- #1
chocopanda
- 15
- 1
- 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.
$$\langle n+1|b^\dagger bb^\dagger + \frac 12 |n \rangle = \langle n+1|b^\dagger bb^\dagger |n \rangle + \langle n+1| \frac 12 |n \rangle $$
$$ = \langle n+1|b^\dagger b \sqrt{n+1} |n+1 \rangle + \frac 12 \langle n+1|n \rangle $$
$$ = \sqrt{n+1} \quad \langle n+1|b^\dagger \sqrt{n} |n \rangle + \frac 12 \langle n+1|n \rangle $$
$$ = \sqrt{(n+1)n} \quad \langle n+1|\sqrt{n+1} |n+1 \rangle + \frac 12 \langle n+1|n \rangle $$
$$ = \sqrt{(n+1)^2 n} + \frac 12 \langle n+1|n \rangle $$
Can I simplify the last expression? Provided it's correct.
Many thanks in advance.
)