How do electron wavelengths relate to orbitals and probability density? I'm doing a physics research project and I am a bit confused. We haven't learnt much of this on our course so I'm sorry if this is a stupid question, I couldn't seem to find an explaination that I understood online.
I understand stationary waves, and why electron wavelengths (with wave/particle duality) mean that they can only be at certain energy levels, like this:

(I still don't really know why they can't/ what would happen if they deconstructively interfered) What I don't understand is how this relates to the electron probability density- what happens at the nodes on this diagram, and why the wave direction in the second diagram is away from the nucleus.
 
'If the electron interfering with itself in the diagram- as it moves around the nucleus- is what causes the stationary wave, what is moving both towards and away from the nucleus in the second diagram in order to create a stationary wave with nodes? 

Again, sorry if this is a stupid question but any help is appreciated!
Thank you for your time.
 A: The two diagrams come from two different, incompatible models. The second set of diagrams is much more accurate.
The "electron as wave on circular string" diagram comes from the Bohr model, which assumes that electrons occupy certain spatially-ordered "shells" corresponding to particular energies. The discreteness of the shells is explained with an analogy to standing waves for a wave of a given wavelength on circles of varying radius: standing waves are only possible on circles of certain discrete radii. This model is not meant to be an accurate representation of where the electron actually is or how it behaves. Just about the only thing it's good for is roughly capturing the behavior of the energy levels of hydrogen-like atoms.
The second set of diagrams is showing the magnitude of the wavefunction for different orbitals. This actually is related to where you'll find the electron (in particular, the magnitude of the wavefunction squared is the probability density). So these provide a more accurate picture of what the orbitals actually look like.
As for the nodes, they are generated by the solutions of the differential equation that determines the shape of the wavefunction (the Schrodinger equation). In general, any solution to the Schrodinger equation under the Coulomb potential $V=\frac{kZe^2}{r^2}$ (for an atom with $Z$ protons and $1$ electron) can be decomposed in the following way:
$$\text{Wavefunction for energy level }n = (\text{Constant})\times(\text{Polynomial in }r)\times e^{-Zr/na_0}\times(\text{Spherical harmonic in }\theta,\phi)$$
The nodes don't come from wave interference, per se; rather, they are the zeros of that polynomial in $r$ that comes directly from the differential equation solution.
A: Your first sketch (upper left) represents a resonant condition for a 1D wave wrapped around a circle. (Keep in mind that the "waves" in the sketch are a mathematical graph representing the probability density at points on the circle.) An electron orbital is a resonant 3D standing wave, bounded by the electric field of the nucleus. For a simple hydrogen atom its shape and properties can described by the solution of the Schrodinger equation as described by “probably-someone”. I would say that the nodes in the wave are a result of wave interference and are predicted by the wave equation.  The energy of each resonant pattern seems to well defined.  I don't know that the same can be said about the wavelength in a 3D pattern.
