Undergrad Hamiltonian of a particle moving on the surface of a sphere

[Salmone]

In a quantum mechanical exercise, I found the following Hamiltonian:

Consider a particle of spin 1 constrained to move on the surface of a sphere of radius R with Hamiltonian $H=\frac{\omega}{\hbar}L^2$. I knew that the Hamiltonian of a particle bound to move on the surface of a sphere was $H=\frac{L^2}{2mR^2}$ and then to get the same Hamiltonian should be $\omega=\frac{\hbar}{2mR^2}$ which dimensionally fits, but is it right? How can this equality be proved?

 

[vanhees71]

As you write yourself, it's just a definition for a parameter $\omega$. So what should be right or wrong with it?

 

[Salmone]

@vanhees71 So $\omega$ is just a parameter which should be equal to $\frac{\hbar}{2mR^2}$ and with dimensions $s^{-1}$