Download or read book Effective Hamiltonians for Constrained Quantum Systems written by Jakob Wachsmuth and published by . This book was released on 2014-10-03 with total page 96 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors consider the time-dependent Schrodinger equation on a Riemannian manifold $\mathcal{A}$ with a potential that localizes a certain subspace of states close to a fixed submanifold $\mathcal{C}$. When the authors scale the potential in the directions normal to $\mathcal{C}$ by a parameter $\varepsilon\ll 1$ the solutions concentrate in an $\varepsilon$-neighborhood of $\mathcal{C}$. This situation occurs for example in quantum wave guides and for the motion of nuclei in electronic potential surfaces in quantum molecular dynamics. The authors derive an effective Schrodinger equation on the submanifold $\mathcal{C}$ and show that its solutions suitably lifted to $\mathcal{A}$ approximate the solutions of the original equation on $\mathcal{A}$ up to errors of order $\varepsilon DEGREES3t$ at time $t$. Furthermore the authors prove that the eigenvalues of the corresponding effective Hamiltonian below a certain energy coincide up to errors of order $\varepsilon DEGREES3$ with those of the full Hamiltonian under reasonab