How can i calculate orbital angular momentum of an electron?

The straight up answer is to use the equation ##L=Iw## where L is the angular momentum, I is the moment of inertia of the electron and w is the angular velocity of the electron.

Since the electron is like a point mass, you can find its moment of inertia using ##I = mr^2## where m is the mass of the electron and r is the radius of the electron's orbit.

The angular velocity is found by using w = v/r. Where v is the speed of the electron in meter per second and r, again, is the radius of the orbit.

Putting these three paragraphs (equations) together gives.

##L = Iw##

##L = mr^2w##

##L = mr^2(v/r)##

##L=mrv##

Now all of this is boring and you might ask, "Why do I care what the angular momentum of an electron orbiting the nucleus is?" But things got interesting when Bohr realized that the values for angular momentum of an electron were restricted to only certain values and other values were impossible. It would be as if your car could travel at 5 m/s or 17.5 m/s but all other speeds were impossible.

Bohr realized that the angular momentum was some integer multiple of Planck's constant divided by 2π. Mathematically, ##L = nh/(2π)##.

This oddity makes physics kind of neat, but then when deBroglie realized that if we combine the equation for angular momentum with the restricted equation notice by Bohr, something amazing happens.

##mrv = nh/(2π)## Which can be rewritten as ##2πr = nh/(mv)##.

What makes this new way of writing this amazing is that 2πr is the circumference of the orbit of the electron and mv is the momentum of the electron. So the right side of the equation becomes Planck's constant over momentum, which for light would equal wavelength.

deBroglie posited that the electrons also had a wavelength associated with them and that this was telling us that the electrons were acting like waves and only when the wavelength times and integer equaled the circumference of the orbit could the orbit be a valid orbit. This forever changed our way of looking at matter.