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Propagation methods [2]

The problem is to calculate the propagation operator

$\displaystyle \widehat{U}(t_{e},t_{s})=\widehat{T}e^{-\frac{i}{\hbar }\int\limits _{t_{s}}^{t_{e}}\widehat{H}(t')dt'}$

with the time order operator $ \widehat{T} $ which solves the Schroedinger equation. If $ \widehat{H} $ is time dependent, one can extrapololate by dividing the considered time period into smaller sections over which $ \widehat{H} $ is assumed to be constant. Then the propagation operator assumes the form

$\displaystyle \widehat{U}(t_{e},t_{s})=e^{-\frac{i}{\hbar }\widehat{H}_{s}(t_{e}-t_{s})}$

with different $ \widehat{H}_{s} $ dependent on $ t_{s} $. This extrapolation makes use of the relation $ \widehat{U}(t_{1}+t_{2},t_{0})=\widehat{U}(t_{2},t_{1})\widehat{U}(t_{1},t_{0}). $

A function f applied to an operator $ \widehat{O} $ can be spectrally decomposed to give

$\displaystyle f\left( \widehat{O}\right) =\sum _{n}f(\lambda _{n})\widehat{P}_{n},$

where $ \lambda _{n} $ are the eigenvalues and $ \widehat{P}_{n}=\left\vert u_{n}\right\rangle \left\langle u_{n}\right\vert $ are the projection operators to the eigenspaces with

$\displaystyle \widehat{P}_{n}\widehat{P}_{m}=\delta _{mn}\widehat{P}_{n}.$

If the eigenvalues are known, these projection operators can be formed by successively applying operators of the form $ \widehat{Q}_{i}=\left( \widehat{O}-\lambda _{i}\widehat{I}\right) $, where $ \widehat{I} $ denotes unity:

$\displaystyle \widehat{P}_{n}=\frac{1}{\aleph }\widehat{Q}_{N}\cdots \widehat{Q}_{n+1}\widehat{Q}_{n-1}\cdots \widehat{Q}_{1}$

with the normalization factor

$\displaystyle \aleph =\left( \lambda _{n}-\lambda _{N}\right) \cdots \left( \la...
...bda _{n}-\lambda _{n-1}\right) \cdots \left( \lambda _{n}-\lambda _{1}\right) .$

However, solving the eigenvalue problem is computationally prohibitively expensive $ \left( O\left( N^{2}\right) \right) $ and the number of eigenvalues may be infinite, so approximate methods have to be found to confine the problem to finding a small subset of the set of eigenvalues. Applying the above formulae directly leads to the Lagrangian interpolation of the operator (as the operators $ \widehat{Q}_{i} $ are evaluated recursively) which has numerical disadvantages because adding another sampling point results in the necessity of calculating a completely new interpolation polynomial.



Subsections
next up previous
Next: Newtonian Interpolation Up: Propagation Methods Previous: Comparison
Andreas Markmann 2003-10-22