So when learning about the Bohr model of hydrogen and de Broglie waves, it was shown that treating the electron of hydrogen as a de Broglie wave results in the relationship $$L=n\hbar, \qquad n\in\mathbb{N}.$$ However, when learning about the azimuthal quantum number, it was stated that $$L=\sqrt{\ell(\ell+1)}\hbar.$$ So how come in the ground state ($n=1, \ell=0$), these two equations give different values for angular momentum? I feel like I'm missing something really important here. If it's the case that the Bohr model doesn't accurately describe the angular momentum of the electron in the ground state, why is the angular momentum zero?
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TLDR: $\sqrt{l(l+1)} \rightarrow l$ for large values of $l$, but the largest value it can take within an orbital is $l=n-1$, and $n-1 \approx n$ for large values of $n$. Thus $\sqrt{l(l+1)} \hbar \rightarrow n \hbar$ for $l, n \gg 1$.
Long answer:
The Bohr model was a bridge between the Rutherford's and the quantum mechanical atomic model we know today. The greatest achievement of the Bohr model is the prediction of the QM energy levels in hydrogen all the way to the ground state $n=1$, something that might qualify as a coincidence, but it is more like the Bohr quantization rules being educated postulates.
In this sense, Bohr model is less "wrong" than classical physics. Nevertheless, there is the correspondence principle (also by Bohr) from the "correct" quantum world to this "wrong" classical one. Furthermore, while not being a correspondence principle as such, it is not surprising that few aspects from the QM theory can be corresponded to Bohr's angular momentum quantization $L=n \hbar$ in some cases where Bohr's model did a good job. Ultimately, this viewpoint also helps with the intuition of QM as a student.
One example of this correspondence (QM $\rightarrow L=n \hbar$) applies for the rigid rotor of moment of inertia $I$. While the full QM solution gives $E_l = \frac{\hbar l(l+1)}{2 I}$, Bohr's model predicts $E_n = \frac{\hbar n^2}{2 I}$.
In conclusion, it is not surprising that Bohr's angular momentum rule coincides with the QM angular momentum for large quantum numbers. Although not correct in general, it serves as a way to reconcile the old Bohr model in particular cases with the larger QM. In this way, Bohr's model is not seen as being just so inconsistent with QM.
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In your formulas $n$ doesn't have the same meaning. The 1st formula means that the orbital angular momentum is an integer (or zero) multiple of $\hbar$. But for a level with principal quantum number $n$, the angular momentum varies from $(n-1)\hbar$ to $0$, not from $n\hbar$ to $0$. Thus, you don't have a contradiction.
See the page in Wikipedia https://en.wikipedia.org/wiki/Atomic_orbital, and go to the issue "Complex orbitals".
