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{{Short description|Concept in quantum mechanics}}
{{Quantum book backlink|Quantum dynamics and evolution}}
{{about|the quantum mechanics concept|the thermodynamics concept|Adiabatic process}}

The '''adiabatic theorem''' is a concept in [[Physics:Quantum mechanics|quantum mechanics]] stating that a physical system remains in its instantaneous [[Physics:Eigenstate|eigenstate]] if a given [[Physics:Perturbation theory (quantum mechanics)|perturbation]] acts on it slowly enough and if there is a gap between the [[eigenvalue]] and the rest of the [[Physics:Hamiltonian (quantum mechanics)|Hamiltonian]]'s [[Spectrum of an operator|spectrum]]. In simpler terms, a quantum-mechanical system subjected to gradually changing external conditions adapts its functional form, whereas under rapidly varying conditions there is insufficient time for the state to adapt, so the spatial probability density remains unchanged.<ref name="Born-Fock">{{cite journal |author=Born |first1=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>

[[File:Avoided_crossing_in_linear_field-y.svg|thumb|400px|Avoided crossing in a two-level quantum system: under slow variation the system follows an adiabatic eigenstate, while rapid variation can produce diabatic transition between states.]]

== Introduction ==
The theorem in its original form was given by [[Biography:Max Born|Max Born]] and [[Biography:Vladimir Fock|Vladimir Fock]] in 1928.<ref name="Born-Fock" /> It is one of the central results of quantum mechanics because it explains how quantum states behave when the Hamiltonian changes slowly in time. If the evolution is sufficiently gradual and the relevant eigenvalue remains separated from the rest of the spectrum, the system stays in the corresponding instantaneous eigenstate, acquiring only phase factors.

The theorem is closely related to the notion of [[adiabatic invariant]] in the old quantum theory, but its meaning in quantum mechanics differs from the thermodynamic meaning of the word ''adiabatic''. In quantum mechanics, adiabatic generally means ''slow'' variation compared with the system’s internal timescales.<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>

== Adiabatic pendulum ==
At the 1911 [[Solvay Conference|Solvay conference]], [[Biography:Albert Einstein|Albert Einstein]] discussed the quantum hypothesis that for atomic oscillators <math>E = nh\nu</math>. After Einstein's lecture, [[Biography:Hendrik Lorentz|Hendrik Lorentz]] pointed out that if a classical pendulum is shortened gradually, its energy appears to change smoothly. Einstein replied that although both the energy <math>E</math> and the frequency <math>\nu</math> vary, their ratio <math>\frac{E}{\nu}</math> remains conserved, preserving the quantum hypothesis.<ref>{{Cite book |last1=Instituts Solvay |first1=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 read a paper by [[Biography:Paul Ehrenfest|Paul Ehrenfest]] on the adiabatic hypothesis, and later referred to it in correspondence with Michele Besso.<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><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 |bibcode=1994CaJCh..72..936L |issn=0008-4042}}</ref>

== Diabatic vs. adiabatic processes ==
Rapidly changing conditions prevent the system from adapting its configuration during the process, so the spatial probability density remains unchanged. The system then generally ends in a [[linear superposition]] of eigenstates of the final Hamiltonian. By contrast, gradually changing conditions allow the probability density to change continuously, and if the system begins in an eigenstate of the initial Hamiltonian it ends 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 an initial time <math>t_0</math> a quantum-mechanical system has Hamiltonian <math>\hat{H}(t_0)</math> and is in an eigenstate <math>\psi(x,t_0)</math>. A continuous change in conditions modifies the Hamiltonian to <math>\hat{H}(t_1)</math> at time <math>t_1</math>. The theorem states that the behavior depends critically on the interval <math>\tau = t_1 - t_0</math> over which the change occurs.

For a truly adiabatic process, <math>\tau \to \infty</math>, and the final state becomes an eigenstate of the final Hamiltonian with a modified spatial form:
<math display="block">|\psi(x,t_1)|^2 \neq |\psi(x,t_0)|^2 .</math>

Conversely, in the limit <math>\tau \to 0</math>, corresponding to infinitely rapid or diabatic passage, the configuration remains unchanged:
<math display="block">|\psi(x,t_1)|^2 = |\psi(x,t_0)|^2 .</math>

The approximation depends on both the energy gap separating the relevant state from nearby states and the ratio of the process duration <math>\tau</math> to the internal timescale <math>\tau_\text{int} = 2\pi\hbar/E_0</math>.

The original theorem assumed a discrete and nondegenerate spectrum, so that one can identify which final eigenstate corresponds to the initial one. Later work generalized the theorem to cases without a strict gap condition.<ref name="Avron-Elgart">{{cite journal |author=Avron |first1=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 ===
In [[Physics:Thermodynamics|thermodynamics]], an [[adiabatic process]] is one in which there is no exchange of heat between system and environment. In practice, such processes are often relatively fast compared with the timescale of heat transfer.

In classical and quantum mechanics, however, ''adiabatic'' means something closer to a [[quasistatic process]]: a process slow enough that the system remains near equilibrium or stays in the corresponding instantaneous eigenstate.<ref name=Griffiths />

Thus, in quantum mechanics the adiabatic theorem states that quantum jumps are suppressed when the Hamiltonian changes sufficiently slowly, and the system tends to preserve its state label and 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>

== Example systems ==
=== Simple pendulum ===
A useful classical analogy is a [[pendulum]] whose support is moved slowly. If the change is gradual enough, the mode of oscillation relative to the support remains essentially unchanged, showing how a slow external variation allows the system to adapt continuously.<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 ===
A more specifically quantum example is the [[Physics:Quantum harmonic oscillator|quantum harmonic oscillator]] with increasing [[spring constant]] <math>k</math>. If <math>k</math> increases adiabatically, the system remains in the corresponding instantaneous eigenstate of the current Hamiltonian. For a state initially in the ground state, the wavefunction narrows as the potential steepens, but the quantum number remains unchanged.

If <math>k</math> increases rapidly, the process becomes diabatic. The system then has no time to adapt to the changing potential and the final state becomes a superposition of many eigenstates of the new Hamiltonian, whose combination reproduces the original probability density.

=== Avoided curve crossing ===
{{main|Avoided crossing}}
A particularly important example is a two-level atom in an external [[magnetic field]]. The system may be described in terms of diabatic states <math>|1\rangle</math> and <math>|2\rangle</math> with wavefunction
<math display="block">|\Psi\rangle = c_1(t)|1\rangle + c_2(t)|2\rangle.</math>

If the magnetic-field dependence is linear, the Hamiltonian matrix may be written
<math display="block">\mathbf{H} = \begin{pmatrix}
\mu B(t)-\hbar\omega_0/2 & a \\
a^* & \hbar\omega_0/2-\mu B(t)
\end{pmatrix}</math>

with 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>

Because of the off-diagonal coupling, the eigenvalues do not cross but exhibit an [[avoided crossing]]. Under slow variation of the magnetic field, the system follows an adiabatic eigenstate. Under rapid variation it instead follows a diabatic path, giving a finite probability of transition between the two eigenstates.<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>

== 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
<math display="block">| \psi(0) \rangle = \sum_n c_n(0) | n(0) \rangle</math>
to
<math display="block">| \psi(t) \rangle = \sum_n c_n(t) | n(t) \rangle ,</math>
where the coefficients change only by phase factors,
<math display="block">c_n(t) = c_n(0) e^{i \theta_n(t)} e^{i \gamma_n(t)}.</math>

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

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

Hence <math>|c_n(t)|^2 = |c_n(0)|^2</math>. In particular, if the system starts in an eigenstate of <math>H(0)</math>, it remains in the corresponding eigenstate of <math>H(t)</math> throughout the evolution, apart from phase changes only.

== Proofs ==
A standard derivation expands the full state in the instantaneous eigenbasis of the time-dependent Hamiltonian and inserts that expansion into the time-dependent [[Physics:Schrödinger equation|Schrödinger equation]]. This yields an exact set of coupled first-order differential equations for the coefficients. The off-diagonal couplings are proportional to matrix elements of <math>\dot{H}(t)</math> divided by energy differences between eigenstates. If the Hamiltonian changes slowly enough and a finite gap is maintained, these off-diagonal terms can be neglected, leading directly to the adiabatic approximation and to the phase factors above.<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>

A more detailed formulation expresses the slowly varying Hamiltonian as <math>\hat{H}(t/T)</math> and studies the limit <math>T \to \infty</math>. In this limit, oscillatory off-diagonal terms are suppressed and the exact solution approaches the adiabatic one with an error of order <math>O(T^{-1})</math>. For cyclic adiabatic evolution, the geometric phase becomes a gauge-invariant physical observable, known as the [[Physics:Berry phase|Berry phase]].<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>

A parameter-space formulation writes the Hamiltonian as <math>H(\vec{R}(t))</math>, where <math>\vec{R}</math> varies slowly in time. In this approach the Berry phase can be written as a line integral along a path in parameter space,
<math display="block">\gamma_m(t) = i \int_C \langle \psi_m(\vec{R})|\partial_{\vec{R}} |\psi_m(\vec{R})\rangle d\vec{R},</math>
which highlights its geometric character and its relation to topological ideas.<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><ref>{{Cite web | last=Haldane | title=Nobel Lecture | url=https://www.nobelprize.org/uploads/2018/06/haldane-lecture-slides.pdf}}</ref>

== Example applications ==
The adiabatic theorem underlies many approximation schemes in quantum physics, especially those involving a separation of fast and slow degrees of freedom. A prominent example is the [[Physics:Born–Oppenheimer approximation|Born–Oppenheimer approximation]], where fast electronic motion and slow ionic motion in crystals or molecules are treated separately.<ref name="Bottani">{{cite book |author=Bottani |first=Carlo E. |title=Solid State Physics Lecture Notes |year=2017–2018 |pages=64–67}}</ref>

This helps explain a variety of phenomena in:
* '''thermodynamics''': temperature dependence of [[specific heat]], [[thermal expansion]], and [[melting]]
* '''transport phenomena''': temperature dependence of [[electric resistivity]] in [[Electrical conductor|conductors]], temperature dependence of [[electric conductivity]] in [[insulator (electricity)|insulators]], and properties of low-temperature [[superconductivity]]
* '''optics''': infrared [[Absorption (electromagnetic radiation)|absorption]] in ionic crystals, [[Brillouin scattering]], and [[Raman scattering]]

== Deriving conditions for diabatic vs adiabatic passage ==
One way to analyze the validity of the adiabatic approximation is through the time-evolution operator <math>\hat{U}(t,t_0)</math>, defined by
<math display="block">i\hbar\frac{\partial}{\partial t}\hat{U}(t,t_0) = \hat{H}(t)\hat{U}(t,t_0),</math>
with
<math display="block">\hat{U}(t_0,t_0) = 1.</math>

The system evolves as
<math display="block">|\psi(t)\rangle = \hat{U}(t,t_0)|\psi(t_0)\rangle.</math>

To test whether a process is sudden or adiabatic, one can define the probability of leaving the initial state. In the perturbative limit this leads to
<math display="block">\zeta = \frac{\tau^2\Delta\bar{H}^2}{\hbar^2},</math>
where <math>\bar{H}</math> is the Hamiltonian averaged over the interval and <math>\Delta\bar{H}</math> is its root-mean-square deviation.

The sudden approximation is valid when
<math display="block">\tau \ll {\hbar \over \Delta\bar{H}},</math>
which is a form of the [[Heisenberg uncertainty principle#Energy-time uncertainty principle|time-energy uncertainty relation]].<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>

=== Diabatic passage ===
In the limit <math>\tau \to 0</math>,
<math display="block">\lim_{\tau \to 0}\hat{U}(t_1,t_0) = 1.</math>

The functional form of the state remains unchanged:
<math display="block">|\langle x|\psi(t_1)\rangle|^2 = \left|\langle x|\psi(t_0)\rangle\right|^2 .</math>

This is the sudden approximation, and the probability that the system remains unchanged is
<math display="block">P_D = 1 - \zeta.</math>

=== Adiabatic passage ===
In the limit <math>\tau \to \infty</math>, the system has time to adapt continuously to the changing Hamiltonian:
<math display="block">|\langle x|\psi(t_1)\rangle|^2 \neq |\langle x|\psi(t_0)\rangle|^2 .</math>

If the system begins in an eigenstate of <math>\hat{H}(t_0)</math>, it evolves into the corresponding eigenstate of <math>\hat{H}(t_1)</math>. The probability associated with adiabatic passage is
<math display="block">P_A = \zeta.</math>

== Calculating adiabatic passage probabilities ==
=== The Landau–Zener formula ===
{{main|Landau–Zener formula}}
For a two-level system with a linearly changing perturbation, the probability of a diabatic transition can be estimated using the [[Physics:Landau–Zener formula|Landau–Zener formula]]. The key quantity 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 parameter.

The diabatic transition probability is then
<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>

A large sweep rate gives a large diabatic transition probability, whereas a slow sweep favors adiabatic following.<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>

=== The numerical approach ===
For nonlinear changes in the perturbation variable or time-dependent couplings, the system generally cannot be solved analytically and one must use numerical methods for ordinary differential equations. The amplitudes satisfy
<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 the vector of adiabatic-state amplitudes and <math>\mathbf{H}_A(t)</math> is the adiabatic Hamiltonian.<ref name="Stenholm" />

For a two-state system beginning in state 1, the diabatic transition probability can be obtained from
<math display="block">P_D = |c^A_2(t_1)|^2.</math>

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

== References ==
{{reflist|3}}

{{Author|Harold Foppele}}

{{Sourceattribution|Adiabatic theorem|1}}

Physics:Quantum Adiabatic theorem - Revision history

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