#wave functions

Okay, so for a transition between energy eigenstates, there needs to be an exchange of a photon with the correct energy. I'm assuming you know that.

Since photons are waves of the electromagnetic field, they impose an electric moment on charged particles, which in a vast majority of cases can be modeled as a simple dipole moment.

Now here's where the quantum mechanics starts: The dipole moment is expressed as a linear operator, which when applied to the wave function of a particular state gives you back the eigenvalue for the dipole moment of that state. However, since we want to describe the transition between states, and the operator only applies to the ket of the wave function, the bra and the ket which it is nestled in between are of the different states, aka the starting and the final electron state.

The operator applies to the starting state ket, which can then be completed on the left with the final state bra, and then integrated over to obtain the transition dipole moment integral.

This integral will tell you the expectation value for the transition. For the selection rules, you don't actually have to precisely calculate this integral, you just have to find out whether or not it is zero, because if it is, that means you have an impossible transition on your hands.

Depending on your representation, the dipole operator as well as the wave functions will look different, as will the space you integrate over.

So first, find which representation (spatial, spherical, momentum space, etc.) you are working with, how the dipole operator looks for that representation, and then pick the two states you want to see if a dipole transition exists between them.

The tricky part is usually to get the wave function representation right, and then to leverage the symmetries of that function to determine if the value of the integral is zero or not. The representation that i find most common for tasks like this is this one, which separates the wave function into a radial and two angular parts. It is also already conveniently expressed in terms of 3 quantum numbers, that being the main quantum number n, the orbital angular momentum number l, and the magnetic quantum number m.

I am afraid you'll have to learn the quirks and symmetries of the generalized Laguerre polynomials as well as the spherical harmonics, in order to make statements about the transition dipole moment integral. However, once you get a feel for their symmetries and remember in what special cases integrals vanish (like integrating an odd function over a symmetric interval, etc.) you will be able to derive the selection rules.

Entropy Criteria for Corpuscular Wave Interactions by Korablev GA* in Crimson Publishers: Peer Reviewed Material Science Journals 

The concept of entropy for space-energy interactions is used similarly to the ideas of thermodynamics of statistic entropy. This concept emerged based on the second law of thermodynamics and ideas on reduced amount of heat. Such ideas are general statements independent of microscopic models. Therefore, their application and consideration can have a large number of results, which are also most fruitfully used in statistic thermodynamics. In this investigation the attempt is made to apply such regularities to assess the degree of spatial-energy interactions using their graphical dependence in other fields as well. The nomogram to assess the entropy of various processes is obtained. The diversity of entropy manifestations is discussed, including the entropy in biophysical processes and technical systems. 

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I am going to start off with explaining the Heisenberg Uncertainty Principle, which explains electrons very well.  It basically says we can either know where an electron is at, or where it is going, but never at the same time.

This plays a big part in our understanding of electron orbitals.  Because we could not know the exact orbital of an electron, we would have to find orbitals, or areas that have a probability of having an electron in them around an atom.  

As you increased the energy of the atom and cause the principle quantum number to jump up, you create a space between the two orbitals called a node.  In the case of the s-orbitals, it is a space in-between each sphere shaped orbital.  An electron essentially has zero probability of being found in this area.  This idea gets complicated after the s orbitals, which are the simplest.

Next are the p-orbitals.  These orbitals feature two areas opposite of each other on the same plane, and they always have a node between them that runs through the nucleus.  How can electron be in either side if it can't be found in-between crossing over?

A very simple explanation can be given.  Violin strings, like electrons, vibrate at certain frequencies, and follow their own wave function, and nodes occur when the amplitude of a specific spot hits zero, which is a natural result.  Zero amplitude does not mean the string isn't there, because it is, it just means the magnitude of the vibration is not there.  Likewise, a node does not necessarily mean an electron isn't there, its just that the effects of the electron, its own specific properties just don't show up.  Wherever a node is, the wave function is equal to zero, which means that for that brief moment, it is not moving or acting like a wave, which is why you can't "find" it, or predict a probability there.