The time-evolution operator

The Schrödinger equation (81) can be integrated in a formal sense to obtain:

$$\psi(x,t)=U(t)\psi(x,t=0)=e^{-iHt}\psi(x,t=0).$$ (82)

From here we deduce that the wave function can be evolved forward in time by applying the time-evolution operator $U(t)=\exp{(-iHt)}$:

$$\psi(t+\Delta t)= e^{-iH\Delta t}\psi(t).$$

Likewise, the inverse of the time-evolution operator moves the wave function back in time:

$$\psi(t-\Delta t)=e^{iH\Delta t}\psi(t),$$

where we have use the property

$$U^{-1}(t)=U(-t).$$

Although it would be nice to have an algorithm based on the direct application of $U$, it has been shown that this is not stable. Hence, we apply the following relation:

$$\psi(t+\Delta t)=\psi(t-\Delta t)+\left[e^{-iH\Delta t}-e^{iH\Delta t}\right]\psi(t).$$

Now, the derivatives with recpect to $x$ can be approximated by

$$\frac{\partial \psi}{\partial t} \sim \frac{\psi(x,t+\Delta t)-\psi(x, \Delta t)}{\Delta t},$$ (83)

$$\frac{\partial ^2 \psi}{\partial x^2} \sim \frac{\psi(x+\Delta % x,t)+\psi(x-\Delta x,t)-2\psi(x,t)}{(\Delta x)^2}.$$ (84)

The time evolution operator is approximated by:

$$U(\Delta t)=e^{-iH\Delta t} \sim 1+iH\Delta t,$$ (85)

with an error of the order of $(\Delta t)^2$. Replacing the expression (82) for $H$, we obtain:

$$\psi(x,t+\Delta t)= \psi(x,t-\Delta t)-2i[(4\alpha+\Delta t V(x))\psi(x,t)-$$ (86)

$$\alpha(\psi(x+\Delta x,t)+\psi(x-\Delta x,t))],$$ (87)

with $\alpha=\frac{\Delta t}{2(\Delta x)^2}$.

The probability of finding an electron at $(x,t)$ is given by $\vert\psi(x,t)\vert^2$. This equations do no conserve this probability exactly, but the error is of the order of $(\Delta t)^2$. The convergence can be determined by using smaller steps.