In 1926 the Austrian physicist Erwin Schroedinger (1887-1961) published an equation for a wave function that describes the probability of finding an electron at a particular position. It agrees with Bohr’s model in that the most probable radius for an electron is that given by the Bohr model, and the energy of the electron is the same as Bohr calculated, but its results are fundamentally different.

The solution of Schroedinger’s equation can be shown as a probability cloud that shows the most probable locations for the electron.

The n = 1 state of the hydrogen atom is small and spherical. There are two n = 2 states. The s state has angular momentum zero and another spherically symmetric cloud. The p state, with angular momentum 1 has two most probable locations, the top and bottom. The n = 3 state has three possible angular momenta: 0, 1, and 2. The d state, with angular momentum = 2, has four angular regions with high probability.

Red light is emitted when the electron goes from the n = 3 “p” state to the n = 2 “s” state. The lifetime of the higher-energy state is very short, less than one billionth of a second, so when many atoms emit the light, the energy they emit is spread out— the energy from any particular atom cannot be precisely predicted.