# How is the energy of an electron-shell related to the speed of electrons in that shell?

I am trying to gain an intuitive picture of what is referred to by "electron-shell energy". I have read that outer electron shells have higher energy than inner electron shells, and this seemed to make sense to me by analogy to a wheel — a point on the rim of a wheel moves faster than a point on the hub. However, I have also read that electrons in inner shells move faster than those in outer shells, that in particularly heavy atoms, relativistic effects have to be taken into account for the inner shells to determine the chemical behavior of the element. And if inner shells are smaller, doesn't that mean that for an electron to be in such a shell, it would need a shorter wavelength, thus higher energy? Can anybody shed some light on my confusion?

Edit: I posted my question because there is an aspect to this that I feel has not been addressed in that other question about the speed of electrons, but I did not express it clearly enough. I recognize that the billiard-ball picture of electrons is not right, and getting beyond that picture is not my difficulty. It is that for all I read that speed is not a relevant concept in this context, I nevertheless keep running into mention of electron "speed" in technical literature, like in a recent SA article, Cracks in the Periodic Table (Scientific American, June 2013), and in the Wikipedia article Electron Configuration:

For the heavier elements, it is also necessary to take account of the effects of Special Relativity on the energies of the atomic orbitals, as the inner-shell electrons are moving at speeds approaching the speed of light.

So, speed is not a relevant concept in the context of electron shell energies, except when it is? Can somebody help me sort out the conflicting messages here?

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The very first thing you need to do is to stop imagining that bound electrons follow paths and have speeds along them. They have position distributions and momentum distributions. –  dmckee Jun 2 '13 at 19:33

Your confusion arises from trying to understand the electron as a classical billiard ball.

The electron is a quantum mechanical entity, when free it displays a dual nature, that of a particle whose location is described by a specific $(x,y,z,t)$ and that of a probability wave.

In particularly an electron caught in a potential well has its energy described by the solutions of the quantum mechanical equations. It is not in an orbit around the nucleus it is in an orbital and best understood as a probability wave:

The shapes of the first five atomic orbitals: $1s, 2s, 2px, 2py$, and $2pz$. The colors show the wave function phase. These are graphs of $\psi(x, y, z)$ functions which depend on the coordinates of one electron.

The square of $\psi(x, y, z)$ gives the probability of finding the electron at that space point.

The five d orbitals in $\psi(x, y, z)**2$ form, with a combination diagram showing how they fit together to fill space around an atomic nucleus.

These are orbitals with higher angular momentum l=2

So velocity of the electron has no meaning here . It is pure quantum mechanical probability distributions. The shell energy defines the amount of energy needed to kick the electron out of its orbital, with a photon or another interaction. The lower the n quantum number the tighter the binding. When free it can exhibit its particle nature and have a velocity assigned to it .

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![Might orbitals appear as depicted here][1]

[1]: http://i.stack.imgur.com/0MYZ6.jpg This image appeared during a test and impressed me as possibly displaying orbitals or energy fluxes generated by them.

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