5.C: Nodes of the hydrogen atom eigenstates
The eigenstates, or stationary states, of the hydrogen atom are described at length in almost every general chemistry text book. If you haven't studied the hydrogen atom yet, you should turn to your text book before proceeding further with this tutorial. Often stationary states or eigenstates are simply referred to as quantum states in text books because they do not delve as deeply into the subject of quantum mechanics as we do here.
Hydrogen atom stationary states are labeled by a set of quantum numbers (n,l,m,ms), and also have the more informal designations 1s, 2s, 2p, 3s, 3p, 3d, .... The number of nodes in a hydrogen atom ground state is n-1, just by accident the same formula as for the particle between two walls. In hydrogen the situation can be more complicated because it is a 3-dimensional system while the particle between two walls is a simple 1-dimensional example. In hydrogen, nodes can be either radial or angular.
The energy of hydrogen atom eigenstates increases with principle quantum number n and the number of nodes according to the formula shown under the figures. Again, more nodes means more waviness and higher energy. Radial nodes mean more energy exists in motion where the electron moves closer and farther from the nucleus. Angular nodes mean the energy is stored in rotation of the electron around the nucleus.