# Hermitian transformation

I am studying Quantum Mechanics, and the book by Griffths introduces some concepts that I have never come across in my Math courses. I will try to summarize my questions, and hopefully someone will be able to give me some directions.

First of all, the inner product $\langle .,.\rangle$ is defined as the operation with the following properties:

$\langle \alpha,\beta \rangle = \langle \beta,\alpha \rangle^{*}$

$\langle \alpha,\alpha \rangle \geq 0$ and $\langle \alpha,\alpha \rangle=0 \iff | \alpha \rangle = |0\rangle$

$\langle\alpha|(b|\beta\rangle+c|\gamma\rangle)=b\langle \alpha,\beta\rangle + c\langle \alpha,\gamma\rangle$

Then, for an orthonormal basis $\langle \alpha,\beta\rangle = a_{1}^{*}b_{1}+a_{2}^{*}b_{2}+...+a_{n}^{*}b_{n}$. So is this just a convention? Could $\langle \alpha,\beta\rangle$ also be $b_{1}^{*}a_{1}+...+b_{n}^{*}a_{n}$?

Then a Hermitian Transformation is defined as a transformation such that: $$\langle \hat{T}^{\dagger}\alpha|\beta\rangle = \langle\alpha|\hat{T}\beta\rangle$$ Then the author goes on to say that, in particular: $\langle \alpha|c\beta\rangle = c\langle\alpha|\beta\rangle$ but $\langle c\alpha|\beta\rangle = c^{*}\langle\alpha|\beta\rangle$ for any scalar $c$. My other question is: how can these last two equalities be derived from the definition of scalar product and Hermitian Transformation?

• The first equation’s is written in page 92.You can check it yourself(after understanding former section). – Harry May 27 at 14:06

Then, for an orthonormal basis $⟨α,β⟩=a_{∗1}b_1+a_{∗2}b_2+...+a_{∗n}b_n$. So is this just a convention? Could $⟨α,β⟩$ also be $b_∗1a1+...+b_{∗n}a_n$?
For the first, use the third axiom of the inner product with $|\gamma\rangle=0$. Then we get the second by $\langle c\alpha|\beta\rangle=(\langle \beta|c\alpha\rangle)^*=c^*\langle \beta|\alpha\rangle^*=c^*\langle\alpha|\beta\rangle$. These identities don't depend on the notion of Hermitian transformation at all.