2nd quantisation of the Harmonic Oscillator

The Hamiltonian is described by $H=\frac{P^2}{2m}+\frac{1}{2}m{\omega }^2{X}^2$, where

• $P$ is the Momentum Operator
• $X$ is the Position Operator

We factorise $H$ into the Adjoint Operators $a_{\pm }=\frac{1}{\sqrt{2m\omega \hslash }}\mp iP+m\omega X$. $a_{+}$ is the raising operator, and $a_{-}$ is the lowering operator. Defining $N=a_{+}{a}_{-}$, we have the following relations:

1. $H=(N+\frac{1}{2}1)\hslash \omega$
2. $N^{†}=N$
3. $[a_{-},a_{+}]=1$
4. $[N,a_{\pm }]=\pm a_{\pm }$
5. $\Vert a_{-}\psi {\Vert }^2=⟨\psi |N|\psi ⟩$
6. $\Vert a_{+}\psi {\Vert }^2=⟨\psi |(N+1)|\psi ⟩$
7. $N\psi =\lambda \psi \Rightarrow N{a}_{\pm }\psi =(\lambda \pm 1)\psi$
8. $\text{Spec}(N)\subset {\mathbb{R}}_0^{+}$, where $\text{Spec}$ is the spectrum of the Linear Operator $N$. where $[\cdot ,\cdot ]$ is the Commutator and $1$ is the identity map.