Quantum Canonical Transformations and Exact Solution of the Schrödinger Equation
نویسنده
چکیده
Time-dependent unitary transformations are used to study the Schrödinger equation for explicitly timedependent Hamiltonians of the form H(t) = ~ R(t) · ~ J , where ~ R is an arbitrary real vector-valued function of time and ~ J is the angular momentum operator. The solution of the Schrödinger equation for the most general Hamiltonian of this form is shown to be equivalent to the special case ~ R = (1, 0, ν(t)). This corresponds to the problem of driven two-level atom for the spin half representation of ~ J . It is also shown that by requiring the magnitude of ~ R to depend on its direction in a particular way, one can solve the Schrödinger equation exactly. In particular, it is shown that for every Hamiltonian of the form H(t) = ~ R(t) · ~ J there is another Hamiltonian with the same eigenstates for which the Schrödinger equation is exactly solved. The application of the results to the exact solution of the parallel transport equation and exact holonomy calculation for SU(2) principal bundles (Yang-Mills gauge theory) is also pointed out.
منابع مشابه
Generalized Adiabatic Product Expansion: A nonperturbative method of solving time-dependent Schrödinger equation
We outline a method based on successive canonical transformations which yields a product expansion for the evolution operator of a general (possibly non-Hermitian) Hamiltonian. For a class of such Hamiltonians this expansion involves a finite number of terms, and our method gives the exact solution of the corresponding timedependent Schrödinger equation. We apply this method to study the dynami...
متن کاملLinear Canonical Transformations and the Hamilton-Jacobi Theory in Quantum Mechanics
We investigate two methods of constructing a solution of the Schrödinger equation from the canonical transformation in classical mechanics. One method shows that we can formulate the solution of the Schrödinger equation from linear canonical transformations, the other focuses on the generating function which satisfies the Hamilton-Jacobi equation in classical mechanics. We also show that these ...
متن کاملQuantum Exchange Algebra and Locality in Liouville Theory
Exact operator solution for quantum Liouville theory is investigated based on the canonical free field. Locality, the field equation and the canonical commutation relations are examined based on the exchange algebra hidden in the theory. The exact solution proposed by Otto and Weigt is shown to be correct to all order in the cosmological constant. PACS: 11.10.Kk, 11.25.Hf, 11.25.Pm
متن کاملBorel Summable Solutions to 1D Schrödinger equation
It is shown that so called fundamental solutions the semiclassical expansions of which have been established earlier to be Borel summable to the solutions themselves appear also to be the unique solutions to the 1D Schrödinger equation having this property. Namely, it is shown in this paper that for the polynomial potentials the Borel function defined by the fundamental solutions can be conside...
متن کاملExact solution of Effective mass Schrödinger Equation for the Hulthen potential
A general form of the effective mass Schrödinger equation is solved exactly for Hulthen potential. Nikiforov-Uvarov method is used to obtain energy eigenvalues and the corresponding wave functions. A free parameter is used in the transformation of the wave function. PACS numbers: 03.65.-w; 03.65.Ge; 12.39.Fd
متن کامل