38.6.4 Quantization of Angular Momentum

The magnitude of the angular momentum of a particle in circular motion, whether it is a planet or an electron, is

You will recall that angular momentum is conserved in orbital motion because a central force exerts no torque on the particle. Bohr used conservation of energy explicitly in his analysis of the hydrogen atom, but what role does conservation of angular momentum play?

The condition that a de Broglie wave for the electron set up a standing wave around the circumference was given, in Equation 38.26, as

Multiplying by mv and dividing by $2\pi$, we can rewrite this equation as

But mvr is the angular momentum $L$ for a particle in a circular orbit. It appears that the angular momentum of an orbiting electron cannot have just any value. Instead, it must satisfy

Thus angular momentum also is quantized! The electron’s angular momentum must be an integer multiple of Planck’s constant $\hslash$.

The quantization of angular momentum is a direct consequence of this wave-like nature of the electron. We will find that the quantization of angular momentum plays a major role in the behavior of more complex atoms, leading to the idea of electron shells that you likely have studied in chemistry.