2
$\begingroup$

How to solve the Schrödinger equation with time dependent potential in 1D or 3D (if it is easier):

$$i\hbar\dfrac{\partial \Psi}{\partial t}(x,t)=\left(-\dfrac{\hbar}{2m}\nabla^2-\frac{e^2}{x+\alpha}-exE(t)\right)\Psi(x,t)$$

where $E(t) = E_0 \exp(-t/\tau^2)sin(\omega_0 t)$ is a Gaussian pulse in time, $\alpha$ is a constant and $e$ is a constant (the electron charge). $\Psi(x,0)$ is hydrogen ground state.

What would it mean to find the solution in a self consistent manner?

$\endgroup$
  • 1
    $\begingroup$ This is a Stark effect case. In general there is no exact solution but perturbation theory is well-developed. $\endgroup$ – Urgje Feb 24 '16 at 10:55
4
$\begingroup$

I'm not sure exactly what problem you are trying to solve (or even if you are still trying to solve it...) but I have used the Crank-Nicolson method to solve something similar. Rather than using a ground-state potential explicitly you might be able to approximate it using a soft Coulomb potential (similar to the second term in the brackets in your question).

Taylor expanding the wavefunction to approximate the time evolution and comparing the result with a time propagator (given by $\hat{T}=\exp\{-(i/\hbar)\hat{H}(t-t_{0})\}$) allows for forward temporal evolution of the wavefunction in the potential to be estimated:

\begin{equation} |\psi(x, t_{0}+t)\rangle =\left[\hat{I}-\frac{i\Delta t}{\hbar}\left(-\frac{\hbar^{2}}{2m_{e}}\frac{\partial^{2}}{\partial x^{2}}+V(x,t)\right)\right]|\psi(x, t_{0})\rangle \end{equation}

$\hat{I}$ is the identity operator, and $\Delta t=(t-t_{0})$. To calculate numerically the second spatial derivative in the Schrödinger equation take the Taylor expansion of both $|\psi(x_{0}+x,t)\rangle$ and $|\psi(x_{0}-x,t)\rangle$ and add them. Rearrange the result for the estimated spatial derivative:

\begin{equation} \frac{\partial^{2}\psi}{\partial x^{2}} = \frac{|\psi(x_{0}+x,t)\rangle - 2|\psi(x,t)\rangle + |\psi(x_{0}-x, t)\rangle}{\Delta x^{2}} \end{equation}

where $\Delta x$ is the numerical spatial step size. You can substitute the spatial derivative in the Schrödinger equation for:

\begin{equation} \psi_{n+1}^{i} = \left[\hat{I}-\frac{i\Delta t}{\hbar}\left(-\frac{\hbar^{2}}{2m_{e}}\frac{\psi_{n}^{i+1} - 2\psi_{n}^{i} + \psi_{n}^{i-1}}{\Delta x^{2}}+V(x,t)\right)\right]\psi_{n}^{i}. \end{equation}

This is the "forward time central step" part of the Crank-Nicolson algorithm, i.e., the forward Euler method (I have simplified the notation so that $|\psi(x_{i},t_{n})\rangle = \psi_{n}^{i}$ here to prevent line wrapping). To get the "backward time central step" term, simply change the sign after $\hat{I}$, which is equivalent to reversing the time propagator (hence "backward time"). The Crank-Nicolson solution to the time-dependent Schrödinger equation is then found by solving:

\begin{equation} \psi_{n+1}^{i} = \left(\hat{I} + \frac{i\Delta t}{2\hbar}\hat{H}\right)^{-1}\left(\hat{I}-\frac{i\Delta t}{2\hbar}\hat{H}\right)\psi_{n}^{i}. \end{equation}

It is a good idea to use sparse matrices if you do intend to solve this numerically as you need a lot of grid points in the time domain and a large spatial window.

This can be quite a cool toy for modelling any time-dependent wavefunction motion for reasonably simple wavefunction evolution, e.g., free-electron wavepacket motion in DC and AC fields, but it can also be used for highly complex light-matter interactions such as high-harmonic generation. For example, the figure below shows a simple single atom model of high-harmonic generation by a very high intensity laser pulse (white line) which is contributing enormously to an electronic potential given by $V(x, t)=V_{\text{soft Coulomb}}(x,t) + V_{\rm{E-Field}}(x,t)$. The colourmap is $\text{log}_{10}(|\psi|^{2})$.

enter image description here

$\endgroup$
0
$\begingroup$

this is not an easy topic to discus, since it would take some writing-up.. you can find in Google the detailed methods. $1D$ is simpler. You can use the split operator time-propagator, where the $x$ and $d/dx$-alike operators are treated differently. When applying the derivative operators, usually one makes use of the Fourier Transform, for which very good software packages are available as open source (FFTW - the Fastest Fourier Transform in the West). Not sure there is any meaning to "consistent manner" in this context . You need to specify the initial wave-function though.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.