values within the same framework by means of a numerical evaluation of the Whittaker functions. However, the Whittaker functions are not very well suited to large scale computations, because of the time required to evaluate each individual function. We choose instead to perform most of our calculations in a Fourier sine basis, which is numerically much faster, and use the Whittaker basis as a check on our results. Choosing the (orthonormalized) basis functions as, $$\label{eq:eq1}
\chi_{n\ell} = \sqrt{\frac{2}{r}} \sin(n\pi r)  
\frac{e^{i\ell\phi}}{\sqrt{2\pi}},$$ one can show, after a straightforward calculation, that the matrix elements of $\tilde H_{dc}$ are given by, $$\begin{aligned}
\label{eq:eq2}
(\tilde H_{dc})_{mn} &=& \Bigg\{ \frac{{\tilde\hbar}^2}{2}(n\pi)^2 + 
n\pi (\ell^2-\frac{1}{4}) 
{\tilde\hbar}^2{\rm Si}(2n\pi) + \frac{1}{2}\left
(\frac{{\tilde\omega_c}}{2}\right)^2 
\left(\frac{1}{3}-\frac{1}{2n^2\pi^2}\right) + \ell{\tilde\hbar}
\frac{{\tilde\omega_c}}{2} 
\Bigg\} ~ \delta_{mn} \nonumber \\
&+& \Bigg\{ \frac{\pi}{2}(\ell^2-\frac{1}{4}){\tilde\hbar}^2\left
\{ (m+n){\rm Si}
\left[(m+n)\pi\right] - (m-n){\rm Si}\left[(m-n)\pi\right]\right\} 
\Bigg . \nonumber \\
&+& \Bigg . \frac{1}{2}\left(\frac{{\tilde\omega_c}}{2}\right)^2 
\frac{(-)^{m+n}}{\pi^2} \frac{8mn}{(m^2-n^2)^2} \Bigg\}~ 
(1-\delta_{mn})\end{aligned}$$ where ${\rm Si}(x)$ is the Sine integral. One can similarly compute matrix elements of other needed operators.

Having worked out a suitable set of basis functions, we now proceed to tackle the full time-dependent problem. The Schrödinger equation for the time evolution operator is, $$\label{eq:n}
i {\tilde\hbar}\,\frac{\partial}{\partial\tau}\,U(\tau,\tau_0) = 
( \tilde H_{dc} + 
\tilde H_1(\tau) )\,
U(\tau,\tau_0).$$ 