Proof of equation 2.64 for harmonic oscillator ladder operators

[SherLOCKed]

I am self studying the Book- Introduction to Quantum Mechanics , 2e. Griffith. Page 47.
While the book has given a proof for eq. 2.64 but its not very ellaborate
Integral(infinity,-infinity) [f*(a±g(x)).dx] = Integral(infinity,-infinity) [(a±f)* g(x).dx] . It would be great help if somebody could provide me a more step by step proof of the same.
Where a+ and a- are roots of Hamiltonian of harmonic oscillator problem.

 

[Jilang]

Are a+ and a- real? If so it would be justified.

 

[paralleltransport]

The operator

$${d \over dx}$$ picks up a minus sign under hermitian conjugation when the hilbert space is that of functions that vanish fast enough at infinity. The reason why that's true is because:

$$\int f {d g \over dx} = 0 - \int g {df \over dx} dx$$

This implies that when you take the hermitian conjugate of the $\hat{a}_+$ operator, just replace ${d \over dx}$ by $-{d \over dx}$ which gives the $\hat{a}_-$ operator

 

[BvU]

Where a+ and a- are roots of Hamiltonian of harmonic oscillator

Be careful there: they are not! There is a constant $\hbar\omega$ difference !