I have been trying to learn QM and it went well (all untill harmonic oscilator) until i had to face the formalism:
Hilbert space- As a novice to QM i am very sad that in none of the books i have read i found the reasons for using Hilbert space $\mathcal H$ at first place followed by a full geometrical explaination of this space and how we build this space out of $\mathbb R$. It goes same for for its dual space $\mathcal H_d$. Where can i get this? Every single author starts this topic by just bombard novices with bunch of rules for $\mathcal H, \mathcal H_d$ which i can't just trust and this is only pointless learning by heart.
Dirac's notation ... because i don't understand Hilbert space i don't understand what i am allowed to do with kets $|~ \rangle$ and bras $\langle~ |$ and for example:
- Why do we have to write an operator on the left side of kets $| \hat{A} \psi\rangle = \hat{A}|\psi\rangle$ but for bras it is vice versa (we write it on the right) $\langle \hat{A} \psi| = \langle \psi|\hat{A}$.
- Why can we factor out a constant from kets $| a\psi \rangle = a |\psi \rangle$ and we can only factor out complex conjugate from bras $\langle a \psi | = a^* \langle \psi |$.
Hermitean stuff (which i don't even know what it means as it has too many names which totally confuse me - authors should really start writing in a unified language (1 word 1 meaning) otherwise it is a big mess here. For eample here. Just check how many names there is for an conjugate transpose: $A^*,A^{*T},A^{T*},A^\dagger,A^+, A^H$ $\rightarrow$ this leads to a confusion). It seems to me that $A^\dagger$ is the most spread in QM by far, but $A^{*T}$ makes much more reason.
And finally i need all above to just understand this one line: $a a^\dagger = n$. I encountered this in my other topic where i tried to explain a harmonic oscilator to myself. How am i suppose to believe this? Well @Eugene B provided a proff but i don't understand it... The proff was:
By definition,
\begin{equation} \hat{a}^\dagger \left| n \right\rangle = \sqrt{n + 1} \left| n + 1 \right\rangle , \end{equation} \begin{equation} \hat{a} \left| n \right\rangle = \sqrt{n} \left| n - 1 \right\rangle , \end{equation} where $\left| n \right\rangle$ is the eigenstate of creation and annihilation operators, as well as of the Hamiltonian (due to the fact that they commute - homework to prove).
Now \begin{equation} \hat{a}^\dagger\hat{a} \left| n \right\rangle = \hat{a}^\dagger \sqrt{n} \left| n - 1 \right\rangle = \sqrt{n} \sqrt{n} \left| n \right\rangle = n \left| n \right\rangle , \end{equation} so conclude that the eigenvalue of a number operator, $\hat{N}$, is just $n$...
I don't understand the meaning of definitions for $a$ and $a^\dagger$ and even if i did, how can the line above even be true?
\begin{equation} \hat{a}^\dagger\hat{a} \left| n \right\rangle = \hat{a}^\dagger \sqrt{n} \left| n - 1 \right\rangle = \sqrt{n} \sqrt{n} \left| n \right\rangle = n \left| n \right\rangle , \end{equation}
If i changed this equation just a bit and used a $|\psi\rangle$ in place of $|n\rangle$, i would get a different result and couldn't conclude that $a a^\dagger = n$.
Long story short i am totaly confused and i need some guidance to understand this. The best for me would be any book with a geometrical explaination of Hilbert space... Is there any?