A quantum system, with Hamiltonian $H_{0}$, has continuous energy eigenstates $|E\rangle$ for all $E \geq 0$, and also a discrete eigenstate $|0\rangle$, with $H_{0}|0\rangle=E_{0}|0\rangle,\langle 0 \mid 0\rangle=1, E_{0}>0$. A time-independent perturbation $H_{1}$, such that $\left\langle E\left|H_{1}\right| 0\right\rangle \neq 0$, is added to $H_{0}$. If the system is initially in the state $|0\rangle$ obtain the formula for the decay rate

$$w=\frac{2 \pi}{\hbar} \rho\left(E_{0}\right)\left|\left\langle E_{0}\left|H_{1}\right| 0\right\rangle\right|^{2},$$

where $\rho$ is the density of states.

[You may assume that $\frac{1}{t}\left(\frac{\sin \frac{1}{2} \omega t}{\frac{1}{2} \omega}\right)^{2}$ behaves like $2 \pi \delta(\omega)$ for large $t$.]

Assume that, for a particle moving in one dimension,

$$H_{0}=E_{0}|0\rangle\left\langle 0\left|+\int_{-\infty}^{\infty} p^{2}\right| p\right\rangle\langle p| d p, \quad H_{1}=f \int_{-\infty}^{\infty}(|p\rangle\langle 0|+| 0\rangle\langle p|) d p$$

where $\left\langle p^{\prime} \mid p\right\rangle=\delta\left(p^{\prime}-p\right)$, and $f$ is constant. Obtain $w$ in this case.