The time dependent Schroedinger equation

The development of the algorithm for the Schroedinger equation will parallel that for the Euler and Navier-Stokes equations. The Wavelet-Galerkin solution of the Euler and Navier-Stokes equations with periodic boundary conditions is presented in reference [47,58].

For complex wave function $\psi$ and real potential $U$ the Schroedinger equation is:

$$\imath \psi_t = \Delta \psi + U\psi$$

We use the implicit time differencing

$$\imath (\psi_{n+1}-\psi_n)/dt = .5\Delta(\psi_{n+1}+\psi_n) + .5(\psi_{n+1}+\psi_n)U,$$

which conserves the energy of the wave function, $\int\psi\overline{\psi}$. The implicit time differencing is solved for $\psi_{n+1}$ by iteration

$$\imath (f_{k+1}-\psi_n)/dt = .5\Delta(f_{k+1}+\psi_n) + .5(f_{k}+\psi_n)U,$$

until $\sup\vert f_{k+1}-f_{k}\vert < \epsilon$ where $f_0=\psi_n$.

In applying the Wavelet-Galerkin method to the two dimensional Schroedinger equation we expand and approximate the field $\psi$ and the potential in terms of the scaling function $\varphi$,

$$\psi(x,y) = \sum_{j=1}^N\sum_{k=1}^N\Psi_{j,k}\varphi(x-j)\varphi(y-k),$$

$$U(x,y) = \sum_{j=1}^N\sum_{k=1}^N U_{j,k}\varphi(x-j)\varphi(y-k).$$

Since we assume periodic boundary conditions there is a periodic wrap around in $(x,y)$ and we let the period scale with the number of terms in the expansion. This allows neglect of the dilation factor in the scaling function. Substituting into the equation and projecting the result onto the subspace spanned by $\{\varphi(x-j)\varphi(y-k): \;\; j=1,\cdots,N ; \; k=1,\cdots,N\}$ requires evaluating terms of the form

$$\int\varphi_{xx}(x)\varphi(x-k)dx.$$

This uniquely determines the $\Psi_{j,k}$ as solutions of the Wavelet-Galerkin ordinary differential equations.

Define the connection coefficients

$$\Omega_j^{02} = \int\varphi(x)\varphi_{xx}(x-j)dx$$

$$\Omega_{jl}^{000} = \int\varphi(x)\varphi(x-j)\varphi(x-l)dx$$

With the summation convention on the indices $(p,q,j,l,k,m)$ the Wavelet-Galerkin equations are

$$\imath \Psi_t(p,q) = U(j+p,k+q)\Psi(l+p,m+q)\Omega_{j,l}^{000}\Omega_{k,m}^{000} +$$

$$\left(\Omega_j^{02}\Psi(j+p,q)+\Omega_k^{02}\Psi(p,k+q)\right).$$