Three-dimensional problems: The time-independent Schrödinger equation for three-dimensional problems is given by: .
Particle in a three-dimensional box: The wave function for a particle in a cubical box is the product of a function of only, a function of only, and a function of only. Each stationary state is described by three quantum numbers : , . Most of the energy levels given by this equation exhibit degeneracy: More than one quantum state has the same energy.
The hydrogen atom: The Schrödinger equation for the hydrogen atom gives the same energy levels as the Bohr model: . If the nucleus has charge , there is an additional factor of in the numerator. The possible magnitudes of orbital angular momentum are given by equation: , . The possible values of the -component of orbital angular momentum are given by equation: , .
The probability that an atomic electron is between and from the nucleus is , given by equation: . Atomic distances are often measured in units of , the smallest distance between the electron and the nucleus in the Bohr model: .
The Zeeman effect: The interaction energy of an electron (mass ) with magnetic quantum number in a magnetic field along the -direction is given by equation: , where is called the Bohr magneton.
Electron spin: An electron has an intrinsic spin angular momentum of magnitude , given by equation . The possible values of the -component of the spin angular momentum are .
An orbiting electron experience an interaction between its spin and the effective magnetic field produced by the relative motion of electron and nucleus. This spin-orbit coupling, along with relativistic effects, splits the energy levels according to their total angular momentum quantum number : .