Dimensional analysis - quantum and classical lengthscale ratio

AI Thread Summary
Dimensional analysis reveals that the classical length scale is approximately 2.8 x 10^-15 m, while the quantum length scale is about 2.4 x 10^-12 m. The discussion centers on the implications of assuming that the quantum length scale (l_q) is approximately equal to the classical length scale (l_c). It is noted that if l_q is close to l_c, electromagnetic effects must be accounted for, as they significantly influence the relationship between the two scales. The ratio of l_c to l_q is small and primarily determined by the fine structure constant, highlighting the importance of electromagnetic interactions in this context. Understanding these effects is crucial for accurate estimations in quantum mechanics.
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Homework Statement


I was supposed to use dimensional analysis to approximate the length scale (in classical and quantum mechanics). The results I got(same as those in the answer sheet): classicalyl_c≈\frac{e^2}{4πm_ec^2ε_0}≈2.8*10^-15m In quantum mechanicsl_q≈\frac{h}{m_ec}≈2.4*10^-12m


Homework Equations


The next question is: how would your analysis in case of l_qhave to change if l_q≈l_c?


The Attempt at a Solution


I have the answer to question b), but I don't think I understand it. The answer they give is:
'Ifl_q≈l_c it would not be possible to estimate l_q without considering electromagnetic effects and write l_q=f(\frac{l_c}{l_q}) In fact ratio l_c/l_q is is small (apart from a factor of 2π it is just the fine structure constant).'

Could anyone please explain or comment on the given answer?
 
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Could anyone explain why the electromagnetic effects need to be considered and how it influences the ratio?
 

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