Given a problem, how does one know whether to use quantum mechanics or classical mechanics?

Take for example electron scattering from a nucleus. The electrons are given a wavefunction in this case. Why not use the classical approach?


You can decide whether you can use classical mechanics or need to use quantum mechanics by estimating the dimensions:

From the particle's speed $v$ you can estimate its de-Broglie wavelength by $\lambda = \frac{h}{mv}$. Let $d$ be the size of the structures involved.

When $\lambda$ turns out to be much smaller than $d$, then the particle's wave nature has no big effect and you can safely use classical mechanics as a good approximation.
Otherwise, you need to use quantum mechanics.

This is essentially the same decision as how to handle light in optics:
When $\lambda \ll d$, then you can safely use geometric optics (with light rays).
Otherwise, you need to use wave optics (with light waves).

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