Find the Eigenfunctions to the following Hamiltonian:
$$\hat H = \frac{1}{2}\left(-\frac{\partial^2}{\partial x^2} + x^2 + \Theta(x)\right),\,<br /> \Theta(x) = \begin{cases}1&amp;x\ge 0\\0&amp;x&lt;0\end{cases}$$
Good luck.