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| The '''adiabatic theorem''' is a concept in [[quantum mechanics]]. Its original form, due to [[Max Born]] and [[Vladimir Fock]] (1928), was stated as follows:
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| :''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=M. Born and V. A. Fock |title=Beweis des Adiabatensatzes |journal=Zeitschrift für Physik A |volume=51 |issue=3–4 |pages=165–180 |year=1928|doi=10.1007/BF01343193}}</ref>
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| 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 and the probability density remains unchanged.
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| == Diabatic vs. adiabatic processes ==
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| <div style="background-color:#deb887; border: ridge; border-color:black; padding-left:10px; padding-right:10px; margin-bottom:0.5em;">
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| <p align="left">
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| '''Diabatic process:''' Rapidly changing conditions prevent the system from adapting its configuration during the process, hence the 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.
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| </div>
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| <div style="background-color:#deb887; border: ridge; border-color:black; padding-left:10px; padding-right:10px; margin-bottom:0.75em;">
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| <p align="left">
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| '''Adiabatic process:''' 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=T. Kato |title=On the Adiabatic Theorem of Quantum Mechanics |journal=Journal of the Physical Society of Japan |volume=5 |issue=6 |pages=435–439 |year=1950 |doi=10.1143/JPSJ.5.435|bibcode = 1950JPSJ....5..435K }}</ref>
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| </div>
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| At some initial time <math>\scriptstyle{t_0}</math> a quantum-mechanical system has an energy given by the Hamiltonian <math>\scriptstyle{\hat{H}(t_0)}</math>; the system is in an eigenstate of <math>\scriptstyle{\hat{H}(t_0)}</math> labelled <math>\scriptstyle{\psi(x,t_0)}</math>. Changing conditions modify the Hamiltonian in a continuous manner, resulting in a final Hamiltonian <math>\scriptstyle{\hat{H}(t_1)}</math> at some later time <math>\scriptstyle{t_1}</math>. The system will evolve according to the Schrödinger equation, to reach a final state <math>\scriptstyle{\psi(x,t_1)}</math>. The adiabatic theorem states that the modification to the system depends critically on the time <math>\scriptstyle{\tau = t_1 - t_0}</math> during which the modification takes place.
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| For a truly adiabatic process we require <math>\scriptstyle{\tau \rightarrow \infty}</math>; in this case the final state <math>\scriptstyle{\psi(x,t_1)}</math> will be an eigenstate of the final Hamiltonian <math>\scriptstyle{\hat{H}(t_1)}</math>, with a modified configuration:
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| :<math>|\psi(x,t_1)|^2 \neq |\psi(x,t_0)|^2</math>.
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| The degree to which a given change approximates an adiabatic process depends on both the energy separation between <math>\scriptstyle{\psi(x,t_0)}</math> and adjacent states, and the ratio of the interval <math>\scriptstyle{\tau}</math> to the characteristic time-scale of the evolution of <math>\scriptstyle{\psi(x,t_0)}</math> for a time-independent Hamiltonian, <math>\scriptstyle{\tau_{int} = 2\pi\hbar/E_0}</math>, where <math>\scriptstyle{E_0}</math> is the energy of <math>\scriptstyle{\psi(x,t_0)}</math>.
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| Conversely, in the limit <math>\scriptstyle{\tau \rightarrow 0}</math> we have infinitely rapid, or diabatic passage; the configuration of the state remains unchanged:
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| :<math>|\psi(x,t_1)|^2 = |\psi(x,t_0)|^2\quad</math>.
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| 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>\scriptstyle{\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>\scriptstyle{\hat{H}(t_1)}</math> ''corresponds'' to <math>\scriptstyle{\psi(t_0)}</math>). In 1999 J. E. Avron and A. Elgart reformulated the adiabatic theorem, eliminating the gap condition.<ref name="Avron-Elgart">{{cite journal |author=J. E. Avron and A. Elgart |title=Adiabatic Theorem without a Gap Condition |journal=Communications in Mathematical Physics |volume=203 |issue=2 |pages=445–463 |year=1999 |url=http://www.springerlink.com/content/ad0jyug24jg97nt6/fulltext.pdf|doi=10.1007/s002200050620|format=PDF|arxiv = math-ph/9805022 |bibcode = 1999CMaPh.203..445A }}</ref>
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| Note that the term "adiabatic" is traditionally used in [[thermodynamics]] to describe processes without the exchange of heat between system and environment (see [[adiabatic process]]). The quantum mechanical definition is closer to the thermodynamical concept of a [[quasistatic process]], and has no direct relation with heat exchange.
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| == Example systems ==
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| === Simple pendulum ===
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| 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. This is referred to as an [[Adiabatic process (quantum mechanics)|adiabatic process]].<ref name=Griffiths>{{cite book |last=Griffiths |first=David J. |title=Introduction to Quantum Mechanics |year=2005 |publisher=Pearson Prentice Hall |location= |isbn=0-13-111892-7 |chapter=10 }}</ref>
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| === Quantum harmonic oscillator ===
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| [[Image:HO adiabatic process.gif|thumb|right|300px|'''Figure 1.''' Change in the probability density, <math>\scriptstyle{|\psi(t)|^2}</math>, of a ground state quantum harmonic oscillator, due to an adiabatic increase in spring constant.]]
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| 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>\scriptstyle{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]].
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| If <math>\scriptstyle{k}</math> is increased adiabatically <math>\scriptstyle{\left(\frac{dk}{dt} \rightarrow 0\right)}</math> then the system at time <math>\scriptstyle{t}</math> will be in an instantaneous eigenstate <math>\scriptstyle{\psi(t)}</math> of the ''current'' Hamiltonian <math>\scriptstyle{\hat{H}(t)}</math>, corresponding to the initial eigenstate of <math>\scriptstyle{\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>\scriptstyle{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.
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| For a rapidly increased spring constant, the system undergoes a diabatic process <math>\scriptstyle{\left(\frac{dk}{dt} \rightarrow \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>\scriptstyle{\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>\scriptstyle{\hat{H}(t)}</math>, that resembles the initial state. The final state is composed of a [[linear superposition]] of many different eigenstates of <math>\scriptstyle{\hat{H}(t)}</math> which sum to reproduce the form of the initial state.
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| === Avoided curve crossing ===
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| {{main|Avoided crossing}}
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| [[Image:Avoided crossing.png|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>\scriptstyle{|1\rangle}</math> and <math>\scriptstyle{|2\rangle}</math> and the [[eigenvalues]] of the Hamiltonian, giving the energies of the eigenstates <math>\scriptstyle{|\phi_1\rangle}</math> and <math>\scriptstyle{|\phi_2\rangle}</math> (the adiabatic states).]]
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| 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=S. Stenholm |title=Quantum Dynamics of Simple Systems |author-link=Stig Stenholm |journal=The 44th Scottish Universities Summer School in Physics |volume= |issue= |pages=267–313 |year=1994 |url= |pmid= |doi=}}</ref> The states, labelled <math>\scriptstyle{|1\rangle}</math> and <math>\scriptstyle{|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:
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| :<math>|\Psi\rangle = c_1(t)|1\rangle + c_2(t)|2\rangle.</math>
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| With the field absent, the energetic separation of the diabatic states is equal to <math>\scriptstyle{\hbar\omega_0}</math>; the energy of state <math>\scriptstyle{|1\rangle}</math> increases with increasing magnetic field (a low-field-seeking state), while the energy of state <math>\scriptstyle{|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
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| :<math>\mathbf{H} = \begin{pmatrix}
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| \mu B(t)-\hbar\omega_0/2 & a \\
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| a^* & \hbar\omega_0/2-\mu B(t) \end{pmatrix}</math>
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| where <math>\scriptstyle{\mu}</math> is the [[magnetic moment]] of the atom, assumed to be the same for the two diabatic states, and <math>\scriptstyle{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>\scriptstyle{E_1(t)}</math> and <math>\scriptstyle{E_2(t)}</math>), however, as <math>\scriptstyle{\mathbf{H}}</math> is not a [[diagonal matrix]], it is clear that these states are not eigenstates of the new Hamiltonian that includes the magnetic field contribution.
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| The eigenvectors of the matrix <math>\scriptstyle{\mathbf{H}}</math> are the eigenstates of the system, which we will label <math>\scriptstyle{|\phi_1(t)\rangle}</math> and <math>\scriptstyle{|\phi_2(t)\rangle}</math>, with corresponding eigenvalues
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| :<math>\begin{align}
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| \varepsilon_1(t) &= -\frac{1}{2}\sqrt{4a^2 + (\hbar\omega_0 - 2\mu B(t))^2}\\
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| \varepsilon_2(t) &= +\frac{1}{2}\sqrt{4a^2 + (\hbar\omega_0 - 2\mu B(t))^2}.\\
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| \end{align}</math>
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| It is important to realise that the eigenvalues <math>\scriptstyle{\varepsilon_1(t)}</math> and <math>\scriptstyle{\varepsilon_2(t)}</math> are the only allowed outputs for any individual measurement of the system energy, whereas the diabatic energies <math>\scriptstyle{E_1(t)}</math> and <math>\scriptstyle{E_2(t)}</math> correspond to the [[expectation value]]s for the energy of the system in the diabatic states <math>\scriptstyle{|1\rangle}</math> and <math>\scriptstyle{|2\rangle}</math>.
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| '''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>\scriptstyle{|\phi_1(t_0)\rangle}</math> in zero magnetic field (on the red curve, at the extreme left), an adiabatic increase in magnetic field <math>\scriptstyle{\left(\frac{dB}{dt}\rightarrow0\right)}</math> will ensure the system remains in an eigenstate of the Hamiltonian <math>\scriptstyle{|\phi_1(t)\rangle}</math> throughout the process(follows the red curve). A diabatic increase in magnetic field <math>\scriptstyle{\left(\frac{dB}{dt}\rightarrow\infty\right)}</math> will ensure the system follows the diabatic path (the solid black line), such that the system undergoes a transition to state <math>\scriptstyle{|\phi_2(t_1)\rangle}</math>. For finite magnetic field slew rates <math>\scriptstyle{\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 [[Adiabatic theorem#Calculating diabatic passage probabilities|below]] for approaches to calculating these probabilities.
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| 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.
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| == Proof of the Adiabatic theorem ==
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| The first proof of this theorem was given by [[Max Born]] and [[Vladimir Fock]], in [[European Physical Journal|Zeitschrift für Physik]] '''51''', 165 (1928). The concept of this theorem deals with the ''time-dependent'' [[Hamiltonian (quantum mechanics)|Hamiltonian]] (which might be called a subject of [[Quantum dynamics]]) where the Hamiltonian changes with time.
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| :For the case of a ''time-independent'' [[Hamiltonian (quantum mechanics)|Hamiltonian]] or in a broad sense time-independent potential (subjects of [[Quantum statics]]) the [[Schrödinger equation]]: | |
| ::<math>i \hbar{\partial \over \partial t} \Psi(x,t) = \hat H \Psi(x,t)</math>
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| :can be simplified to the ''[[Schrödinger equation#Time independent equation|time-independent Schrödinger equation]]'',
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| ::<math>\hat H\psi_n(x) = E_n\psi_n(x)</math>
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| :as the general solution of the Schrödinger equation then can be found by the method of [[Separation of variables]] to give the [[Wave function|wavefunction]] of the form:
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| ::<math> \Psi(x,t)= \sum_n c_n\Psi_n(x,t) = \sum_n c_n\psi_n(x) e^{-iE_nt/\hbar}</math>
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| :or, for ''n''th [[eigenstate]] only : <math>\ \Psi_n(x,t)=\psi_n(x) e^{-iE_nt/\hbar}</math>
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| :This signifies that a particle which starts from the ''n''<sup>th</sup> eigenstate remains in the ''n''<sup>th</sup> eigenstate, simply picking up a [[phase factor]] <math>\scriptstyle{(-E_nt/\hbar)}</math>.
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| In an adiabatic process the Hamiltonian is ''time-dependent'' ''i.e,'' the Hamiltonian changes with time (not to be confused with [[Perturbation theory (quantum mechanics)|Perturbation theory]], as here the change in the Hamiltonian is ''not small''; it's ''huge'', although it happens gradually). As the Hamiltonian changes with time, the eigenvalues and the eigenfunctions are time dependent.
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| :<math>\hat H(t)\psi_n(x,t) = E_n(t)\psi_n(x,t)</math>
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| But at any particular instant of time the states still form a [[Complete orthogonal system]]. ''i.e,''
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| :<math>\langle\psi_n(t)|\psi_m(t)\rangle = \delta_{nm}</math>
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| Notice that: The dependence on position is ''tacitly'' implied, as the ''time dependence'' part will be in more concern. <math>\scriptstyle{\Psi(t)}</math> will considered to be the state of the system at time ''t'' no-matter how it depends on its position.
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| The general solution of time dependent Schrödinger equation now can be expressed as
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| :<math> \Psi(t)= \sum_n c_n(t)\psi_n(t) e^{i\theta_n(t)}</math> where <math>\scriptstyle{\theta_n(t) = -\frac{1}{\hbar}\int\limits_{0}^{t}E_n(t')dt'}</math>.
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| The phase <math>\scriptstyle{\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
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| :<math>i \hbar \sum_n (\dot{c_n}\psi_n + c_n\dot{\psi_n} + i c_n\psi_n\dot{\theta_n})e^{i\theta_n} = \sum_n c_n\hat H\psi_n e^{i\theta_n}</math>
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| The term <math>\scriptstyle{\dot{\theta_n}}</math> gives <math>\scriptstyle{-E/\hbar}</math> and so the third term of left hand side cancels out with the right hand side leaving
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| :<math>\sum_n \dot{c_n}\psi_n e^{i\theta_n} = - \sum_n c_n\dot{\psi_n}e^{i\theta_n}</math>
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| now taking the inner product with an arbitrary eigenfunction <math>\scriptstyle{\langle\psi_m|}</math>, the on the left <math>\scriptstyle{\langle\psi_m|\psi_n\rangle}</math> gives <math>\delta_{nm}</math> which is 1 only for m = n otherwise vanishes. The remaining part gives
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| :<math>\dot{c}_m(t) = - \sum_n c_n\langle\psi_m|\dot{\psi_n}\rangle e^{i(\theta_n-\theta_m)}</math>
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| calculating the expression for <math>\scriptstyle{\langle\psi_m|\dot{\psi_n}\rangle}</math> from differentiating the modified time independent Schrödinger equation above it can have the form
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| :<math>\dot{c}_m(t) = - c_m\langle\psi_m|\dot{\psi_m}\rangle - \sum_{n\ne m}c_n\frac{\langle\psi_m|\dot{\hat H}|\psi_n\rangle}{E_n - E_m} e^{i(\theta_n-\theta_m)}</math>
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| This is also ''exact''.<br>For the '''adiabatic approximation''' which says the time derivative of Hamiltonian ''i.e,'' <math>\scriptstyle{\dot{\hat H}}</math> is extremely small as a long time is taken, the last term will drop out and one has
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| :<math>\dot{c}_m(t) = - c_m\langle\psi_m|\dot{\psi_m}\rangle</math>
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| that gives, after solving, | |
| :<math>c_m(t) = c_m(0)\exp[-\textstyle\int\limits_{0}^{t}\langle\psi_m(t')|\dot{\psi_m}(t')\rangle dt'] = c_m(0)e^{i\gamma_m(t)},</math>
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| having defined the [[geometric phase]] as <math>\scriptstyle{\dot\gamma_m(t)=i\langle\psi_m(t)|\dot{\psi_m}(t)\rangle}</math>. Putting it in the expression for n<sup>th</sup> eigenstate one has
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| :<math>\Psi_n(t)=\psi_n(t) e^{i\theta_n(t)}e^{i\gamma_n(t)}.</math>
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| So, for an adiabatic process, a particle starting from n<sup>th</sup> eigenstate also remains in that n<sup>th</sup> eigenstate like it does for the time-independent processes, only picking up a couple of phase factors. The new phase factor <math>\scriptstyle{\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>\scriptstyle{\gamma_n(t)}</math> becomes a gauge-invariant physical quantity, known as the [[Berry phase]].
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| == Deriving conditions for diabatic vs adiabatic passage ==
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| 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>\scriptstyle{t}</math> can be written
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| :<math>|\psi(t)\rangle = \sum_n c^A_n(t)e^{-iE_nt/\hbar}|\phi_n\rangle</math>,
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| where the spatial wavefunction alluded to earlier is the projection of the state vector onto the eigenstates of the [[position operator]]
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| :<math>\psi(x,t) = \langle x|\psi(t)\rangle</math>.
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| It is instructive to examine the limiting cases, in which <math>\scriptstyle{\tau}</math> is very large (adiabatic, or gradual change) and very small (diabatic, or sudden change).
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| Consider a system Hamiltonian undergoing continuous change from an initial value <math>\scriptstyle{\hat{H}_0}</math>, at time <math>\scriptstyle{t_0}</math>, to a final value <math>\scriptstyle{\hat{H}_1}</math>, at time <math>\scriptstyle{t_1}</math>, where <math>\scriptstyle{\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]]
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| :<math>\hat{U}(t,t_0) = 1 - \frac{i}{\hbar}\int_{t_0}^t\hat{H}(t^\prime)\hat{U}(t^\prime,t_0)dt^\prime</math>,
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| which is equivalent to the [[Schrödinger equation]].
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| :<math>i\hbar\frac{\partial}{\partial t}\hat{U}(t,t_0) = \hat{H}(t)\hat{U}(t,t_0)</math>,
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| along with the initial condition <math>\scriptstyle{\hat{U}(t_0,t_0) = 1}</math>. Given knowledge of the system [[wave function]] at <math>\scriptstyle{t_0}</math>, the evolution of the system up to a later time <math>\scriptstyle{t}</math> can be obtained using
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| :<math>|\psi(t)\rangle = \hat{U}(t,t_0)|\psi(t_0)\rangle.</math>
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| The problem of determining the ''adiabaticity'' of a given process is equivalent to establishing the dependence of <math>\scriptstyle{\hat{U}(t_1,t_0)}</math> on <math>\scriptstyle{\tau}</math>.
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| 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>\scriptstyle{|0\rangle \equiv |\psi(t_0)\rangle}</math>, we have:
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| :<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>.
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| We can expand <math>\scriptstyle{\hat{U}(t_1,t_0)}</math>
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| :<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^\prime\int_{t_0}^{t^\prime}dt^{\prime\prime}\hat{H}(t^\prime)\hat{H}(t^{\prime\prime}) + \ldots</math>.
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| In the [[Perturbation theory|perturbative limit]] we can take just the first two terms and substitute them into our equation for <math>\scriptstyle{\zeta}</math>, recognizing that
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| :<math>{1 \over \tau}\int_{t_0}^{t_1}\hat{H}(t)dt \equiv \bar{H}</math>
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| is the system Hamiltonian, averaged over the interval <math>\scriptstyle{t_0 \rightarrow t_1}</math>, we have:
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| :<math>\zeta = \langle 0|(1 + \frac{i}{\hbar}\tau\bar{H})(1 - {i \over \hbar}\tau\bar{H})|0\rangle - \langle 0|(1 + {i \over \hbar}\tau\bar{H})|0\rangle\langle 0|(1 - {i \over \hbar}\tau\bar{H})|0\rangle</math>.
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| After expanding the products and making the appropriate cancellations, we are left with:
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| :<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>,
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| giving
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| :<math>\zeta = \frac{\tau^2\Delta\bar{H}^2}{\hbar^2}</math>,
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| where <math>\scriptstyle{\Delta\bar{H}}</math> is the [[root mean square]] deviation of the system Hamiltonian averaged over the interval of interest.
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| The sudden approximation is valid when <math>\scriptstyle{\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
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| :<math>\tau \ll {\hbar \over \Delta\bar{H}}</math>,
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| which is a statement of the [[Heisenberg uncertainty principle#Energy-time uncertainty principle|time-energy form of the Heisenberg uncertainty principle]].
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| === Diabatic passage ===
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| In the limit <math>\scriptstyle{\tau \rightarrow 0}</math> we have infinitely rapid, or diabatic passage:
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| :<math>\lim_{\tau \rightarrow 0}\hat{U}(t_1,t_0) = 1</math>.
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| The functional form of the system remains unchanged:
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| :<math>|\langle x|\psi(t_1)\rangle|^2 = |\langle x|\psi(t_0)\rangle|^2\quad</math>.
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| 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:
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| :<math>P_D = 1 - \zeta\quad</math>.
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| === Adiabatic passage ===
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| In the limit <math>\scriptstyle{\tau \rightarrow \infty}</math> we have infinitely slow, or adiabatic passage. The system evolves, adapting its form to the changing conditions,
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| :<math>|\langle x|\psi(t_1)\rangle|^2 \neq |\langle x|\psi(t_0)\rangle|^2</math>.
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| If the system is initially in an [[eigenstate]] of <math>\scriptstyle{\hat{H}(t_0)}</math>, after a period <math>\scriptstyle{\tau}</math> it will have passed into the ''corresponding'' eigenstate of <math>\scriptstyle{\hat{H}(t_1)}</math>.
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| 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:
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| :<math>P_A = \zeta\quad</math>.
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| == Calculating diabatic passage probabilities ==
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| === The Landau-Zener formula ===
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| {{main|Landau-Zener transition}}
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| 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=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> 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).
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| The key figure of merit in this approach is the Landau-Zener velocity:
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| :<math>v_{LZ} = {\frac{\partial}{\partial t}|E_2 - E_1| \over \frac{\partial}{\partial q}|E_2 - E_1|} \approx \frac{dq}{dt}</math>,
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| where <math>\scriptstyle{q}</math> is the perturbation variable (electric or magnetic field, molecular bond-length, or any other perturbation to the system), and <math>\scriptstyle{E_1}</math> and <math>\scriptstyle{E_2}</math> are the energies of the two diabatic (crossing) states. A large <math>\scriptstyle{v_{LZ}}</math> results in a large diabatic transition probability and vice versa.
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| Using the Landau-Zener formula the probability, <math>\scriptstyle{P_D}</math>, of a diabatic transition is given by
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| :<math>\begin{align}
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| P_D &= e^{-2\pi\Gamma}\\
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| \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|}\\
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| &= {a^2 \over \hbar|\alpha|}\\
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| \end{align}</math>
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| <!--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.
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| For a fully quantum–mechanical treatment of a general system, the equations of motion
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| for the coefficients, <math>\scriptstyle{c_1(t)}</math> and <math>\scriptstyle{c_2(t)}</math> of the diabatic states, <math>\scriptstyle{|1\rangle}</math> and <math>\scriptstyle{|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:
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| # The perturbation parameter is a known, linear function of time
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| # The energy separation of the diabatic states varies linearly with time
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| # The coupling <math>\scriptstyle{a}</math> in the diabatic Hamiltonian matrix is independent of time
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| 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.
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| The second simplification allows us to make the substitution <math>\scriptstyle{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.
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| The final simplification requires that the time–dependent perturbation does not
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| couple the diabatic states; rather, the coupling must be due to a static deviation from
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| a <math>\scriptstyle{1/r}</math> [[coulomb potential]], commonly described by a [[quantum defect]].
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| The details of Zener’s solution are somewhat opaque, relying on a set of substitutions
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| to put the equation of motion into the form of the Weber equation and using
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| 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]].-->
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| === The numerical approach ===
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| {{main|Numerical ordinary differential equations|l1=Numerical solution of ordinary differential equations}}
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| 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 variety of [[Numerical ordinary differential equations|numerical solution algorithms for ordinary differential equations]].
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| The equations to be solved can be obtained from the time-dependent Schrödinger equation:
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| :<math>i\hbar\dot{\underline{c}}^A(t) = \mathbf{H}_A(t)\underline{c}^A(t)</math>,
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| where <math>\scriptstyle{\underline{c}^A(t)}</math> is a [[Column vector|vector]] containing the adiabatic state amplitudes, <math>\scriptstyle{\mathbf{H}_A(t)}</math> is the time-dependent adiabatic Hamiltonian,<ref name="Stenholm" /> and the overdot represents a time-derivative.
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| 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:
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| :<math>P_D = |c^A_2(t_1)|^2\quad</math>
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| for a system that began with <math>\scriptstyle{|c^A_1(t_0)|^2 = 1}</math>.
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| == See also ==
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| * [[Landau–Zener formula]]
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| * [[Berry phase]]
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| * [[Quantum stirring, ratchets, and pumping]]
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| * [[Born–Oppenheimer approximation]]
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| == References ==
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| {{reflist|2}}
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| {{DEFAULTSORT:Adiabatic Theorem}}
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| [[Category:Theorems in quantum physics]]
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| [[ru:Адиабатическое приближение]]
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| [[uk:Адіабатичне наближення]]
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