Raising and lowering operators

[Moston-Duggan]

Homework Statement

The quantum simple harmonic operator is described by the Hamiltonian:

$\hat{H}$ = -$\frac{h^{2}}{2m}$$\frac{d^{2}}{dx^{2}}$ + $\frac{1}{2}$m$\omega^{2}$x$^{2}$

Show how this hamiltonian can be written in terms of the raising and lowering operators:

$\widehat{a}$$_{+}$ = -$\frac{h}{\sqrt{2m}}$$\frac{d}{dx}$ + $\sqrt{\frac{m}{2}}\omega$x

$\widehat{a}$$_{-}$ = $\frac{h}{\sqrt{2m}}$$\frac{d}{dx}$ + $\sqrt{\frac{m}{2}}\omega$x

The "h" in the above eqns are actually "h-bars"

Homework Equations

Above

The Attempt at a Solution

$\widehat{a}$$_{+}$$\widehat{a}$$_{-}$ = (-$\frac{h}{\sqrt{2m}}$$\frac{d}{dx}$ + $\sqrt{\frac{m}{2}}\omega$)( $\frac{h}{\sqrt{2m}}$$\frac{d}{dx}$ + $\sqrt{\frac{m}{2}}\omega$x) = $\hat{H}$

But the solution is in the picture with a red highlight of where my solution differs and i cannot work out how that extra highlighted part is added

 

[George Jones]

Let the second line of your expression for $\widehat{a}$$_{+}\widehat{a}$$_{-}$ operate on a function $f\left(x\right)$.

 

[Moston-Duggan]

Oh that's so simple haha thankyou