Quantum Mechanics: creation and annihilation operators

[chocopanda]

Homework Statement

Calculate the following with the creation and annihilation operators

Relevant Equations

$$\langle n+1|b^\dagger bb^\dagger + \frac 12 |n \rangle$$

Hello everyone, I'm new here and I'm struggling with the mathematical formalities in quantum mechanics.

$$\langle n+1|b^\dagger bb^\dagger + \frac 12 |n \rangle = \langle n+1|b^\dagger bb^\dagger |n \rangle + \langle n+1| \frac 12 |n \rangle $$
$$ = \langle n+1|b^\dagger b \sqrt{n+1} |n+1 \rangle + \frac 12 \langle n+1|n \rangle $$
$$ = \sqrt{n+1} \quad \langle n+1|b^\dagger \sqrt{n} |n \rangle + \frac 12 \langle n+1|n \rangle $$
$$ = \sqrt{(n+1)n} \quad \langle n+1|\sqrt{n+1} |n+1 \rangle + \frac 12 \langle n+1|n \rangle $$
$$ = \sqrt{(n+1)^2 n} + \frac 12 \langle n+1|n \rangle $$

Can I simplify the last expression? Provided it's correct.

Many thanks in advance.

 

[BvU]

What about the last term?
(Provided the expression is correct )

 

[chocopanda]

I'm wondering if I calculated that correct and if I can still simplify it because I don't know how that would work :)

 

[vela]

I'm wondering if I calculated that correct and if I can still simplify it because I don't know how that would work :)

You made a mistake calculating $b\lvert n+1 \rangle$. And yes, you can simplify the final expression. @BvU is suggesting you look up or recall some basic information about the eigenstates.