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Orbital angular momentum of an electron is $\hbar \sqrt{\ell(\ell+1)}$ where $\ell$ is angular quantum number.

Angular momentum of an electron by Bohr is given by $mvr$ or $\frac{nh}{2\pi}$ (where $v$ is the velocity, $n$ is the orbit in which electron is, $m$ is mass of the electron, and $r$ is the radius of the $n$'th orbit).

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In quantum mechanics, The eigenvalues of the total angular momentum operator $(L^2$) and angular momentum operator ($L_z$) look like $$L^2|lm\rangle =l(l+1)\hbar^2|lm\rangle $$ $$L_z|lm\rangle =m\hbar |lm\rangle $$


The second equation can be recognized as Bohr's condition, that the angular momentum is an integer multiple of ħ was later reinterpreted by de Broglie as a standing wave condition: the electron is described by a wave and a whole number of wavelengths must fit along the circumference of the electron's orbit: $$n\lambda=2\pi r$$ $$\Rightarrow l=n\hbar$$

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  • $\begingroup$ Though correct this answer isn't accessible to most introductory students on this material. $\endgroup$ – shai horowitz Feb 17 at 8:55

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