Prove and demonstrate the existence of ladder operators for the simple harmonic oscillator

Method

 * 1) Find the Hamiltonian that describes a simple harmonic oscillator: \(\mathbf{\hat{H}}=\frac{1}{2m}\hat{p}^2 + \frac{1}{2}m\omega^2\hat{x}^2\)
 * 2) Identify the ladder operators for this Hamiltonian: \(\hat{a} = \sqrt{\frac{m\omega}{2\hbar}} \left ( \hat{x} + i\frac{\hat{p}}{m\omega} \right ) \) \(\hat{a}^{\dagger} = \sqrt{\frac{m\omega}{2\hbar}} \left ( \hat{x} - i\frac{\hat{p}}{m\omega} \right ) \)
 * 3) Write the Hamiltonian in terms of the ladder operators.
 * 4) Rearrange the ladder operators as simultaneous equations to find \(\hat{x}\) and \(\hat{p}\).
 * 5) Add the ladder operators together to cancel the \(\hat{p}\) term and therefore find \(\hat{x}\): \(\hat{x}=\frac{1}{2}\sqrt{\frac{2\hbar}{m\omega}} \left ( \hat{a} + \hat{a}^{\dagger} \right ) \)
 * 6) Subtract \(a^{\dagger}\) from \(a\) to cancel the \(\hat{x}\) term and therefore find \(\hat{p}\): \(\hat{p}=\frac{m\omega}{2i}\sqrt{\frac{2\hbar}{m\omega}} \left ( \hat{a} - \hat{a}^{\dagger} \right ) \)
 * 7) Substitute these into the Hamiltonian: \(\mathbf{\hat{H}}=\frac{1}{2m}\left (\frac{m\omega}{2i}\sqrt{\frac{2\hbar}{m\omega}} \left ( \hat{a} - \hat{a}^{\dagger} \right )\right)^2 + \frac{1}{2}m\omega^2\left ( \frac{1}{2}\sqrt{\frac{2\hbar}{m\omega}} \left ( \hat{a} + \hat{a}^{\dagger} \right ) \right )^2\)
 * 8) Carry out the squaring operation: \(\mathbf{\hat{H}}=\frac{1}{2m}\left (\frac{m\omega\hbar}{-2} \left (\hat{a}\hat{a} + \hat{a}^{\dagger}\hat{a}^{\dagger} - \hat{a}\hat{a}^{\dagger} - \hat{a}^{\dagger}\hat{a} \right )\right) + \frac{1}{2}m\omega^2\left ( \frac{\hbar}{2m\omega} \left ( \hat{a}\hat{a} + \hat{a}^{\dagger}\hat{a}^{\dagger} + \hat{a}\hat{a}^{\dagger} + \hat{a}^{\dagger}\hat{a} \right ) \right )\)
 * 9) Simplify: \(\mathbf{\hat{H}}=\frac{\omega\hbar}{2} \left ( \hat{a}\hat{a}^{\dagger} + \hat{a}^{\dagger}\hat{a} \right )\)
 * 10) Add \(\hat{a}^{\dagger}\hat{a} - \hat{a}^{\dagger}\hat{a}\): \(\mathbf{\hat{H}}=\frac{\omega\hbar}{2} \left ( \hat{a}\hat{a}^{\dagger} - \hat{a}^{\dagger}\hat{a} + \hat{a}^{\dagger}\hat{a} + \hat{a}^{\dagger}\hat{a} \right )\)
 * 11) Use commutation relation to remove \(\hat{a}\hat{a}^{\dagger}\) term.
 * 12) Substitute in the commutator \([\hat{a},\hat{a}^{\dagger}]\): \(\mathbf{\hat{H}}=\frac{\omega\hbar}{2} \left ( [\hat{a},\hat{a}^{\dagger}] + \hat{a}^{\dagger}\hat{a} + \hat{a}^{\dagger}\hat{a} \right )\)
 * 13) Use the commutation relation \([\hat{a},\hat{a}^{\dagger}] = 1\): \(\mathbf{\hat{H}}=\frac{\omega\hbar}{2} \left ( 1 + \hat{a}^{\dagger}\hat{a} + \hat{a}^{\dagger}\hat{a} \right )\)
 * 14) Simplify: \(\mathbf{\hat{H}}=\omega\hbar \left (\hat{a}^{\dagger}\hat{a} + \frac{1}{2} \right )\)