5.2.2. Length gauge
\[V_I(\mathbf{r}, t) = \mathcal{E}(t) \vec{u} \cdot \mathbf{r}\]
Using the integrals over position operators given in (see Spherical harmonics) we have that
\[\begin{split}i \dot{u}_{l^\prime,m^\prime}(r,t) = \left( -\frac{1}{2} \frac{d^2 u_{l^\prime,m^\prime}(r,t)}{dr^2} + V_{l^\prime}(r)u_{l^\prime,m^\prime}(r,t) \right)
+ r \mathcal{E}(t)
\begin{cases}
\frac{1}{2}\sum_{l, m} (b_{l-1,-m-1}\delta_{l^\prime, l-1}\delta_{m^\prime, m+1}
- b_{l,m}\delta_{l^\prime, l+1}\delta_{m^\prime, m+1}
- b_{l-1,m-1}\delta_{l^\prime, l-1}\delta_{m^\prime, m-1}
+ b_{l,-m}\delta_{l^\prime, l+1}\delta_{m^\prime, m-1}) u_{l,m}(r,t), \text{ if } \vec{u} = \vec{e}_x \\
\frac{i}{2}\sum_{l,m} (b_{l-1,-m-1}\delta_{l^\prime, l-1}\delta_{m^\prime, m+1}
- b_{l,m}\delta_{l^\prime, l+1}\delta_{m^\prime, m+1}
+ b_{l-1,m-1}\delta_{l^\prime, l-1}\delta_{m^\prime, m-1}
- b_{l,-m}\delta_{l^\prime, l+1}\delta_{m^\prime, m-1}) u_{l,m}(r,t), \text{ if } \vec{u} = \vec{e}_y \\
\sum_{l,m} (a_{l,m}\delta_{l^\prime, l+1}\delta_{m^\prime, m}
+ a_{l-1,m}\delta_{l^\prime, l-1}\delta_{m^\prime, m}) u_{l,m}(r,t), \text{ if } \vec{u} = \vec{e}_z
\end{cases}\end{split}\]
Similar to the approach we used for the TISE (see The time-independent Schrödinger equation), we introduce the
new wavefunction \(\phi_{l,m}(x) = \dot{r}(x)^{1/2}u_{l,m}(r(x))\) and discretrize the EOMs with the pseudospectral method, which yields
\[\begin{split}i \dot{\phi}_{l^\prime, m^\prime}(x_i) =
\sum_{j=1}^{N-1} \left(-\frac{1}{2} \frac{\tilde{g}^{\prime \prime}_j(x_i)}{\dot{r}(x_i) \dot{r}(x_j)} \tilde{\phi}_{l^\prime,m^\prime}(x_j) \right)
+ V_{l^\prime}(r(x_i))\tilde{\phi}_{l^\prime,m^\prime}(x_i)
+ r(x_i) \mathcal{E}(t)
\begin{cases}
\frac{1}{2}\sum_{l, m} (b_{l-1,-m-1}\delta_{l^\prime, l-1}\delta_{m^\prime, m+1}
- b_{l,m}\delta_{l^\prime, l+1}\delta_{m^\prime, m+1}
- b_{l-1,m-1}\delta_{l^\prime, l-1}\delta_{m^\prime, m-1}
+ b_{l,-m}\delta_{l^\prime, l+1}\delta_{m^\prime, m-1}) \tilde{\phi}_{l,m}(x_i), \text{ if } \vec{u} = \vec{e}_x \\
\frac{i}{2}\sum_{l,m} (b_{l-1,-m-1}\delta_{l^\prime, l-1}\delta_{m^\prime, m+1}
- b_{l,m}\delta_{l^\prime, l+1}\delta_{m^\prime, m+1}
+ b_{l-1,m-1}\delta_{l^\prime, l-1}\delta_{m^\prime, m-1}
- b_{l,-m}\delta_{l^\prime, l+1}\delta_{m^\prime, m-1}) \tilde{\phi}_{l,m}(x_i), \text{ if } \vec{u} = \vec{e}_y \\
\sum_{l,m} (a_{l,m}\delta_{l^\prime, l+1}\delta_{m^\prime, m}
+ a_{l-1,m}\delta_{l^\prime, l-1}\delta_{m^\prime, m}) \tilde{\phi}_{l,m}(x_i), \text{ if } \vec{u} = \vec{e}_z
\end{cases}\end{split}\]