Imaginary time method to find the ground state wave function
Here is a numerical method for solving the ground state wave function. If the potential energy in the Schrodinger equation does not contain time, the result of solving the Schrodinger equation by the method of separating variables is
\begin{equation}
\Psi( \boldsymbol{\mathbf{x}} , t) = \sum_i \psi_i( \boldsymbol{\mathbf{x}} ) \mathrm{e} ^{- \mathrm{i} E_i t}
\end{equation}
Among them, $\psi_i( \boldsymbol{\mathbf{x}} )$ is the energy eigenstate with energy $E_i$.
Now if the ground state is required, we can use imaginary time, namely $t' = - \mathrm{i} t$, to make the time-dependent wave function become
\begin{equation}
\Psi( \boldsymbol{\mathbf{x}} , t) = \sum_i \psi_i( \boldsymbol{\mathbf{x}} ) \mathrm{e} ^{- E_i t}
\end{equation}
In this way, the excited state decays faster than the ground state. When $t \to +\infty$, only the ground state wave function is left.
Suppose we have a numerical method for solving TDSE, then we only need to use it to solve the equation
\begin{equation}
\boldsymbol{\mathbf{H}} \Psi = - \frac{\partial \Psi}{\partial t}
\end{equation}
Then normalize the wave function in each cycle. Because the separated variable solution of this equation is
eq. 2 .