Editing Adiabatic theorem (section)

== 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>.