Editing Adiabatic theorem (section)

=== 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>.