I attempt to understand how nondimensionalization works from this Wiki page for a second-order ordinary differential equation (ODE). The process is demonstrated for the Schrödinger equation with a harmonic oscillator potential:
$$\left(-\frac{\hbar^2}{2m} \frac{d^2}{d x^2} + \frac{1}{2}m \omega^2 x^2\right) \psi(x) = E \psi(x).$$
To nondimensionalize the above equation, at first the position coordinate $x$ is divided by a constant $x_c$ with the same dimension to introduce the dimensionless position variable $\tilde{x} = x/x_c$. With this substitution, the equation is then simplified and $x_c$ is determined.
My Question
What I don't understand is the following line:
This gives us a dimensionless wave function $\tilde \psi$ defined via $\psi(x) = \psi(\tilde x x_{\text{c}}) = \psi(x(x_{\text{c}})) = \tilde \psi(\tilde x).$
Originally $\psi(x)$ had its own dimension. How do I motivate myself that if I substitute $x$ by $\tilde{x} x_c$ in $\psi(x)$, then $\psi(\tilde{x} x_c)$ would become nondimeinsionalized so that we can denote it by $\tilde \psi(\tilde x)$? What am I missing here?