Determine numerical infinity for Schrodinger equation $−\psi''(x) + x^ 2 \psi(x) = E\psi(x)$

Question

Consider the following Schrodinger equation for the harmonic oscillator with real $x$: $$ −ψ''(x) + x^ 2 ψ(x) = Eψ(x). $$

I solve the last equation using shooting method and implicit Runge-Kutta integration. For the eigenvalue $E=1$, the corresponding eigenfunction looks like:

where I set $x=4$ as $+∞$ and $x=-4$ as $-∞$ . From the Fig, we see that the eigenfunction exponentially decay at $x=\pm 4$ .

However, when eigenvalue $E=17$, the corresponding eigenfunction is too complicated so that it won't exponentially decays at $x=\pm 4$ as shown on the following Fig.

In fact, the eigenfunction decays at $x=\pm 6$ as shown on the following Fig.

My Question

The numerical infinity should be large (like $x=6$) for large eigenvalue $E$, and small (like $x=4$) for small eigenvalue. If independent from the eigenvalue, I uniformly set numerical infinity as a very large number, it would waste computational resource. So, is there any formula to determine the exact numerical infinity based on different inputted eigenvalue please?

Answer 1

You look at the question the wrong way. First, let us consider the exact problem and its solution: $$ -\psi''(x) + x^2 \psi(x) = E\psi(x), $$ with boundary values $\psi(\pm\infty)=0$. Now you want to consider truncating your domain somewhere, i.e., you want to consider a different problem that reads $$ -\tilde\psi''(x) + x^2 \tilde\psi(x) = E\tilde\psi(x), $$ with boundary values $\tilde\psi(\pm x_\infty)=0$ where $x_\infty$ is a finite termination point of your domain.

Clearly, these two problems will have different answers, $\psi$ and $\tilde\psi$. The question you are asking is, in essence, how large you have to choose $x_\infty$. But this cannot be answered without also saying how small you want the difference $\psi-\tilde\psi$ to be!

If you choose $x_\infty$ small, then you can solve the problem of course, but $\psi-\tilde\psi$ will be large. On the other hand, if you want the difference to be small, you will likely have to choose $x_\infty$ to be large. In general, if you require that $\|\psi-\tilde\psi\|\le\varepsilon$, then there will be some value $\bar x(\varepsilon)$ so that any $x_\infty\ge \bar x(\varepsilon)$ will be a valid choice of the domain size.

Answer 2

You should maybe look at the analytical form of these solutions (here's the wikipedia page), this would probably help you to derive a relation between the energy level $n$ and the typical length of your state (I would look at the $H_n$ functions given in the link above).