adiabatic theorem

{{See also|Rayleigh–Lorentz pendulum}}

At the 1911 Solvay conference, Einstein gave a lecture on the quantum hypothesis, which states that $E\; =\; nh\; \backslash nu$ 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 $E$ and the frequency $\backslash nu$ would change, their ratio $\backslash frac\{E\}\{\backslash nu\}$ would still be conserved, thus saving the quantum hypothesis.{{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}}

Before the conference, Einstein had just read a paper by Paul Ehrenfest on the adiabatic hypothesis.EHRENFEST, P. (1911): ``Welche Züge der Lichtquantenhypothese spielen in der Theorie der Wärmestrahlung eine wesentliche Rolle?'' Annalen der Physik 36, pp. 91–118. Reprinted in KLEIN (1959), pp. 185–212. We know that he had read it because he mentioned it in a letter to Michele Besso written before the conference.{{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}}{{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}}

At some initial time $t\_0$ a quantum-mechanical system has an energy given by the Hamiltonian $\backslash hat\{H\}(t\_0)$; the system is in an eigenstate of $\backslash hat\{H\}(t\_0)$ labelled $\backslash psi(x,t\_0)$. Changing conditions modify the Hamiltonian in a continuous manner, resulting in a final Hamiltonian $\backslash hat\{H\}(t\_1)$ at some later time $t\_1$. The system will evolve according to the time-dependent Schrödinger equation, to reach a final state $\backslash psi(x,t\_1)$. The adiabatic theorem states that the modification to the system depends critically on the time $\backslash tau\; =\; t\_1\; -\; t\_0$ during which the modification takes place.

For a truly adiabatic process we require $\backslash tau\; \backslash to\; \backslash infty$; in this case the final state $\backslash psi(x,t\_1)$ will be an eigenstate of the final Hamiltonian $\backslash hat\{H\}(t\_1)$, with a modified configuration:

:$|\backslash psi(x,t\_1)|^2\; \backslash neq\; |\backslash psi(x,t\_0)|^2\; .$

The degree to which a given change approximates an adiabatic process depends on both the energy separation between $\backslash psi(x,t\_0)$ and adjacent states, and the ratio of the interval $\backslash tau$ to the characteristic timescale of the evolution of $\backslash psi(x,t\_0)$ for a time-independent Hamiltonian, $\backslash tau\_\backslash text\{int\}\; =\; 2\backslash pi\backslash hbar/E\_0$, where $E\_0$ is the energy of $\backslash psi(x,t\_0)$.

Conversely, in the limit $\backslash tau\; \backslash to\; 0$ we have infinitely rapid, or diabatic passage; the configuration of the state remains unchanged:

:$|\backslash psi(x,t\_1)|^2\; =\; |\backslash psi(x,t\_0)|^2\; .$

The so-called "gap condition" included in Born and Fock's original definition given above refers to a requirement that the spectrum of $\backslash hat\{H\}$ is discrete and nondegenerate, such that there is no ambiguity in the ordering of the states (one can easily establish which eigenstate of $\backslash hat\{H\}(t\_1)$ corresponds to $\backslash psi(t\_0)$). In 1999 J. E. Avron and A. Elgart reformulated the adiabatic theorem to adapt it to situations without a gap.{{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}}

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 and quantum mechanics definition{{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 }} 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.{{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}}

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.

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 page and here.{{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}}

Image:HO adiabatic process.gif

The 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 $k$ 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.

If $k$ is increased adiabatically $\backslash left(\backslash frac\{dk\}\{dt\}\; \backslash to\; 0\backslash right)$ then the system at time $t$ will be in an instantaneous eigenstate $\backslash psi(t)$ of the current Hamiltonian $\backslash hat\{H\}(t)$, corresponding to the initial eigenstate of $\backslash hat\{H\}(0)$. 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, $n\; =\; 0$, 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 $\backslash left(\backslash frac\{dk\}\{dt\}\; \backslash to\; \backslash infty\backslash right)$ 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 $\backslash left(|\backslash psi(t)|^2\; =\; |\backslash psi(0)|^2\backslash right)$ for a process occurring over a vanishing time period, there is no eigenstate of the new Hamiltonian, $\backslash hat\{H\}(t)$, that resembles the initial state. The final state is composed of a linear superposition of many different eigenstates of $\backslash hat\{H\}(t)$ which sum to reproduce the form of the initial state.

{{main|Avoided crossing}}

File:Avoided_crossing_in_linear_field.svg of the Hamiltonian, giving the energies of the eigenstates $|\backslash phi\_1\backslash rangle$ and $|\backslash phi\_2\backslash rangle$ (the adiabatic states). (Actually, $|\backslash phi\_1\backslash rangle$ and $|\backslash phi\_2\backslash rangle$ should be switched in this picture.)]]

For a more widely applicable example, consider a 2-level atom subjected to an external magnetic field.{{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}} The states, labelled $|1\backslash rangle$ and $|2\backslash rangle$ using bra–ket notation, can be thought of as atomic 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:

:$|\backslash Psi\backslash rangle\; =\; c\_1(t)|1\backslash rangle\; +\; c\_2(t)|2\backslash rangle.$

With the field absent, the energetic separation of the diabatic states is equal to $\backslash hbar\backslash omega\_0$; the energy of state $|1\backslash rangle$ increases with increasing magnetic field (a low-field-seeking state), while the energy of state $|2\backslash rangle$ 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

:$\backslash mathbf\{H\}\; =\; \backslash begin\{pmatrix\}$

\mu B(t)-\hbar\omega_0/2 & a \\

a^* & \hbar\omega_0/2-\mu B(t)

\end{pmatrix}

where $\backslash mu$ is the magnetic moment of the atom, assumed to be the same for the two diabatic states, and $a$ is some time-independent coupling between the two states. The diagonal elements are the energies of the diabatic states ($E\_1(t)$ and $E\_2(t)$), however, as $\backslash mathbf\{H\}$ is not a diagonal matrix, it is clear that these states are not eigenstates of $\backslash mathbf\{H\}$ due to the off-diagonal coupling constant.

The eigenvectors of the matrix $\backslash mathbf\{H\}$ are the eigenstates of the system, which we will label $|\backslash phi\_1(t)\backslash rangle$ and $|\backslash phi\_2(t)\backslash rangle$, with corresponding eigenvalues

$$\backslash begin\{align\}$$

\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}

It is important to realise that the eigenvalues $\backslash varepsilon\_1(t)$ and $\backslash varepsilon\_2(t)$ are the only allowed outputs for any individual measurement of the system energy, whereas the diabatic energies $E\_1(t)$ and $E\_2(t)$ correspond to the expectation values for the energy of the system in the diabatic states $|1\backslash rangle$ and $|2\backslash rangle$.

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, and thus we have an avoided crossing. If an atom is initially in state $|\backslash phi\_2(t\_0)\backslash rangle$ in zero magnetic field (on the red curve, at the extreme left), an adiabatic increase in magnetic field $\backslash left(\backslash frac\{dB\}\{dt\}\; \backslash to\; 0\backslash right)$ will ensure the system remains in an eigenstate of the Hamiltonian $|\backslash phi\_2(t)\backslash rangle$ throughout the process (follows the red curve). A diabatic increase in magnetic field $\backslash left(\backslash frac\{dB\}\{dt\}\backslash to\; \backslash infty\backslash right)$ will ensure the system follows the diabatic path (the dotted blue line), such that the system undergoes a transition to state $|\backslash phi\_1(t\_1)\backslash rangle$. For finite magnetic field slew rates there will be a finite probability of finding the system in either of the two eigenstates. See below for approaches to calculating these probabilities.

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

Under a slowly changing Hamiltonian $H(t)$ with instantaneous eigenstates $|\; n(t)\; \backslash rangle$ and corresponding energies $E\_n(t)$, a quantum system evolves from the initial state

$$|\; \backslash psi(0)\; \backslash rangle\; =\; \backslash sum\_n\; c\_n(0)\; |\; n(0)\; \backslash rangle$$

to the final state

$$|\; \backslash psi(t)\; \backslash rangle\; =\; \backslash sum\_n\; c\_n(t)\; |\; n(t)\; \backslash rangle\; ,$$

where the coefficients undergo the change of phase

$$c\_n(t)\; =\; c\_n(0)\; e^\{i\; \backslash theta\_n(t)\}\; e^\{i\; \backslash gamma\_n(t)\}$$

with the dynamical phase

$$\backslash theta\_m(t)\; =\; -\backslash frac\{1\}\{\backslash hbar\}\; \backslash int\_0^t\; E\_m(t\text{'})\; dt\text{'}$$

and geometric phase

$$\backslash gamma\_m(t)\; =\; i\; \backslash int\_0^t\; \backslash langle\; m(t\text{'})\; |\; \backslash dot\{m\}(t\text{'})\; \backslash rangle\; dt\text{'}\; .$$

In particular, $|c\_n(t)|^2\; =\; |c\_n(0)|^2$, so if the system begins in an eigenstate of $H(0)$, it remains in an eigenstate of $H(t)$ during the evolution with a change of phase only.

:

:

:

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.{{cite book |author=Bottani |first=Carlo E. |title=Solid State Physics Lecture Notes |year=2017–2018 |pages=64–67}}

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 conductors, the temperature dependence of electric conductivity in insulators, Some properties of low temperature superconductivity
• optics: optic absorption in the infrared for ionic crystals, Brillouin scattering, Raman scattering

{{Disputed section|errors in the technical section|date = January 2016}}

We will now pursue a more rigorous analysis.{{cite book |last=Messiah |first=Albert |title=Quantum Mechanics |year=1999 |publisher=Dover Publications |isbn=0-486-40924-4 |chapter=XVII }} Making use of bra–ket notation, the state vector of the system at time $t$ can be written

:$|\backslash psi(t)\backslash rangle\; =\; \backslash sum\_n\; c^A\_n(t)e^\{-iE\_nt/\backslash hbar\}|\backslash phi\_n\backslash rangle\; ,$

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

:$\backslash psi(x,t)\; =\; \backslash langle\; x|\backslash psi(t)\backslash rangle\; .$

It is instructive to examine the limiting cases, in which $\backslash tau$ 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 $\backslash hat\{H\}\_0$, at time $t\_0$, to a final value $\backslash hat\{H\}\_1$, at time $t\_1$, where $\backslash tau\; =\; t\_1\; -\; t\_0$. The evolution of the system can be described in the Schrödinger picture by the time-evolution operator, defined by the integral equation

:$\backslash hat\{U\}(t,t\_0)\; =\; 1\; -\; \backslash frac\{i\}\{\backslash hbar\}\backslash int\_\{t\_0\}^t\backslash hat\{H\}(t\text{'})\backslash hat\{U\}(t\text{'},t\_0)dt\text{'}\; ,$

which is equivalent to the Schrödinger equation.

:$i\backslash hbar\backslash frac\{\backslash partial\}\{\backslash partial\; t\}\backslash hat\{U\}(t,t\_0)\; =\; \backslash hat\{H\}(t)\backslash hat\{U\}(t,t\_0),$

along with the initial condition $\backslash hat\{U\}(t\_0,t\_0)\; =\; 1$. Given knowledge of the system wave function at $t\_0$, the evolution of the system up to a later time $t$ can be obtained using

:$|\backslash psi(t)\backslash rangle\; =\; \backslash hat\{U\}(t,t\_0)|\backslash psi(t\_0)\backslash rangle.$

The problem of determining the adiabaticity of a given process is equivalent to establishing the dependence of $\backslash hat\{U\}(t\_1,t\_0)$ on $\backslash tau$.

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 $|0\backslash rangle\; \backslash equiv\; |\backslash psi(t\_0)\backslash rangle$, we have:

:$\backslash zeta\; =\; \backslash langle\; 0|\backslash hat\{U\}^\backslash dagger(t\_1,t\_0)\backslash hat\{U\}(t\_1,t\_0)|0\backslash rangle\; -\; \backslash langle\; 0|\backslash hat\{U\}^\backslash dagger(t\_1,t\_0)|0\backslash rangle\backslash langle\; 0\; |\; \backslash hat\{U\}(t\_1,t\_0)\; |\; 0\; \backslash rangle.$

We can expand $\backslash hat\{U\}(t\_1,t\_0)$

:$\backslash hat\{U\}(t\_1,t\_0)\; =\; 1\; +\; \{1\; \backslash over\; i\backslash hbar\}\; \backslash int\_\{t\_0\}^\{t\_1\}\backslash hat\{H\}(t)dt\; +\; \{1\; \backslash over\; (i\backslash hbar)^2\}\; \backslash int\_\{t\_0\}^\{t\_1\}dt\text{'}\; \backslash int\_\{t\_0\}^\{t\text{'}\}dt$ \hat{H}(t')\hat{H}(t) + \cdots.

In the perturbative limit we can take just the first two terms and substitute them into our equation for $\backslash zeta$, recognizing that

:$\{1\; \backslash over\; \backslash tau\}\backslash int\_\{t\_0\}^\{t\_1\}\backslash hat\{H\}(t)dt\; \backslash equiv\; \backslash bar\{H\}$

is the system Hamiltonian, averaged over the interval $t\_0\; \backslash to\; t\_1$, we have:

:$\backslash zeta\; =\; \backslash langle\; 0|(1\; +\; \backslash tfrac\{i\}\{\backslash hbar\}\backslash tau\backslash bar\{H\})(1\; -\; \backslash tfrac\{i\}\{\backslash hbar\}\backslash tau\backslash bar\{H\})|0\backslash rangle\; -\; \backslash langle\; 0|(1\; +\; \backslash tfrac\{i\}\{\backslash hbar\}\backslash tau\backslash bar\{H\})|0\backslash rangle\; \backslash langle\; 0|(1\; -\; \backslash tfrac\{i\}\{\backslash hbar\}\backslash tau\backslash bar\{H\})|0\backslash rangle\; .$

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

:$\backslash zeta\; =\; \backslash frac\{\backslash tau^2\}\{\backslash hbar^2\}\backslash left(\backslash langle\; 0|\backslash bar\{H\}^2|0\backslash rangle\; -\; \backslash langle\; 0|\backslash bar\{H\}|0\backslash rangle\backslash langle\; 0|\backslash bar\{H\}|0\backslash rangle\backslash right)\; ,$

giving

:$\backslash zeta\; =\; \backslash frac\{\backslash tau^2\backslash Delta\backslash bar\{H\}^2\}\{\backslash hbar^2\}\; ,$

where $\backslash Delta\backslash bar\{H\}$ is the root mean square deviation of the system Hamiltonian averaged over the interval of interest.

The sudden approximation is valid when $\backslash zeta\; \backslash ll\; 1$ (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

:$\backslash tau\; \backslash ll\; \{\backslash hbar\; \backslash over\; \backslash Delta\backslash bar\{H\}\}\; ,$

which is a statement of the time-energy form of the Heisenberg uncertainty principle.

In the limit $\backslash tau\; \backslash to\; 0$ we have infinitely rapid, or diabatic passage:

:$\backslash lim\_\{\backslash tau\; \backslash to\; 0\}\backslash hat\{U\}(t\_1,t\_0)\; =\; 1\; .$

The functional form of the system remains unchanged:

:$|\backslash langle\; x|\backslash psi(t\_1)\backslash rangle|^2\; =\; \backslash left|\backslash langle\; x|\backslash psi(t\_0)\backslash rangle\backslash right|^2\; .$

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:

:$P\_D\; =\; 1\; -\; \backslash zeta.$

In the limit $\backslash tau\; \backslash to\; \backslash infty$ we have infinitely slow, or adiabatic passage. The system evolves, adapting its form to the changing conditions,

:$|\backslash langle\; x|\backslash psi(t\_1)\backslash rangle|^2\; \backslash neq\; |\backslash langle\; x|\backslash psi(t\_0)\backslash rangle|^2\; .$

If the system is initially in an eigenstate of $\backslash hat\{H\}(t\_0)$, after a period $\backslash tau$ it will have passed into the corresponding eigenstate of $\backslash hat\{H\}(t\_1)$.

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:

:$P\_A\; =\; \backslash zeta\; .$

{{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,{{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}} 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:

$$v\_\backslash text\{LZ\}\; =\; \{\backslash frac\{\backslash partial\}\{\backslash partial\; t\}|E\_2\; -\; E\_1|\; \backslash over\; \backslash frac\{\backslash partial\}\{\backslash partial\; q\}|E\_2\; -\; E\_1|\}\; \backslash approx\; \backslash frac\{dq\}\{dt\}\; ,$$

where $q$ is the perturbation variable (electric or magnetic field, molecular bond-length, or any other perturbation to the system), and $E\_1$ and $E\_2$ are the energies of the two diabatic (crossing) states. A large $v\_\backslash text\{LZ\}$ results in a large diabatic transition probability and vice versa.

Using the Landau–Zener formula the probability, $P\_\{\backslash rm\; D\}$, of a diabatic transition is given by

$$\backslash begin\{align\}$$

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}

{{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 solution algorithms for ordinary differential equations.

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

$$i\backslash hbar\backslash dot\{\backslash underline\{c\}\}^A(t)\; =\; \backslash mathbf\{H\}\_A(t)\backslash underline\{c\}^A(t)\; ,$$

where $\backslash underline\{c\}^A(t)$ is a vector containing the adiabatic state amplitudes, $\backslash mathbf\{H\}\_A(t)$ is the time-dependent adiabatic Hamiltonian, 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:

$$P\_D\; =\; |c^A\_2(t\_1)|^2$$

for a system that began with $|c^A\_1(t\_0)|^2\; =\; 1$.

• Landau–Zener formula
• Berry phase
• Quantum stirring, ratchets, and pumping
• Adiabatic quantum motor
• Born–Oppenheimer approximation
• Eigenstate thermalization hypothesis
• Adiabatic process

{{reflist|2}}

Category:Theorems in quantum mechanics