Commuting Operators, Quantum Numbers, Symmetry Properties
F. S. Levin
Chapters 5–7 have shown us why and how states play such an essential role in quantum theory. States are distinguished through their labeling, which consists of a set of quantum numbers specific to the state. In general, each degree of freedom gives rise to a quantum number, where, in the present context, the phrase “degrees of freedom” refers to the number of spatial coordinates characterizing a given quantum system. The 1-D systems employed in the two chapters illustrating the postulates were thus single-degree-of-freedom systems: all the eigenstates encountered there were labeled by a single quantum number ( n for bound states and k for continuum states) related to the energy of the system. Having studied systems with a single degree of freedom, it might seem that our next step would be to consider systems with more than one degree of freedom, for example, two 1-D particles or a particle in 3-D subject to a potential, etc. Among the goals of such studies would be the assigning of labels to the eigenstates. If the assignment of quantum numbers were based solely on spatial degrees of freedom, then the preceding endeavor would indeed be a way to proceed with our development of quantum theory. However, spatial degrees of freedom are not the only source of the labels which specify eigenstates. An additional source is the sets of eigenvalues of the operators that commute with Ĥ and with each other. Such operators are the main subject of this chapter.
Cambridge University Press eBooks•Dec 3, 2001•0 citations