Hamiltonian of the Half Harmonic Oscillator

[Raxonan]

Given the half harmonic potential:

\begin{equation}V=\begin{cases}1/2\omega^2mx^2 & x > 0\\\infty & x < 0\end{cases}\end{equation}What will be the Hamiltonian of the half oscillator?I understand that for x>0 the Hamiltonian will be:

\begin{equation}H=\frac{1}{2}\omega^2mx^2+\frac{P^2}{2m}\end{equation}

But how do I find the Hamiltonian for the part when V = ∞?

 

[PeroK]

The infinite potential simply constrains the particle to the region x > 0. The wave-equation for x < 0 must be 0. I can't see how you could define a Hamiltonian for this.