(t) |\dot{n}(t)\rangle .</math>

Again take the inner product with <math>| m(t) \rangle</math> and use <math>\langle m(t) | H(t) = E_m(t) \langle m(t) |</math> and orthonormality to find
<math display="block">\langle m(t) | \dot{n}(t) \rangle = - \frac{\langle m(t) | \dot{H}(t) | n(t) \rangle}{E_m(t) - E_n(t)}
\qquad (m \neq n).</math>

Insert this into the differential equation for the coefficients to obtain
<math display="block">\dot{c}_m(t) + \left(\frac{i}{\hbar} E_m(t) + \langle m(t) | \dot{m}(t) \rangle \right) c_m(t) = \sum_{n \neq m} \frac{\langle m(t) | \dot{H} | n(t) \rangle}{E_m(t) - E_n(t)} c_n(t).</math>

This differential equation describes the time-evolution of the coefficients, but now in terms of matrix elements of <math>\dot{H}(t)</math>. To arrive at the adiabatic theorem, neglect the right hand side. This is valid if the rate of change of the Hamiltonian <math>\dot{H}(t)</math> is small '''and''' there is a finite gap <math>E_m(t) - E_n(t) \neq 0</math> between the energies. This is known as the '''adiabatic approximation'''. Under the adiabatic approximation,
<math display="block">\dot{c}_m(t) = i \left(-\frac{E_m(t)}{\hbar} + i \langle m(t) | \dot{m}(t) \rangle \right) c_m(t)</math>
which integrates precisely to the adiabatic theorem
<math display="block">c_m(t) = c_m(0) e^{i \theta_m(t)} e^{i \gamma_m(t)}</math>
with the phases defined in the statement of the theorem.

The dynamical phase <math>\theta_m(t)</math> is real because it involves an integral over a real energy. To see that the geometric phase <math>\gamma_m(t)</math> is purely real, differentiate the normalization <math>\langle m(t) | m(t) \rangle = 1</math> of the eigenstates and use the product rule to find that

<math display="block">0
= \frac{d}{dt} \Bigl ( \langle m(t) | m(t) \rangle \Bigr )
= \langle \dot{m}(t) | m(t) \rangle + \langle m(t)) | \dot{m}(t) \rangle
= \langle m(t)) | \dot{m}(t) \rangle^* + \langle m(t)) | \dot{m}(t) \rangle
= 2 \, \operatorname{Re} \Bigl ( \langle m(t)) | \dot{m}(t) \rangle \Bigr )
. </math>

Thus, <math>\langle m(t)) | \dot{m}(t) \rangle </math> is purely imaginary, so the geometric phase <math>\gamma_m(t) </math> is purely real.
|}

:{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
!Adiabatic approximation<ref name="Zwiebach">{{Cite web |last=Zwiebach |first=Barton |url=https://www.youtube.com/watch?v=pgEFvhkEp-c |archive-url=https://ghostarchive.org/varchive/youtube/20211221/pgEFvhkEp-c |archive-date=2021-12-21 |url-status=live| title=L16.1 Quantum adiabatic theorem stated| date=Spring 2018| publisher=MIT 8.06 Quantum Physics III}}{{cbignore}}</ref><ref name="MIT 8.06 Quantum Physics III">{{Cite web|title=MIT 8.06 Quantum Physics III| url=https://ocw.mit.edu/8-06S18}}</ref>
|-
|
Proof with the details of the adiabatic approximation<ref name="Zwiebach"/><ref name="MIT 8.06 Quantum Physics III"/>
We are going to formulate the statement of the theorem as follows:

: For a slowly varying Hamiltonian <math>\hat{H}</math> in the time range T the solution of the Schrödinger equation <math>\Psi(t)</math> with initial conditions <math>\Psi(0) = \psi_{n}(0)</math>
: where <math>\psi_{n}(t)</math> is the eigenvector of the instantaneous Schrödinger equation <math>\hat{H}(t)\psi_{n}(t)=E_{n}(t)\psi_{n}(t)</math> can be approximated as: <math display="block">\left\| {\Psi(t)-\psi_\text{adiabatic}(t)} \right\| \approx O(\frac{1}{T})</math> where the adiabatic approximation is: <math display="block"> |\psi_\text{adiabatic}(t)\rangle = e^{i\theta_{n}(t)}e^{i\gamma_{n}(t)}|\psi_n(t)\rangle</math> and <math display="block">\theta_{n}(t) = - \frac{1}{\hbar} \int_{0}^{t}E_{n}(t') dt'</math> <math display="block">\gamma_{n}(t) =  \int_{0}^{t}\nu_{n}(t')dt'</math> also called [[Berry phase]] <math display="block">\nu_{n}(t) = i \langle\psi_{n}(t) | \dot{\psi}_{n}(t)\rangle</math>

And now we are going to prove the theorem.

Consider the ''time-dependent'' [[Schrödinger equation]]
<math display="block">i \hbar{\partial \over \partial t} |\psi(t)\rangle = \hat{H}(\tfrac{t}{T}) |\psi(t)\rangle</math>
with [[Hamiltonian (quantum mechanics)|Hamiltonian]] <math>\hat{H}(t).</math>
We would like to know the relation between an initial state <math>|\psi(0)\rangle</math> and its final state <math>|\psi(T)\rangle</math> at <math>t = T</math> in the adiabatic limit <math>T \to \infty.</math>

First redefine time as <math>\lambda = \tfrac{t}{T} \in [0,1]</math>:
<math display="block">i \hbar{\partial \over \partial \lambda} |\psi(\lambda)\rangle = T \hat{H}(\lambda) |\psi(\lambda)\rangle.</math>
At every point in time <math>\hat{H}(\lambda)</math> can be diagonalized <math>\hat H(\lambda)|\psi_n(\lambda)\rangle = E_n(\lambda)|\psi_n(\lambda)\rangle</math> with eigenvalues <math>E_n</math> and eigenvectors <math>|\psi_n(\lambda)\rangle</math>. Since the eigenvectors form a complete basis at any time we can expand <math>|\psi(\lambda)\rangle</math> as:
<math display="block"> |\psi(\lambda)\rangle = \sum_n c_n(\lambda)|\psi_n(\lambda)\rangle e^{iT\theta_n(\lambda)},</math> where <math display="block">\theta_n(\lambda) = -\frac{1}{\hbar}\int_0^\lambda E_n(\lambda')d\lambda'.</math>
The phase <math>\theta_n(t)</math> is called the ''dynamic phase factor''. By substitution into the Schrödinger equation, another equation for the variation of the coefficients can be obtained:
<math display="block">i \hbar \sum_n (\dot{c}_n|\psi_n\rangle + c_n|\dot{\psi}_n\rangle + i c_n|\psi_n\rangle T\dot{\theta}_n)e^{iT\theta_n} = \sum_n c_n T E_n|\psi_n\rangle e^{iT\theta_n}.</math>
The term <math>\dot{\theta}_n</math> gives <math>-E_n/\hbar</math>, and so the third term of left side cancels out with the right side, leaving
<math display="block">\sum_n \dot{c}_n|\psi_n\rangle e^{iT\theta_n} = -\sum_n c_n|\dot{\psi}_n\rangle e^{iT\theta_n}.</math>

Now taking the inner product with an arbitrary eigenfunction <math>\langle\psi_m|</math>, the <math>\langle\psi_m|\psi_n\rangle</math> on the left gives <math>\delta_{nm}</math>, which is 1 only for ''m'' = ''n'' and otherwise vanishes. The remaining part gives 
<math display="block">\dot{c}_m = -\sum_n c_n\langle\psi_m|\dot{\psi}_n\rangle e^{iT(\theta_n-\theta_m)}.</math>

For <math>T \to \infty</math> the <math>e^{iT(\theta_n-\theta_m)}</math> will oscillate faster and faster and intuitively will eventually suppress nearly all terms on the right side. The only exceptions are when <math>\theta_n-\theta_m</math> has a critical point, i.e. <math>E_n(\lambda) = E_m(\lambda)</math>. This is trivially true for <math>m = n</math>. Since the adiabatic theorem assumes a gap between the eigenenergies at any time this cannot hold for <math>m \neq n</math>. Therefore, only the <math>m = n</math> term will remain in the limit <math>T \to \infty</math>.

In order to show this more rigorously we first need to remove the <math>m = n</math> term.
This can be done by defining <math display="block">d_m(\lambda) = c_m(\lambda) e^{\int_0^\lambda\langle\psi_m|\dot{\psi}_m\rangle d\lambda} = c_m(\lambda) e^{-i\gamma_m(\lambda)}.</math>

We obtain:
<math display="block">\dot{d}_m = -\sum_{n\neq m} d_n\langle\psi_m|\dot{\psi}_n\rangle e^{iT(\theta_n-\theta_m)-i(\gamma_m-\gamma_n)}.</math>
This equation can be integrated: 
<math display="block">\begin{align}
d_m(1)-d_m(0) &= -\int_0^1 \sum_{n\neq m} d_n\langle\psi_m|\dot{\psi}_n\rangle e^{iT(\theta_n-\theta_m)-i(\gamma_m-\gamma_n)} d\lambda\\
&= -\int_0^1 \sum_{n\neq m} (d_n-d_n(0))\langle\psi_m|\dot{\psi}_n\rangle e^{iT(\theta_n-\theta_m)-i(\gamma_m-\gamma_n)} d\lambda - \int_0^1 \sum_{n\neq m} d_n(0)\langle\psi_m|\dot{\psi}_n\rangle e^{iT(\theta_n-\theta_m)-i(\gamma_m-\gamma_n)}d\lambda
\end{align}</math> 
or written in vector notation
<math display="block">\vec{d}(1)-\vec{d}(0) = -\int_0^1 \hat{A}(T, \lambda) (\vec{d}(\lambda)-\vec{d}(0)) d\lambda - \vec{\alpha}(T).</math>
Here <math>\hat{A}(T, \lambda)</math> is a matrix and
<math display="block">\alpha_m(T) = \int_0^1 \sum_{n\neq m} d_n(0)\langle\psi_m|\dot{\psi}_n\rangle e^{iT(\theta_n-\theta_m)- i(\gamma_m-\gamma_n)}d\lambda</math> is basically a Fourier transform.
It follows from the [[Riemann–Lebesgue lemma|Riemann-Lebesgue lemma]] that <math>\vec{\alpha}(T) \to 0 </math> as <math>T \to \infty</math>. As last step take the norm on both sides of the above equation:
<math display="block">\Vert\vec{d}(1)- \vec{d}(0)\Vert \leq \Vert\vec{\alpha}(T)\Vert + \int_0^1 \Vert\hat{A}(T, \lambda)\Vert \Vert\vec{d}(\lambda)-\vec{d}(0)\Vert d\lambda</math>
and apply [[Grönwall's inequality]] to obtain 
<math display="block">\Vert\vec{d}(1)-\vec{d}(0)\Vert \leq \Vert\vec{\alpha}(T)\Vert e^{\int_0^1 \Vert\hat{A}(T, \lambda)\Vert d\lambda}.</math>
Since <math>\vec{\alpha}(T) \to 0</math> it follows <math>\Vert\vec{d}(1)-\vec{d}(0)\Vert \to 0</math> for <math>T \to \infty</math>. This concludes the proof of the adiabatic theorem.

In the adiabatic limit the eigenstates of the Hamiltonian evolve independently of each other. If the system is prepared in an eigenstate <math>|\psi(0)\rangle = |\psi_n(0)\rangle</math> its time evolution is given by:
<math display="block">|\psi(\lambda)\rangle = |\psi_n(\lambda)\rangle e^{iT\theta_n(\lambda)}e^{i \gamma_n(\lambda)}.</math>

So, for an adiabatic process, a system starting from ''n''th eigenstate also remains in that ''n''th eigenstate like it does for the time-independent processes, only picking up a couple of phase factors. The new phase factor <math>\gamma_n(t)</math> can be canceled out by an appropriate choice of gauge for the eigenfunctions. However, if the adiabatic evolution is [[Berry connection and curvature#Berry phase and cyclic adiabatic evolution|cyclic]], then <math>\gamma_n(t)</math> becomes a gauge-invariant physical quantity, known as the [[Berry phase]].
|}

:{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
!Generic proof in parameter space
|-
|
Let's start from a parametric Hamiltonian <math>H(\vec{R}(t))</math>, where the parameters are slowly varying in time, the definition of slow here is defined essentially by the distance in energy by the eigenstates (through the uncertainty principle, we can define a timescale that shall be always much lower than the time scale considered).

This way we clearly also identify that while slowly varying the eigenstates remains clearly separated in energy (e.g. also when we generalize this to the case of bands as in the [[TKNN formula]] the bands shall remain clearly separated). Given they do not intersect the states are ordered and in this sense this is also one of the meanings of the name [[topological order]].

We do have the instantaneous Schrödinger equation:
<math display="block">H(\vec{R}(t))| \psi_m(t)\rangle = E_m(t)| \psi_m(t)\rangle </math>
And instantaneous eigenstates:
<math display="block">\langle\psi_m(t)|\psi_n(t)\rangle = \delta_{mn}</math>
The generic solution:
<math display="block">|\Psi(t)\rangle = \sum a_n(t)|\psi_n(t)\rangle</math>
plugging in the full Schrödinger equation and multiplying by a generic eigenvector:
<math display="block">\langle \psi_m(t)|i\hbar\partial_t|\Psi(t)\rangle = \langle \psi_m(t)|H(\vec{R}(t))|\Psi(t)\rangle </math>
<math display="block">\dot{a}_m + \sum_n\langle \psi_m(t)|\partial_{\vec{R}} |\psi_n(t)\rangle\dot{\vec{R}}a_n = -\frac{i}{\hbar}E_m(t)a_m </math>
And if we introduce the adiabatic approximation:
<math display="block"> | \langle \psi_m(t)|\partial_{\vec{R}} |\psi_n(t)\rangle\dot{\vec{R}}a_n | \ll |a_m|</math> for each <math>m\ne n</math> 
We have 
<math display="block">\dot{a}_m = - \langle \psi_m(t)|\partial_{\vec{R}} |\psi_m(t)\rangle\dot{\vec{R}}a_m -\frac{i}{\hbar}E_m(t)a_m</math>
and
<math display="block">a_m(t) = e^{-\frac{i}{\hbar} \int_{t_0}^t E_m(t')dt'} e^{i\gamma_m(t)}a_m(t_0)</math>
where 
<math display="block">\gamma_m(t) = i \int_{t_0}^t \langle \psi_m(t)|\partial_{\vec{R}} |\psi_m(t)\rangle\dot{\vec{R}}dt' = i \int_C \langle \psi_m(\vec{R})|\partial_{\vec{R}} |\psi_m(\vec{R})\rangle d\vec{R}  </math>
And C is the path in the parameter space,

This is the same as the statement of the theorem but in terms of the coefficients of the total wave function and its initial state.<ref>{{Cite book| title=Topological insulators and Topological superconductors|last1=Bernevig| first1=B. Andrei|last2=Hughes|first2=Taylor L.| year=2013| pages=Ch. 1|publisher=Princeton university press}}</ref>

Now this is slightly more general than the other proofs given we consider a generic set of parameters, and we see that the Berry phase acts as a local geometric quantity in the parameter space.
Finally integrals of local geometric quantities can give topological invariants as in the case of the [[Gauss-Bonnet theorem]].<ref>{{Cite web | last=Haldane | title=Nobel Lecture | url=https://www.nobelprize.org/uploads/2018/06/haldane-lecture-slides.pdf}}</ref> 
In fact if the path C is closed then the Berry phase persists to gauge transformation and becomes a physical quantity.
|}

== Example applications ==
Often a solid crystal is modeled as a set of independent valence electrons moving in a mean perfectly periodic potential generated by a rigid lattice of ions. With the Adiabatic theorem we can also include instead the motion of the valence electrons across the crystal and the thermal motion of the ions as in the [[Born–Oppenheimer approximation]].<ref name="Bottani">{{cite book |author=Bottani |first=Carlo E. |title=Solid State Physics Lecture Notes |year=2017–2018 |pages=64–67}}</ref>

This does explain many phenomena in the scope of:
* '''thermodynamics''': Temperature dependence of [[specific heat]], [[thermal expansion]], [[melting]]
* '''transport phenomena''': the temperature dependence of [[electric resistivity]] of [[Electrical conductor|conductor]]s, the temperature dependence of [[electric conductivity]] in [[insulator (electricity)|insulator]]s, Some properties of low temperature [[superconductivity]]
* '''optics''': optic [[Absorption (electromagnetic radiation)|absorption]] in the [[infrared]] for [[ionic crystals]], [[Brillouin scattering]], [[Raman scattering]]

== Deriving conditions for diabatic vs adiabatic passage ==
{{Disputed section|errors in the technical section|date = January 2016}}
We will now pursue a more rigorous analysis.<ref name=Messiah>{{cite book |last=Messiah |first=Albert |title=Quantum Mechanics |year=1999 |publisher=Dover Publications |isbn=0-486-40924-4 |chapter=XVII }}</ref> Making use of [[bra–ket notation]], the [[Quantum state|state vector]] of the system at time <math>t</math> can be written

:<math>|\psi(t)\rangle = \sum_n c^A_n(t)e^{-iE_nt/\hbar}|\phi_n\rangle ,</math>

where the spatial wavefunction alluded to earlier is the projection of the state vector onto the eigenstates of the [[position operator]]

:<math>\psi(x,t) = \langle x|\psi(t)\rangle .</math>

It is instructive to examine the limiting cases, in which <math>\tau</math> is very large (adiabatic, or gradual change) and very small (diabatic, or sudden change).

Consider a system Hamiltonian undergoing continuous change from an initial value <math>\hat{H}_0</math>, at time <math>t_0</math>, to a final value <math>\hat{H}_1</math>, at time <math>t_1</math>, where <math>\tau = t_1 - t_0</math>. The evolution of the system can be described in the [[Schrödinger picture]] by the time-evolution operator, defined by the [[integral equation]]

:<math>\hat{U}(t,t_0) = 1 - \frac{i}{\hbar}\int_{t_0}^t\hat{H}(t')\hat{U}(t',t_0)dt' ,</math>

which is equivalent to the [[Schrödinger equation]].

:<math>i\hbar\frac{\partial}{\partial t}\hat{U}(t,t_0) = \hat{H}(t)\hat{U}(t,t_0),</math>

along with the initial condition <math>\hat{U}(t_0,t_0) = 1</math>. Given knowledge of the system [[wave function]] at <math>t_0</math>, the evolution of the system up to a later time <math>t</math> can be obtained using

:<math>|\psi(t)\rangle = \hat{U}(t,t_0)|\psi(t_0)\rangle.</math>

The problem of determining the ''adiabaticity'' of a given process is equivalent to establishing the dependence of <math>\hat{U}(t_1,t_0)</math> on <math>\tau</math>.

To determine the validity of the adiabatic approximation for a given process, one can calculate the probability of finding the system in a state other than that in which it started. Using [[bra–ket notation]] and using the definition <math>|0\rangle \equiv |\psi(t_0)\rangle</math>, we have:

:<math>\zeta = \langle 0|\hat{U}^\dagger(t_1,t_0)\hat{U}(t_1,t_0)|0\rangle - \langle 0|\hat{U}^\dagger(t_1,t_0)|0\rangle\langle 0 | \hat{U}(t_1,t_0) | 0 \rangle.</math>

We can expand <math>\hat{U}(t_1,t_0)</math>

:<math>\hat{U}(t_1,t_0) = 1 + {1 \over i\hbar} \int_{t_0}^{t_1}\hat{H}(t)dt + {1 \over (i\hbar)^2} \int_{t_0}^{t_1}dt' \int_{t_0}^{t'}dt'' \hat{H}(t')\hat{H}(t'') + \cdots.</math>

In the [[Perturbation theory|perturbative limit]] we can take just the first two terms and substitute them into our equation for <math>\zeta</math>, recognizing that

:<math>{1 \over \tau}\int_{t_0}^{t_1}\hat{H}(t)dt \equiv \bar{H}</math>

is the system Hamiltonian, averaged over the interval <math>t_0 \to t_1</math>, we have:

:<math>\zeta = \langle 0|(1 + \tfrac{i}{\hbar}\tau\bar{H})(1 - \tfrac{i}{\hbar}\tau\bar{H})|0\rangle - \langle 0|(1 + \tfrac{i}{\hbar}\tau\bar{H})|0\rangle \langle 0|(1 - \tfrac{i}{\hbar}\tau\bar{H})|0\rangle .</math>

After expanding the products and making the appropriate cancellations, we are left with:

:<math>\zeta = \frac{\tau^2}{\hbar^2}\left(\langle 0|\bar{H}^2|0\rangle - \langle 0|\bar{H}|0\rangle\langle 0|\bar{H}|0\rangle\right) ,</math>

giving

:<math>\zeta = \frac{\tau^2\Delta\bar{H}^2}{\hbar^2} ,</math>

where <math>\Delta\bar{H}</math> is the [[root mean square]] deviation of the system Hamiltonian averaged over the interval of interest.

The sudden approximation is valid when <math>\zeta \ll 1</math> (the probability of finding the system in a state other than that in which is started approaches zero), thus the validity condition is given by

:<math>\tau \ll {\hbar \over \Delta\bar{H}} ,</math>

which is a statement of the [[Heisenberg uncertainty principle#Energy-time uncertainty principle|time-energy form of the Heisenberg uncertainty principle]].

=== Diabatic passage ===

In the limit <math>\tau \to 0</math> we have infinitely rapid, or diabatic passage:

:<math>\lim_{\tau \to 0}\hat{U}(t_1,t_0) = 1 .</math>

The functional form of the system remains unchanged:

:<math>|\langle x|\psi(t_1)\rangle|^2 = \left|\langle x|\psi(t_0)\rangle\right|^2 .</math>

This is sometimes referred to as the sudden approximation. The validity of the approximation for a given process can be characterized by the probability that the state of the system remains unchanged:

:<math>P_D = 1 - \zeta.</math>

=== Adiabatic passage ===

In the limit <math>\tau \to \infty</math> we have infinitely slow, or adiabatic passage. The system evolves, adapting its form to the changing conditions,

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

If the system is initially in an [[eigenstate]] of <math>\hat{H}(t_0)</math>, after a period <math>\tau</math> it will have passed into the ''corresponding'' eigenstate of <math>\hat{H}(t_1)</math>.

This is referred to as the adiabatic approximation. The validity of the approximation for a given process can be determined from the probability that the final state of the system is different from the initial state:

:<math>P_A = \zeta .</math>

== Calculating adiabatic passage probabilities ==

=== The Landau–Zener formula ===
{{main|Landau–Zener formula}}

In 1932 an analytic solution to the problem of calculating adiabatic transition probabilities was published separately by [[Lev Landau]] and [[Clarence Zener]],<ref name="Zener">{{cite journal |author=Zener |first=C. |year=1932 |title=Non-adiabatic Crossing of Energy Levels |journal=Proceedings of the Royal Society of London, Series A |volume=137 |issue=6 |pages=692–702 |bibcode=1932RSPSA.137..696Z |doi=10.1098/rspa.1932.0165 |jstor=96038 |doi-access=free}}</ref> for the special case of a linearly changing perturbation in which the time-varying component does not couple the relevant states (hence the coupling in the diabatic Hamiltonian matrix is independent of time).

The key figure of merit in this approach is the Landau–Zener velocity:
<math display="block">v_\text{LZ} = {\frac{\partial}{\partial t}|E_2 - E_1| \over \frac{\partial}{\partial q}|E_2 - E_1|} \approx \frac{dq}{dt} ,</math>
where <math>q</math> is the perturbation variable (electric or magnetic field, molecular bond-length, or any other perturbation to the system), and <math>E_1</math> and <math>E_2</math> are the energies of the two diabatic (crossing) states. A large <math>v_\text{LZ}</math> results in a large diabatic transition probability and vice versa.

Using the Landau–Zener formula the probability, <math>P_{\rm D}</math>, of a diabatic transition is given by

<math display="block">\begin{align}
P_{\rm D} &= e^{-2\pi\Gamma}\\
\Gamma &= {a^2/\hbar \over \left|\frac{\partial}{\partial t}(E_2 - E_1)\right|} = {a^2/\hbar \over \left|\frac{dq}{dt}\frac{\partial}{\partial q}(E_2 - E_1)\right|}\\
&= {a^2 \over \hbar|\alpha|}\\
\end{align}</math>
<!--In order to describe this approach we will use as an example a 2-level atom in a magnetic field, as described [[Adiabatic theorem#Avoided curve crossing|above]].  All the same notation will be used.

For a fully quantum–mechanical treatment of a general system, the equations of motion
for the coefficients, <math>c_1(t)</math> and <math>c_2(t)</math> of the diabatic states, <math>|1\rangle</math> and <math>|2\rangle</math>, cannot be solved analytically. In 1932, two closely related papers by Lev Landau and Clarence Zener<ref name="Zener">{{cite journal |author=C. Zener |title=Non-adiabatic Crossing of Energy Levels |journal=Proceedings of the Royal Society of London, Series A |volume=137 |issue=6 |pages=692–702 |year=1932 |doi=10.1098/rspa.1932.0165 |jstor=96038|bibcode = 1932RSPSA.137..696Z }}</ref> were published on the subject of diabatic transitions between quantum states. Such transitions occur between states of the entire system, hence any description of the system must include all external influences, including collisions and external electric and magnetic fields. In order that the equations of motion for the system might be solved analytically, a set of simplifications are made, known collectively as the Landau–Zener approximation. The simplifications are as follows:

# The perturbation parameter is a known, linear function of time
# The energy separation of the diabatic states varies linearly with time
# The coupling <math>a</math> in the diabatic Hamiltonian matrix is independent of time

The first simplification makes this a semi-classical treatment. In the case of an atom in a magnetic field, the field strength becomes a classical variable which can be precisely measured during the transition. This requirement is quite restrictive as a linear change will not, in general, be the optimal profile to achieve the desired transition probability.
The second simplification allows us to make the substitution <math>E_2(t) - E_1(t) \equiv \alpha t</math>; for our model system this corresponds to a linear change in magnetic field. For a linear [[Zeeman effect|Zeeman shift]] this follows directly from point 1.
The final simplification requires that the time–dependent perturbation does not couple the diabatic states; rather, the coupling must be due to a static deviation from a <math>1/r</math> [[Coulomb potential]], commonly described by a [[quantum defect]].
The details of Zener’s solution are somewhat opaque, relying on a set of substitutions to put the equation of motion into the form of the Weber equation and using the known solution. A more transparent solution is provided by Wittig<ref name="Wittig">{{cite journal |author=C. Wittig |title=The Landau–Zener Formula |journal=Journal of Physical Chemistry B |volume=109 |issue=17 |pages=8428–8430 |year=2005 |url=https://pubs.acs.org/secure/login?url=http%3A%2F%2Fpubs.acs.org%2Fcgi-bin%2Farticle.cgi%2Fjpcbfk%2F2005%2F109%2Fi17%2Fpdf%2Fjp040627u.pdf| doi=10.1021/jp040627u|format=PDF |pmid=16851989}}</ref> using [[contour integration]].-->

=== The numerical approach ===
{{main|Numerical ordinary differential equations|l1=Numerical solution of ordinary differential equations}}

For a transition involving a nonlinear change in perturbation variable or time-dependent coupling between the diabatic states, the equations of motion for the system dynamics cannot be solved analytically. The diabatic transition probability can still be obtained using one of the wide varieties of [[Numerical ordinary differential equations|numerical solution algorithms for ordinary differential equations]].

The equations to be solved can be obtained from the time-dependent Schrödinger equation:

<math display="block">i\hbar\dot{\underline{c}}^A(t) = \mathbf{H}_A(t)\underline{c}^A(t) ,</math>

where <math>\underline{c}^A(t)</math> is a [[Column vector|vector]] containing the adiabatic state amplitudes, <math>\mathbf{H}_A(t)</math> is the time-dependent adiabatic Hamiltonian,<ref name="Stenholm" /> and the overdot represents a time derivative.

Comparison of the initial conditions used with the values of the state amplitudes following the transition can yield the diabatic transition probability. In particular, for a two-state system:
<math display="block">P_D = |c^A_2(t_1)|^2</math>
for a system that began with <math>|c^A_1(t_0)|^2 = 1</math>.

== See also ==
* [[Landau–Zener formula]]
* [[Berry phase]]
* [[Quantum stirring, ratchets, and pumping]]
* [[Adiabatic quantum motor]]
* [[Born–Oppenheimer approximation]]
*[[Eigenstate thermalization hypothesis]]
*[[Adiabatic process]]

== References ==
{{reflist|2}}

[[Category:Theorems in quantum mechanics]]