Solution of time-dependent Schrödinger equation in a nontraditional Hilbert space

Kit-Keung Kan, James J. Griffin, T. W. Atwater +1 more

The gauge-invariant periodic quantization (GIPQ) method is applied within the complete Hilbert space of solutions of the exact time-dependent Schr\"odinger equation. The multiplicity of such GIPQ solutions is shown to exceed by far the number of eigenfunctions they are intended to analogize. However, time averaging (or the equivalent averaging over all values of the progressive phase) reduces in this space to a projection upon the exact eigenfunctions under which every one of these GIPQ solutions either vanishes identically or reduces precisely to one of the stationary (but unnormalized) eigenfunctions of the exact Hamiltonian. One concludes that the correspondence between the GIPQ solutions and exact eigenfunctions is well manageable in the complete Hilbert space, a result which encourages the further study of time-averaged GIPQ solutions in parametric subspaces, where time averaging projects not upon an eigenstate, but upon the constant component of the GIPQ periodic solution.

Alejandro Palma, I. Pedraza

Alexander S. Chikhachev

A. Petridis, L. P. Staunton, Jon Vermedahl +1 more

Analytical solutions to the time-dependent Shrödinger equation in one dimension are developed for time-independent potentials, one consisting of an infinite wall and a repulsive delta function. An exact solution is obtained by means of a convolution of time-independent solutions spanning the given Hilbert space with appropriately chosen spectral functions. Square-integrability and the boundary conditions are satisfied. The probability for the particle to be found inside the potential well is calculated and shown to exhibit non-exponential decay decreasing at large times as $t^{-3}$. The result is generalized for all square-integrable solutions to this problem.

Ts. Istatkov, Kh.Ya. Khristov

Jason Twamley

In this paper we solve for the quantum propagator of a general time-dependent system quadratic in both position and momentum, linearly coupled to an infinite bath of harmonic oscillators. We work in the regime where the quantum optical master equation is valid. We map this master equation to a Schr\"odinger equation on super-Hilbert space and utilize Lie-algebraic techniques to solve for the dynamics in this space. We then map back to the original Hilbert space to obtain the solution of the quantum dynamics. The Lie-algebraic techniques used are preferable to the standard Wei-Norman methods in that only coupled systems of first-order ordinary differential equations and purely algebraic equations need only be solved. We look at two examples.