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A free electron is one that is not bound to an atom, whereas a bound electron is one that is bound to the nucleus of atom. An important distinction is that free electrons are not apart of the quantum system, whereas bound electrons are.
Also, note that free electrons are at a higher energy state than bound electrons. While an electron is bound to an atom (via electrostatic forces) the system is most stable when the bound electrons are at their lowest energy levels allowable. That's why excited, bound electrons emit radiation to fall back to their "ground" state.
It may also be important to note that in some cases, like metals, electrons can be free in the sense that they are not bound to any one specific atom, but rather exist as a sea of communication in between existing atoms. For example, when electricity runs through a wire, electrons are free to rush around the wire in waves and patterns.
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