I'm reading from Landau's book about second quantization and I confused about the bra-ket notation for the creation and annhilation operators. From the book, annhilation oparator defined as $$ a_i|N_i\rangle = \sqrt{N_i}|N_i-1\rangle, \tag{eq. 64.5} $$ and it can be represented in the form of a matrix whose only non-zero element is $$ \langle N_i-1|a_i|N_i\rangle=\sqrt{N_i}. \tag{eq. 64.6} $$
The creation operator can be represented by a matrix whose only non-zero element is (eq. 64.7) $$ \langle N_i|a_i^\dagger|N_i-1\rangle=\sqrt{N_i}. \tag{eq. 64.7} $$
From here, it says that by direct multiplication of the matrices (eqs. 64.6 and 64.7) it can be proved that $$ a_i^\dagger a_i = N_i. $$
The questions are:
- What is the form of the matrix from eq. 64.6? Is it correct (for $i = 1,2$)?
$$ \begin{bmatrix}\sqrt{N_1}&0\\0&\sqrt{N_2}\end{bmatrix} $$
- How do I represent the multiplication of 64.6 and 64.7 using bra-ket notation? I'm thinking about this but I don't know if it's correct...
$$ \langle N_i-1|a_i|N\rangle \otimes \langle N_i|a_i^\dagger|N_i-1\rangle = \sqrt{N_i}\cdot\sqrt{N_i}=N_i = \begin{bmatrix}N_1&0\\0&N_2\end{bmatrix}. $$
Thanks for help!