Why is (⟨n|) = (|n⟩)* = (a₊ⁿ|0⟩)* = ⟨0|a₋ⁿ wrong?

[carllacan]

Hi.

This is clearly wrong, but I don't know where is the error:
$\langle n\vert = (\vert n \rangle )^* = (a_+^n\vert 0 \rangle )^* = \langle 0\vert a_{-}^n = 0$

 

[The_Duck]

The error is in the last step. If $a_{-}$ is the lowering operator, $a_{-}|0\rangle = 0$, but $\langle 0 | a_{-} = \langle 1 | \neq 0$. To see this, take the conjugate of the equation $a_{+}|0\rangle = |1\rangle$.

 

[carllacan]

Oh, ok, that explains it. Thank you.