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}}\omegax

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

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}}\omegax) = \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