A particle in one dimension has position and momentum operators $\hat{x}$ and $\hat{p}$. Explain how to introduce the position-space wavefunction $\psi(x)$ for a quantum state $|\psi\rangle$ and use this to derive a formula for $\||\psi\rangle \|^{2}$. Find the wavefunctions for $\hat{x}|\psi\rangle$ and $\hat{p}|\psi\rangle$ in terms of $\psi(x)$, stating clearly any standard properties of position and momentum eigenstates which you require.

Define annihilation and creation operators $a$ and $a^{\dagger}$ for a harmonic oscillator of unit mass and frequency and write the Hamiltonian

$$H=\frac{1}{2} \hat{p}^{2}+\frac{1}{2} \hat{x}^{2}$$

in terms of them. Let $\left|\psi_{\alpha}\right\rangle$ be a normalized eigenstate of $a$ with eigenvalue $\alpha$, a complex number. Show that $\left|\psi_{\alpha}\right\rangle$ cannot be an eigenstate of $H$ unless $\alpha=0$, and that $\left|\psi_{0}\right\rangle$ is an eigenstate of $H$ with the lowest possible energy. Find a normalized wavefunction for $\left|\psi_{\alpha}\right\rangle$ for any $\alpha$. Do there exist normalizable eigenstates of $a^{\dagger}$ ? Justify your answer.