quantum mechanics - Quantization of a particle on a spherical surface

Suppose we have a particle of mass $m$ confined to the surface of a sphere of radius $R$. The classical Lagrangian of the system is

$$L = \frac{1}{2}mR^2 \dot{\theta}^2 + \frac{1}{2}m R^2 \sin^2 \theta \dot{\phi}^2$$

The canonical momenta are

$$P_\theta = \frac{\partial L }{\partial \dot{\theta }} = m R^2 \dot{\theta }$$ and $$P_\phi = \frac{\partial L }{\partial \dot{\phi }} = m R^2 \sin^2 \theta \dot{\phi }$$

The Hamiltonian is

$$H = \frac{P_\theta^2}{2 m R^2} + \frac{P_\phi^2}{2 m R^2 \sin^2\theta }$$

Now start to quantize the system. We replace $P_\theta$ and $P_\phi$ as $-i\hbar \frac{\partial}{\partial \theta}$ and $-i\hbar \frac{\partial}{\partial \phi}$, respectiely, to obtain

$$H = -\frac{ \hbar^2 \partial^2}{2 m R^2 \partial \theta^2} - \frac{\hbar^2 \partial^2 }{2 m R^2 \sin^2\theta \partial \phi^2 }$$

This is apparently wrong, it should be the total angular momentum!

So what is the right procedure to quantize a system, especially a system in curvilinear coordinates?