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 $\bar\psi(\pm x_\infty)=0$ 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.

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.

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 $$ -\bar\psi''(x) + x^2 \bar\psi(x) = E\bar\psi(x), $$ with boundary values $\bar\psi(\pm x_\infty)=0$ where $x_\infty$ is a finite termination point of your domain.

Clearly, these two problems will have different answers, $\psi$ and $\bar\psi$. 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 $\psi-\bar\psi$ to be!

If you choose $x_\infty$ small, then you can solve the problem of course, but $\psi-\bar\psi$ 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 $\|\psi-\bar\psi\|\le\varepsilon$, 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.

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 $\bar\psi(\pm x_\infty)=0$ 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.