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We know that electron(s) is(are) moving around the nucleus, that consists of protons and neutrons. But does nucleus itself has its own motion, its own momentum, or the nucleus is stationary? Now, is undisturbed atom stationary or is it "wandering" in space? If it does is that a spin or vibration or any type of an organized motion?

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Electrons actually don't orbit the nucleus. This is a model that simplifies things. The most correct model of the atom that we have today is the Schrodinger model.

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The denser the the cloud of dots, the more likely it is for an electron to be there. They do not have a predictable, regular motion. The nucleus also has this random positioning. It has a probability density.

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