# Pre-QFT 1: The quantum harmonic oscillator

The classical system called the simple harmonic oscillator involves a particle in a 1-dimensional space (coordinate \(q\)) moving under the influence of a potential which is quadratic in \(q\). We'll write \(p\) for the momentum of the particle along the \(q\)-axis. In suitable units (to make the constants as simple as possible) the Hamiltonian for the system is \[H=\frac{1}{2}p^2+\omega^2q^2,\] where \(\omega\) is a constant. When we quantise this system, it makes life slightly easier if we rewrite this Hamiltonian as \[H=\omega\frac{1}{\sqrt{2\omega}}\left(\omega q-ip\right) \frac{1}{\sqrt{2\omega}}\left(\omega q+ip\right).\] We apply canonical quantisation to this system:

- taking as our Hilbert space of states the space of square-integrable functions in \(q\);
- replacing \(q\) with the operator \(\hat{q}\): \[\hat{q}(\psi(q))=q\psi(q);\]
- replacing \(p\) with the operator \(\hat{p}\): \[\hat{p}(\psi)=-i\partial_q\psi.\]

*canonical commutation relation*.

The Hamiltonian becomes \[\hat{H}=\omega a^\dagger a,\] where \begin{align*} a&=\frac{1}{\sqrt{2\omega}}(\omega\hat{q}+i\hat{p})\\ a^\dagger&=\frac{1}{\sqrt{2\omega}}(\omega\hat{q}-i\hat{p})\\ \end{align*} (here the dagger denotes Hermitian conjugation). Equivalently, \begin{align*} \hat{q}&=\frac{1}{\sqrt{2\omega}}\left(a+a^\dagger\right)\\ \hat{p}&=-i\sqrt{\frac{\omega}{2}}\left(a-a^\dagger\right). \end{align*}

*before*we quantise, the ordering of \(q\) and \(p\) is not important, so we have to make choice about how we order them when we quantise. Different choices will give different Hamiltonians (differing by a constant, which emerges from the canonical commutation relation as the qs and ps move past one another). The choice we made here is called

*normal ordering*, where we have written the Hamiltonian as an expression in \(a\) and \(a^\dagger\), and taken all the \(a^\dagger\) terms to the left. Later, in QFT, we will have an infinite number of simple harmonic oscillators to handle at the same time, and the extra constants that would appear from a different choice of ordering would give rise to an annoying infinite quantity.

*vacuum state*which satisfies \[a|0\rangle=0,\quad\langle 0|0\rangle=1.\] This is an eigenstate of the quantised Hamiltonian with eigenvalue zero: \[\hat{H}|0\rangle=\omega a^\dagger a|0\rangle=0.\]