So I've run into a bit of a notation problem on my coursework.
I have a vector, $\vert A\rangle$ expressed in the orthogonal basis $\vert 1\rangle$ and $\vert 2\rangle$ as
\begin{align}
\vert A\rangle &= \vert 1\rangle + i \vert 2\rangle
\end{align}
I need to convert it into bra form to normalize it. Is the answer
$\langle A\vert = \vert 1\rangle – i\vert 2\rangle$ or
$\langle A \vert = \langle 1\vert – i\langle 2\vert$?
Basically, I get that to convert between bra and ket form, you take the complex conjugate and change a column vector to a row one, but I need to know if you convert the basis vectors into bra form as well.
Best Answer
The comment from @Ellie is what you need to know. This shows the steps. $$ |A\rangle = |1\rangle + i|2\rangle $$ $$ |A\rangle = 1|1\rangle + i|2\rangle $$ $$ \langle A| = |A\rangle^*$$ $$ |A\rangle^* = \alpha^* \langle 1 | + \beta^*\langle 2 |$$ $$ |A\rangle^* = 1^* \langle 1 | + i^* \langle 2 | $$ $$ |A\rangle^* = 1 \langle 1 | + -i \langle 2 | $$ $$ |A\rangle^* = 1 \langle 1 | -i \langle 2 | $$ $$ |A\rangle^* = \langle 1 | -i \langle 2 | $$ $$ |A\rangle^* = \langle A | = \langle 1 | -i \langle 2 | $$
Apologies if it's a bit verbose, but it should be clear.