Rejecting the Semiclassical Bohr Model: Examining the Uncertainty Relation

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"Show that the uncertainty relation forces us to reject the semiclassical Bohr [...]"

Homework Statement

The problem along with the solution is attached as TheProblemAndSolution.jpg.

Homework Equations

Uncertainty principle/relation.

The Attempt at a Solution

To consider an electron's motion in classical terms, the uncertainties in its position and momentum must be negligible when compared to r and p; in other words, Δx << r and Δp << p.

Why is it the consideration of the motion in classical terms that is reponsible for requiring that the uncertainties in position and momentum be negligible when compared to the actual values? Also, is it the case that the uncertainties need not be negligible when the motion is considered to be non-classical? If so, why is this the case?

Eq. (1.7.1) is incompatible with Eq. (1.7.2), unless n >> 1.

Is the reason why Eq. (1.7.1) is incompatible with Eq. (1.7.2) because we can analyze a situation at a specific energy level (and therefore hold n constant for that situation) such that n >> 1 does not hold (for example when n = 1)? Similarly, is it not the case that n >> 1 because, Eq. (1.7.2) does not hold for all values of n (even if it holds for the majority of the values), we can say that the assumption using classical theory is false and we must therefore reject the semiclassical Bohr model for the hydrogen atom?

Even if I got things right, please confirm it for me.

Any help in fully understanding this problem, would be greatly appreciated!

 

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If there is something important I did not mention, please tell me what is is and I will mention it.