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Next: Example 2: Relaxation method. Up: The Chebyshev method Previous: The Chebyshev method

Example 1: Propagation operator.

The operator $ f\left( \widehat{H'}\right) $ to approximate is $ U=e^{-\frac{i}{\hbar }\widehat{H'}t} $ with the maximal and minimal eigenvectors $ \lambda _{min}=E_{min} $, $ \lambda _{max}=E_{max} $, $ \Delta E=E_{max}-E_{min} $.

This means the factor $ \Phi =e^{\frac{i}{\hbar }\left( \Delta E/2+E_{min}\right) t} $ is a phase factor, $ f(z')=e^{-\frac{i}{\hbar }z't} $ and hence

$\displaystyle b_{n}=(-i)^{n}2J_{n}\left( \alpha \right) ,\, \alpha =\frac{\Delta Et}{2\hbar }$

with $ J_{n} $ the Bessel function of first kind of order $ n $ (the proof for this expression is to be found in in appendix 4.2).typeset@protect @@footnote SF@gobble@opt These formulas were occasionally faulty in some of the papers I am citing.

The coefficients go to zero exponentially as $ n $ becomes greater than $ \alpha $ so the method is very fast except for very short times where this exponentially decaying tail becomes dominant in the calculation time.

The same result is produced by

$\displaystyle f(z')=\frac{a_{0}}{2}+\sum ^{\infty }_{n=1}a_{n}\chi _{n}(z')$

with the recursion
$\displaystyle \chi _{0}$ $\displaystyle =$ $\displaystyle \psi ,$  
$\displaystyle \chi _{1}$ $\displaystyle =$ $\displaystyle -i\widehat{H'}\psi ,$  
$\displaystyle \chi _{n+1}$ $\displaystyle =$ $\displaystyle -2i\widehat{H'}\chi _{n}+\chi _{n-1}$  

and the coefficients

$\displaystyle a_{n}=2J_{n}\left( \alpha \right) .$

This can be shown by proving inductively that $ \chi _{n}=(-i)^{n}\phi _{n} $, as obviously $ b_{n}=(-i)^{n}a_{n} $:
$\displaystyle \chi _{n+1}$ $\displaystyle =$ $\displaystyle -2i\widehat{H'}\chi _{n}+\chi _{n-1}$  
  $\displaystyle =$ $\displaystyle (-i)^{n+1}2\widehat{H'}\phi _{n}+(-i)^{n-1}\phi _{n-1}$  
  $\displaystyle =$ $\displaystyle (-i)^{n+1}\left( 2\widehat{H'}\phi _{n}-\phi _{n-1}\right)$  
  $\displaystyle =$ $\displaystyle (-i)^{n+1}\phi _{n+1}.$  


next up previous
Next: Example 2: Relaxation method. Up: The Chebyshev method Previous: The Chebyshev method
Andreas Markmann 2003-10-22