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 : .
Many-electron atoms: In a hydrogen atom, the quantum numbers , , , and of the electron have certain allowed values given by equation: , , , . In a many-electron atom, the allowed quantum numbers for each electron are the same as in hydrogen, but the energy levels depend on both and because of screening, the partial cancellation of the field of the nucleus by inner electrons. If the effective (screened) charge attracting an electron is , the energies of the levels are given approximately by equation: .
X-ray spectra: Moseley’s law states that the frequency of a x ray from a target with atomic number is given by equation . Characteristic x-ray spectra result from transition to a hole in an inner energy level of an atom.
Quantum entanglement: The wave function of two identical particles can be such that neither particle is itself in a definite state. For example, the wave function could be a combination of one term with particle in state and particle in state and one term with particle in state and particle in state . The two particles are said to be entangled, since measuring the state of one particle automatically determines the result of subsequent measurements of the other particle.