Editing Quantum number (section)

==General properties==

Good quantum numbers correspond to [[eigenvalues]] of [[Operator (quantum mechanics)|operators]] that commute with the [[Hamiltonian (quantum mechanics)|Hamiltonian]], quantities that can be known with precision at the same time as the system's energy. Specifically, observables that [[Commutator|commute]] with the Hamiltonian are [[simultaneously diagonalizable]] with it and so the eigenvalues <math>a</math> and the energy (eigenvalues of the Hamiltonian) are not limited by an [[Uncertainty principle|uncertainty relation]] arising from non-commutativity. Together, a specification of all of the quantum numbers of a quantum system fully characterize a [[Basis (linear algebra)|basis]] state of the system, and can in principle be [[Measurement in quantum mechanics|measured]] together. Many observables have discrete [[Spectrum of an operator|spectra (sets of eigenvalues)]] in quantum mechanics, so the quantities can only be measured in discrete values.  In particular, this leads to quantum numbers that take values in [[Discrete mathematics|discrete sets of integers]] or [[half-integers]]; although they could approach [[infinity]] in some cases.

The tally of quantum numbers varies from system to system and has no universal answer. Hence these parameters must be found for each system to be analyzed. A quantized system requires at least one quantum number. The dynamics (i.e. time evolution) of any quantum system are described by a [[quantum operator]] in the form of a [[Hamiltonian (quantum mechanics)|Hamiltonian]], {{mvar|H}}. There is one quantum number of the system corresponding to the system's energy; i.e., one of the [[eigenvalue]]s of the Hamiltonian. There is also one quantum number for each [[linear independence|linearly independent]] operator {{mvar|O}} that [[commutivity|commutes]] with the Hamiltonian. A [[complete set of commuting observables]] (CSCO) that commute with the Hamiltonian characterizes the system with all its quantum numbers. There is a one-to-one relationship between the quantum numbers and the operators of the CSCO, with each quantum number taking one of the eigenvalues of its corresponding operator. As a result of the different [[Basis (linear algebra)|basis]] that may be arbitrarily chosen to form a complete set of commuting operators, different sets of quantum numbers may be used for the description of the same system in different situations.