Why do electrons in lower atomic orbits have smaller wavelengths?

[daisey]

I was previously under the impression that an atomic electron in its lowest orbit has a larger wavelength than an electron in a higher atomic orbit. I read earlier today that lower orbiting electrons actually have smaller wavelengths.

This seems backwards since electrons naturally try to settle into lower orbits, which have lower energies (I thought). And I normally equate energy with wavelength.

 

[PeterDonis]

I was previously under the impression that an atomic electron in its lowest orbit has a larger wavelength than an electron in a higher atomic orbit. I read earlier today that lower orbiting electrons actually have smaller wavelengths.

This seems backwards since electrons naturally try to settle into lower orbits, which have lower energies (I thought). And I normally equate energy with wavelength.

Energy is proportional to frequency, which means it's *inversely* proportional to wavelength.

 

[daisey]

Energy is proportional to frequency, which means it's *inversely* proportional to wavelength.

So electrons in lower orbits have...

* Relatively less energy
- Higher Kinetic Energy
- Lower Potential Energy
* Smaller Wavelength
* Higher Frequency

 

[kanato]

Wavelength isn't something that is well defined for an electron in an atom, for the same reason that momentum $$p = \hbar/\lambda$$ can't be well defined; the potential keeps the electron from settling into a momentum eigenstate.

You can do things like compute the expectation value of kinetic energy $$1/2m \langle p^2 \rangle$$ which will be non-zero. And it will be true that electrons in lower energy levels will have more kinetic energy, but they will also have a much more negative potential energy. In fact, the Virial theorem applies, so $$\langle V \langle = -2\langle T \rangle$$, so an electron in a state with large kinetic energy will have a much larger potential energy than an electron in a state with low kinetic energy.