What are the criteria for a system to be discrete instead of continuous. To have specific energy levels? Or more generally, to follow the laws of quantum mechanics?

I tried looking on internet but the only thing i found is that it needs to be a bound system. And when i say bound, it could be an electron bound to a nucleus but an electron between two walls could also be described as « bound » to the walls.


Everything is quantum mechanical. The dynamics of any physical system will always follows the laws of quantum mechanics. It is only when you zoom out sufficiently far that the laws of quantum mechanics "turn into" the laws of classical mechanics. Classical physics, e.g. Newton's laws, are emergent phenomena in the macroscopic limit of quantum mechanics. One fundamental aspect of quantum mechanics is the "quantization" of certain observable quantities that classically (read macroscopically) we would expect to take values on a continuum. I use quotation marks because it is deeper than one might think classically.

So, maybe your question is, "When do the laws of classical mechanics fail to describe the system of interest, and thus necessitate the employment of the laws of quantum mechanics?" This happens when, roughly speaking, the de Broglie wavelength of the system becomes comparable to the length scale of the system in question. For example, if you are trying to describe the properties of an electron around an atomic nucleus, the de Broglie wavelength of the electron is approximately of the same order as the size as the "radius of its orbit", so you need quantum mechanics.

This is a rough answer to your question. Describing the classical limit of quantum mechanics is an interesting topic of discussion. A common avenue in this direction of historical importance, though by no means the only one, is the WKB approximation. To go deeper, there are the (related) topics of the precise phase-space formulation of quantum mechanics and geometric quantization.


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