\(\displaystyle i\hbar\frac{\partial \psi(x,t)}{\partial t}=-\frac{\hbar^2}{2m_e}\frac{\partial^2\psi(x,t)}{\partial x^2}+V(x)\psi(x,t).\)

I don't want to "carry" constant prefactors and want to deal with

\(\displaystyle i\frac{\partial \psi(x,t)}{\partial t}=-\frac{1}{2}\frac{\partial^2\psi(x,t)}{\partial x^2}+V(x)\psi(x,t),\)

by setting up my units as \(\displaystyle \hbar=m_e=1.\)

Then, after finding the solution (for example, numerically) I want to return back to physical units.

My question is: "How to convert length and time back to physical (for example, SI) units? I know, that I should scale by some factor, e.g. Bohr radius for the length. But, how to prove that?"

In other words, how much meters in SI units correspond to \(\displaystyle x=1\) in my units? How to derive this scaling factor?

You may say, multiply it by some number \(\displaystyle a_0\), but I need the answer, why exactly this number \(\displaystyle a_0\)?

Thank you in advance for your answer!