Raising And Lowering Operators For Angular Momentum Quantum Numbers Of Orbital Angular Momentum

The angular momentum plays vital role in understanding of the atomic and nuclear structure. In quantum mechanics, the ladder operator technique is widely used. For instance, Three-dimensional motion system, the angular momentum operator eigenstate has two quantum numbers, called angular momentum quantum number, and the direction quantum number or the magnetic quantum number. For the eigenvector of angular momentum, the ladder operator of magnetic quantum number is L±. The action of the angular momentum ladder operator L±on spherical harmonics raises and lowers respectively the magnetic quantum number by one while leaving the angular momentum quantum number unaltered. Then is there any ladder operator that shifts the values of the angular momentum quantum number in the spherical harmonics?As far as our knowledge goes, the first attempt to give a direct answer to the problem is due to Prof. Ka in 2001, who presented a derivation and observed that once acting on the spherical harmonics two vector operators, which are good enough to meet our need.Unfortunately, these two vector operators are not full ones because they contain information of the state acted, i.e., the angular momentum quantum number. Prof. Ka left the operator almost finished. In fact, they are full operators once the angular momentum quantum number is replaced the expressions contains the square of the angular momentum. Explicitly, the vector operators we deal with in this paper take following two simple forms. First, we study their basic properties, and discover that the state really has the minimum angular momentum quantum number for the given magnetic quantum number, and get the state with minimum quantum number in whole set of spherical harmonics. Starting from the with minimum quantum number in whole set of spherical harmonics, we can generate the whole set of the spherical harmonics with appropriate action of the raising and lowering operators.