Final answer:
The primary difference is that Bohr's rule quantizes angular momentum as L = nℏ, whereas quantum mechanics uses L = sqrt(l(l + 1))ℏ with l being the azimuthal quantum number. Bohr's model is an approximation that has been enhanced by quantum mechanical theories defining electron behavior with more complexity and accuracy.
Step-by-step explanation:
The biggest difference between Bohr's rule for the quantization of electron orbital angular momentum and the actual quantum mechanical rule lies in the allowed values for the angular momentum of an electron in an atom. According to Bohr's model, the electron's angular momentum is quantized according to the equation L = nℏ (where L is the angular momentum, n is a principal quantum number, and ℏ is the reduced Planck's constant).
In contrast, quantum mechanics shows that the correct quantization condition for the angular momentum is given by L = sqrt(l(l + 1))ℏ where l is the azimuthal or orbital quantum number which can take on integer values from 0 to n-1. This results in a more complex set of possible angular momentum values, especially for electrons in larger orbits (higher values of n). Bohr's model was a stepping stone in the development of quantum theory but has since been superseded by the more accurate Schrödinger equation that defines electron probabilities rather than fixed orbits.