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I've seen people estimate the Bohr radius using the uncertainty principle by assuming that $$\Delta x \sim r$$ and $$\Delta p \sim p$$

then $$p \approx \frac{\hbar}{r}$$

Using this assumption will give the Bohr radius when minimizing the Hamiltonian.

They didn't however justify their original assumptions $$\Delta x \sim r$$ $$\Delta p \sim p$$

I don't see how one can assume this a priori. It does seem reasonable but I could just as well imagine that $$\Delta x \sim r^2$$ or something like that . Can we assume this approximation for any potential? If not, what about the Coulomb potential allows us to make this approximation?

Thanks

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Just a comment: $\Delta x \sim r^2$ is forbidden since LHS and RHS have differing dimensionality. –  ACuriousMind Jul 22 at 12:42

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These relations are based on the fact that both the position and the momentum distributions are centred around zero, which is in turn due to the symmetry of the atom. Given that, the width of the position and momentum distributions ($\Delta x$ and $\Delta p$) is of the same order as a typical position or momentum within those distributions ($r$ and $p$).

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The way this is justified is as follows: We start with the uncertainty principle, which can be roughly stated as $$\Delta x \Delta p \geq \hbar$$

For this rough estimate, we will ignore some factors of perhaps $2$ or $\pi$, but we're interested in some order of magnitude, not the exact result.

Now, our second assumption will be that the ground state of the system will pretty much saturate the lower bound for the uncertainty principle, so we take $$ \Delta x \Delta p \approx \hbar$$ Now, we search for a characteristic length scale in this problem. The obvious choice is the 'effective radius' of the electron's 'orbit' (note that one has to be careful with classical reasoning such as this in a QM context, but it is nicely in line with Bohr's own approach), $r$. We then take $\Delta x \approx r$, so that we find $\Delta p \approx \frac{\hbar}{r}$. $r$ is usually called the Bohr radius. That's as simple as it is.

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