Question

Simple Quantum Mechanics Problem:
http://i.imgur.com/TmQTo.jpg

Answer 1

The solution says:
"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?

The solution says:
"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!

Answer 2

Why is thinking in classical terms reponsible for requiring that the uncertainties in position and momentum be negligible when compared to the actual values?
Because classical mechanics assumes that the uncertainties are 0. Classical mechanics is never exactly correct for the real atom, but it's a good approximation when the uncertainties are essentially=0 compared with the values.

Also, is it the case that the uncertainties need not be negligible when the motion is considered to be non-classical?
That's the point of non classical (i.e. quantum) mechanics- a better approximation (we don't have better) for large uncertainty values (in proportion to the values for p and x themselves).

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)?
Basically. If you set n=1 and make the 2 equations equal, you get \[1 \le \Delta x \Delta p/xp <<1\]Obviously incorrect.

Answer 3

Thank you very much!