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

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

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

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

with the recursion

$$\chi _{0} = \psi ,$$

$$\chi _{1} = -i\widehat{H'}\psi ,$$

$$\chi _{n+1} = -2i\widehat{H'}\chi _{n}+\chi _{n-1}$$

and the coefficients

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

This can be shown by proving inductively that $\chi _{n}=(-i)^{n}\phi _{n}$, as obviously $b_{n}=(-i)^{n}a_{n}$:

$$\chi _{n+1} = -2i\widehat{H'}\chi _{n}+\chi _{n-1}$$

$$= (-i)^{n+1}2\widehat{H'}\phi _{n}+(-i)^{n-1}\phi _{n-1}$$

$$= (-i)^{n+1}\left( 2\widehat{H'}\phi _{n}-\phi _{n-1}\right)$$

$$= (-i)^{n+1}\phi _{n+1}.$$