I'm studying for a test in quantum mechanics and I'm having a hard time understanding how to use ladder operators. There are no examples in my text book, only definitions that I can't understand how to use, so I hope you can help me instead.
So, there's an assignment where you're supposed to calculate first order correction to the ground state using ladder operators. In the right answer they say that
$$E_0^1 = \langle 0|H'|0\rangle$$
where does the zeros come from? I understand that they're coming from the wave function $\psi$. But I don't really understand the theory behind this.
Then, after some calculations it's written in the right answer
$$ \langle 0|aa^\dagger a^\dagger|0\rangle +\langle0|aaa^\dagger a^\dagger|0\rangle =\langle0|aa^\dagger a|1\rangle +\langle0|aaa^\dagger|1\rangle$$
What are the mathematical rules of these ladder operators? How did these zeros become ones?
I would really appreciate if someone could help me understand this a little bit better.