Ben Niehoff said:You missed the
[tex]e^{\pm i \phi}[/tex]
in the definition of [itex]L_{\pm}[/itex]
The Angular Momentum Operator, denoted as L, is a mathematical operator that describes the angular momentum of a quantum system. In terms of ladder operators, it can be expressed as L = a+a-, where a+ and a- are the ladder operators.
The ladder operators act on the eigenstates of the Angular Momentum Operator by changing the angular momentum by one unit, either increasing or decreasing it. Specifically, a+ increases the angular momentum by one unit, while a- decreases it by one unit.
The eigenvalues of the Angular Momentum Operator represent the possible values of the angular momentum of a quantum system. This can have physical significance, as different eigenvalues correspond to different energy levels and can provide information about the rotational dynamics of the system.
The ladder operators can be used to find the eigenvalues and eigenstates of the Angular Momentum Operator by repeatedly applying them to the ground state of the system. The eigenvalues are given by the number of times the ladder operators are applied, while the eigenstates are the resulting states after each application.
Yes, the Angular Momentum Operator can also be used to describe systems with other types of symmetry, such as spherical symmetry or cylindrical symmetry. In these cases, the ladder operators may have different forms, but the overall concept of changing the angular momentum by one unit remains the same.