# Quantum harmonic oscillator substitution

The Schrödinger equation for the harmonic oscillator is $$\left(-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+\frac{m\omega^2}{2}x^2\right)\psi(x)=E\psi(x).$$ Then often $$x$$ gets substituted with $$x=\xi a$$, where $$a=\sqrt{\hbar/m\omega}$$, which leads to $$\frac{\hbar\omega}{2}\left(-\frac{\partial^2}{\partial \xi^2}+\xi^2\right)\psi(\xi)=E\psi(\xi).$$ So my question is if now it should not be $$\psi(a\xi)$$, or do they define $$\psi^\prime(\xi)= \psi(a\xi)$$ and then rename it?

• The question is actually about math. Jan 24, 2022 at 13:36
• Can you give the source of this? Jan 24, 2022 at 13:38
• @RogerVadim I think this question still fits best at physics stack exchange. Many more physicists than mathematicians have worked with the Schrödinger equation and just because a physics concept contains math doesn't mean that it is a math question. Jan 24, 2022 at 14:04
• @AccidentalTaylorExpansion I think that question is about general mathematical technique, which is why proposed moving it to the Math community. But I do agree that here it arises in a specific context, which is mostly of interest to physicists. Jan 24, 2022 at 14:09

Yes, you are right, one just redefines the $$\psi(x)=\psi_{new}(\xi)=\psi(a\xi)$$. The Schrödinger equation is thus $$\frac{\hbar\omega}{2}\left(-\frac{\partial^2}{\partial \xi^2}+\xi^2\right)\psi_{new}(\xi)=E\psi_{new}(\xi)$$ The basic motivation is to non dimensionalize the independent variable. In order to recover the original $$\psi(x)$$, we have $$\psi(x)=\psi_{new}(\frac{x}{a})$$
Indeed, to perform such an operation properly, one has to define a new function $$\phi(\xi)=\phi(ax)=\psi(x).$$ More generally, when defining new variable(s) $$\zeta(x)$$ one has $$\phi(\xi)=\phi(\xi(x))=\psi(x).$$ One can then calculate the derivatives as, e.g., $$\frac{d\psi(x)}{dx}=\frac{d\phi(\xi)}{dx}=\frac{d\phi(\xi)}{d\xi}\frac{d\xi}{dx},\\ \frac{d^2\psi(x)}{dx^2}=\frac{d^2\phi(\xi)}{dx^2}=\frac{d^2\phi(\xi)}{d\xi^2}\left(\frac{d\xi}{dx}\right)^2+\frac{d\phi(\xi)}{d\xi}\frac{d^2\xi}{dx^2},$$ and so on.