Firstly: Please help me find a proper tag for this question. I explicitly didn't post it on academia stack exchange because it has no latex support and to get answers from more people that might write mathematical/physical articles, theses etc. Thank you.
What this question is about:
In my text, I calculate something regarding density matrices in quantum physics. The idea for the concrete calculation comes from a book, which although doesn't do exactly the same - seems like I do the same, but really prove it and prove a more rigorous version.
The book (Marinescu & Marinescu's classical and quantum information: ) states that (it is easy to show that) every self-adjoint 2x2 matrix $\rho$ can be written as $\rho=\frac{1}{2}\left(I+r_x\sigma_x+r_y\sigma_y+r_z\sigma_z\right)$ with $r_x,r_y,r_z\in\mathbb{C}$ and that those factors are $$r_x=\rho_{01}+\rho_{10}, ir_y=\rho_{10}-\rho_{01},r_z=\rho_{00}-\rho_{11}.$$ if $\rho=\begin{pmatrix}\rho_{00}&\rho_{01}\\ \rho_{10}&\rho_{11}\end{pmatrix}$.
I show that every hermitian 2x2 matrix $\rho$ with trace 1 can be written as $\rho=\frac{1}{2}\left(I+r_x\sigma_x+r_y\sigma_y+r_z\sigma_z\right)$ with $r_z,r_y,r_z\in\mathbb{R}$ and that for $\rho=\begin{pmatrix}\rho_{00}&\rho_{01}\\\rho_{01}^*&1-\rho_{00}\end{pmatrix}$ (which is an arbitrary hermitian matrix with trace 1) those factors are $$ r_x=\rho_{01}+\rho_{01}^*, r_y=i\left(\rho_{01}-\rho_{01}^*\right), r_z=2\rho_{00}-1$$.
In short, it is exactly the same, with 3 differences: Firstly, I give a proof. Secondly, I $\textit{use}$ the hermitian property to calculate the coordinates $r_x,r_y,r_z$ which additionally shows that those are real. Thirdly, Marinescu and Marinescu seem to forget to add "every self-adjoint 2x2 matrix $\rho$ with trace 1..." because the representation $\frac{1}{2}\left(I+r_x\sigma_x+r_y\sigma_y+r_z\sigma_z\right)$ has trace 1.
The question:
How do I cite in my text? I clearly used the book as an inspiration, almost did the same thing. I cannot cite it as usual e.g. as [1, p.3] or such because with that I would claim, that the book says what I proved, and that isn't the case. More generally, how do you cite (or how should one cite, using e.g. the standards of the physics journals AIP or APS) when one builds on an idea/formula... from a book, but alters it so far, that one cannot cite it normally, because one would claim the book to be saying something it doesn't? Should I / how do I make clear, that I got a serious amount of "help/inspiration/guidance" by a concrete source?