User:Lpd-Lbr/sandbox

$$\begin{align} \langle v ^ {(0)^{1^{x}}} \left | w ^ {(0)^{2^2}} \right \rangle &= \langle v | \left( \sum_{i \in \mathbb{N}} | e_i \rangle \langle e_i| \right) | w \rangle \\ &= \langle v | \left( \sum_{i \in \mathbb{N}} | e_i \rangle \langle e_i| \right) \left( \sum_{j \in \mathbb{N}} | e_j \rangle \langle e_j |\right)| w \rangle \\ &= \langle v | e_i \rangle \langle e_i | e_j \rangle \langle e_j | w \rangle \,, \end{align}$$

$$ H_0 \left | n^{(0)} \right \rangle = E_n^{(0)} \left |n^{(0)} \right\rang, \qquad n = 1, 2, 3, \cdots $$