# Dual of an HO energy eigenket

1. Jul 7, 2014

### carllacan

Hi.

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

2. Jul 7, 2014

### 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 $a_{+}|0\rangle = |1\rangle$.

3. Jul 8, 2014

### carllacan

Oh, ok, that explains it. Thank you.