Invariant Eigen-Operator Method And Its Applications In Quantum Mechanics

How to obtain the energy spectrum of quantum systems is a basic and important question in quantum mechanics. People usually make use of Schrodinger equation to dealing energy spectrum problems. Whereas Schrodinger equation is a differential equation, it is hard to solve in most situations. On the other hand, Heisenberg equation is seldom employed to solve energy spectrum problems, although it is important in quantum mechanics as well as Schrodinger equation.By researching these circumstances, we combine Heisenberg's idea with Schrodinger operator, and then find a new method to obtain the energy spectrum of quantum systems, which we name as "invariant eigen-operator method", or "IEO method" for short. Applying this method, we mostly deal with the quantum operators, and don't have to refer the quantum states and wave functions, consequently obviate the difficult fluxional equations. Thus IEO method is convenient in solving many quantum systems. The main contents of this paper are to introduce the development of IEO method and its applications in molecule physics, solid state physics, quantum optics and non-commute quantum mechanics.1. First we cast back the origin of Schrodinger quantization scheme, and compare Schrodinger equation with Heisenberg equation, thereby establish the eigen-equation of quantum operators. In virtue of the corresponding relation between the eigen-operators and energy level gaps, we can educe the IEO method to solve the energy spectrum of quantum systems. The kernel of IEO method is to construct eigen-operators of the Hamiltonian of quantum systems, then accordingly obtain the corresponding eigenvalues, namely the energy level gaps of quantum systems. The full energy spectrum can be known by the energy level gaps.2. We shall show the basic process and unique advantage of the IEO method by solving some few-particle systems. Afterward we apply the IEO method to deal with linear chain Hamiltonians which are typically in solid state physics. Where in solid state physics, lattice vibrating modes correspond to the energy level gaps, we can see that the IEO method is quite suitable for this situation. Due to the lattice periodicity, there is actually a standard technique to construct the eigen-operators.3. Some complicated Hamiltonians can also be solved by IEO method, such as semi-infinite chain model and singular oscillator model. Owing to the complication forms, the constructing of eigen-operators have to be determined basing on particular of the Hamiltonians.4. Recently physicists working on superstring theory paid much attention to the quantum mechanics on non-commutative spaces (NCQM). Systems in NCQM are hard to deal with by ordinary methods, since the non-commutation between the coordinate operators of different particles. We apply the IEO method to some models in NCQM, and notice that the non-commutation does not cause any trouble in our way. Thus we can say the IEO method has superiority in NCQM, and we are hopeful that it can be extend to study more models in NCQM.5. Of course the IEO method is not consummate now, there are still many problems to face at. Just like the Schrodinger equation solution, the Hamiltonians which can be figured out by IEO method are also restricted, and we hope to generalize the IEO method to the time-dependent case in the future. In the final of this paper, we introduce some generalization of the standard IEO method, such as pseudo invariant eigen-operator method and quantum operator perturbation method.