Introducing $A$ and $A^\dagger$

The Hamiltonian for the 1D Harmonic Oscillator

$$\bgroup\color{black} H={p^2\over 2m}+{1\over 2}m\omega^2x^2 \egroup$$

can be rewritten in terms of the operator $A$

$$\bgroup\color{black}A\equiv \left(\sqrt{m\omega\over2\hbar}x+i{p\over\sqrt{2m\hbar\omega}}\right)\egroup$$

and its Hermitian conjugate

$$\bgroup\color{black} A^\dagger=\left(\sqrt{m\omega\over2\hbar}x-i{p\over\sqrt{2m\hbar\omega}}\right) \egroup$$

Both terms in the Harmonic Oscillator Hamiltonian are squares of operators. Note that $A$ is chosen so that $A^\dagger A$ is close to the Hamiltonian. First just compute the quantity

$$\begin{eqnarray*} A^\dagger A&=&{m\omega\over2\hbar}x^2+{p^2\over 2m\hbar\omega}... ...&{p^2\over 2m}+{1\over 2}m\omega^2x^2-{1\over 2}\hbar\omega .\\ \end{eqnarray*}$$

From this we can see that the Hamiltonian can be written in terms of $A^\dagger A$ and some constants.

$$\bgroup\color{black} H=\hbar\omega\left(A^\dagger A+{1\over 2}\right) .\egroup$$