How fast does an electron move around the nucleus?

Many answers here claim that the electron is not moving. They are making a simple mistake, without realizing it: they assume you are talking about a state of definite energy. Indeed, such states are time independent, and therefore (in a classical sense) have no motion.
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Elementary problems in quantum mechanics are typically focussed on energy states, and that can mislead the student into thinking that electrons are always in states of well-defined energy.
But electrons don’t have to be in such a state. If they are in a superposition of energy states, then indeed they can move around the nucleus. This is most readily observed when they are in highly excited states. Then they can appear to move around a nucleus, much like a planet orbits the sun.
Such orbits tend to be relatively low velocity. In the (incorrect but informative) Bohr model, we have circular orbits, and the velocity of the electron in the lowest orbit, the fastest electron, moves at 1/137 the speed of light. But that model is not correct. To obtain electron motion, you need to put an electron in at least two orbitals of different energy. In quantum physics, that’s easy to do. If, for example, you put the electron in a superposition of the lowest energy state and the first excited state (s and p orbitals), then in fact the wavefunction will oscillate back and forth from one side to another, and that can be interpreted as the electron moving. The velocities you get from such motion, because they involve less negative energy states, will always be less than 1/137 c.

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