Hi All,
I'm following a derivation for the Quantum Harmonic Oscillator from the textbook 'Quantum Physics' by Gasiorowicz and have attached a print screen of the page in question for clarity. There is a step in the derivation that I would appreciate if someone could clarify for me. Starting from $$\frac{ d^{ 2} u_{0}(y)}{ dy^{2}} - y^{2} u_{0}(y) = 0$$ The author then multiplies this equation by $$2\frac{du_0}{dy}$$ (no idea why he scales it by 2?) which allows the equation to be rearranged to $$\frac{d}{dy}\left(\frac{du_0}{dy}\right)^2-y^2\frac{d}{dy}(u_0)^2 = 0$$ I understand how he arrives at the first term but not the second. I would have thought the correct way to multiply $y^2u_0(y)$ by $\frac{du_0}{dy}$ would be as follows:$$\frac{du_0}{dy}y^2u_0(y)=\frac{d}{dy}(u_0^2y^2)=u_0^2\frac{d}{dy}(y^2)+(y^2)\frac{d}{dy}(u_0^2)$$ How does the author obtain the $$y^2\frac{d}{dy}(u_0)^2$$ term in the second equation? How can he choose not to operate on $y^2$ with the differential operator when performing the multiplication? What are the rules for multiplying equations by derivatives? Thanks