I have a problem understanding a notation written by my professor. He writes:
$$\langle n | a^+ | m \rangle=\langle n | (a^+ | m \rangle) = (\langle n | a) | m \rangle$$
and he wants to demonstrate this equality. To do so, he expands the first and second expressions:
$\langle n | (a^+ | m \rangle) = \langle n | \sqrt{m+1} | m+1 \rangle = \sqrt{m+1} \langle n | m+1 \rangle$
$(\langle n | a) | m \rangle = \sqrt{n} \langle n-1 | m \rangle$
However, I don't understand why $\langle n | (a^+ | m \rangle)$ is equal to letting $a$ act on the left: $(\langle n | a) | m \rangle$.
To explain my confusion, let me give an example with the time evolution operator:
$\langle n | U | m \rangle = \langle n | (U | m \rangle) = e^{-iE_m t / \hbar} \langle n | m \rangle$
If I let it act on the left, I get:
$(\langle n | U) | m \rangle = e^{-iE_n t / \hbar} \langle n | m \rangle$
and not $(\langle n | U^+ )| m \rangle$, because otherwise I would have:
$(\langle n | U^+ )| m \rangle = e^{+iE_n t / \hbar} \langle n | m \rangle$
What I mean is that $U$ acts on both the right and the left as $U$ and not as $U^+$, otherwise, by acting on the left as $U^+$, I would get $e^{+iE_n t / \hbar} \langle n | m \rangle$ and not the correct $e^{-iE_n t / \hbar} \langle n | m \rangle$.
To clarify this with a slight abuse of notation, I would have:
- Acting on the right: $\langle n | U | m \rangle = \langle n | (U | m \rangle) = \langle n | U m \rangle = \langle n | e^{-iE_m t / \hbar} m \rangle = e^{-iE_m t / \hbar} \langle n | m \rangle$
- Acting on the left: $\langle n | U | m \rangle = (\langle n | U |) m \rangle = \langle n U^+ | m \rangle = \langle n e^{+iE_n t / \hbar} | m \rangle = e^{-iE_n t / \hbar} \langle n | m \rangle$
On the left, it acts as $U$ and not as $U^+$, because the resulting eigenvalue is the negative exponential and not the positive one. If I followed the professor's notation:
- $\langle n | U | m \rangle = \langle n | (U | m \rangle) = (\langle n | U^+ |) m \rangle = \langle n U | m \rangle = \langle n e^{-iE_n t / \hbar} | m \rangle = e^{+iE_n t / \hbar} \langle n | m \rangle$
So...
Returning to the case $\langle n | a^+ | m \rangle$, I would have written that $a^+$ acts on both the right and the left as $a^+$, and thus:
$\langle n | (a^+ | m \rangle) = (\langle n | a^+) | m \rangle$
Could you please explain what I am misunderstanding with the time evolution operator? It seems correct to me, and if I understand this, I would also understand $a$ and $a^+$.
[EDIT] @fulis:
Uh thanks so much, so also in the case of evolution operator where i write:
Acting on the right: $\langle n | U | m \rangle = \langle n | (U | m \rangle) = \langle n | U m \rangle = \langle n | e^{-iE_m t / \hbar} m \rangle = e^{-iE_m t / \hbar} \langle n | m \rangle$
Acting on the left: $\langle n | U | m \rangle = (\langle n | U |) m \rangle = \langle n U^+ | m \rangle = \langle n e^{+iE_n t / \hbar} | m \rangle = e^{-iE_n t / \hbar} \langle n | m \rangle$
we properly have (on the left side): $\langle n | U | m \rangle = (U^\dagger | n \rangle )^\dagger | m \rangle = (e^{+iE_n t / \hbar} | n \rangle )^\dagger | m \rangle = e^{-iE_n t / \hbar} \langle n | m \rangle$$\langle n | U | m \rangle = (U^\dagger | n \rangle )^\dagger | m \rangle = (e^{+iE_n t / \hbar} | n \rangle )^\dagger | m \rangle = e^{-iE_n t / \hbar} (|n \rangle)^\dagger | m \rangle=e^{-iE_n t / \hbar} \langle n | m \rangle$
right? I hope :D
