# Why are numerical solutions for the Schrödinger equation necessary to plot this free waves solution?

Suppose a particle in free space given by:

$$\psi(x,t) = Ae^{ik(x-\frac{\hbar k}{2m}t)} + Be^{-ik(x-\frac{\hbar k}{2m}t)}.$$

Why are numerical solutions necessary in order to plot this? Why can't values for $$x$$ be plugged in for each value of $$t$$ and plotted exactly for the real and imaginary components? Implementations I've found seem to use Euler or Runge-Kutta methods to approximately integrate from a differential rather than a solution. I'm missing something in my understanding.

• In general you would need to numerically solve the SE. For this case you of course would not need to. More information would be needed here though. This is certainly a solution, but it is not the solution, as you have not specified initial conditions. Perhaps the implementations you have looked at are involving some additional assumptions you were unaware of. – Aaron Stevens Oct 15 at 0:14
• It's completely unecessary to use numerics to solve for $\psi$ as you have written it. In fact, by virtue of you having written $\psi(x,t)$ in a closed form expression, there is no need for numerics. However, the Schrodinger equation is applicable to far more scenarios than just free space. If there is some external potential energy $V(c)$ than the eigenfunctions will be complicated and $\psi(x,t)$ might not have such a nice closed form expression. – user1379857 Oct 15 at 0:24
• “Implementations I have found ...” Could you provide an example for some context? – J. Murray Oct 15 at 0:31
• @JacksonCapper In that link you aren't looking at a free particle. There is a potential barrier that exists. Since the program has the option of modifying the barrier, it makes sense that it would implement a numerical solver in order to handle the various allowed cases. – Aaron Stevens Oct 15 at 0:59
• Analytic solutions aren't possible for every ODE/PDE, so numerical methods are used – Kyle Kanos Oct 15 at 0:59