In quantum mechanics one applies the time-independent Schroedinger Equation to a system, in order to find steady-state solutions for the wave function of a particle. One of the best-known examples is the Hydrogen atom. Here the procedure gives one the electron orbitals for the various quantum numbers. All the properties of the electron in an orbital can be computed from the corresponding wave function.(adsbygoogle = window.adsbygoogle || []).push({});

For example, the kinetic energy of the electron can be computed for each orbital. It has a positive, clearly non-zero, expectation value. One can also calculate the mean speed of the electron in the Hydrogen atom. It is found to be 1/137th of the speed of light.

Now, as far as I can see it, there arises a sort of paradox.

[1] The electron's orbitals are derived from the time-independent Schroedinger equation. Therefore the probability functions (absolute value of Psi squared) are also time-independent. This has led physicists to conclude that the electron in the orbital is stationary, not involved in any motion. They speak out against the popularized idea that the electron is rapidly moving in a cloud around the nucleus. They think this popular view is in conflict with the true (stationary) character of the orbital.

[2] On the other hand, as I pointed out above, from the mathematical form of the orbital one can compute different properties of the electron. Including its dynamical properties, such as the kinetic energy. This way one finds that the electron has a mean speed of 1/137th of the speed of light. From this fact one must draw the conclusion that the electron is always moving.

I would like to hear your expert opinions on this issue! And so I come to the following question.

How do physicists reconcile:

[1] the stationary (time-independent) nature of an electron's probability function in an orbital, with

[2] the fact that the non-zero value for the mean speed of the electron implies that it is perpetually in motion?

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# How can an electron in a steady-state have a non-zero speed?

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