Variational solution of the Schrödinger equation

Elso Drigo Filho, Regina Maria Ricotta, Natália Fávaro Ribeiro

Selina Burkhard, Benjamin Dörich, Marlis Hochbruck +1 more

In the present paper we consider the semiclassical magnetic Schrödinger equation, which describes the dynamics of particles under the influence of a magnetic field. The solution of the Schrödinger equation is approximated by Gaussian wave packets via the time-dependent variational formulation by Dirac and Frenkel. For the numerical approximation we derive ordinary differential equations for the parameters of the variational solution. Moreover, we prove $L^2$-error bounds and observable error bounds for the approximating Gaussian wave packet.

Guillermo H. Bozzolo, Angel Ricardo Plastino

Д.С. Голиков

We consider equations for a mathematical model system of fermions. Equations for spectrum are determined from the system of variational equations. The eigenvalues of the system of variational equations were defined for a particular solution.

Malik Scheifinger, Kurt Busch, Marlis Hochbruck +1 more

In the present paper we consider the semiclassical magnetic Schrödinger equation, which describes the dynamics of charged particles under the influence of a electro-magnetic field. The solution of the time-dependent Schrödinger equation is approximated by a single Gaussian wave packet via the time-dependent Dirac--Frenkel variational principle. For the approximation we use ordinary differential equations of motion for the parameters of the variational solution and extend the second-order Boris algorithm for classical mechanics to the quantum mechanical case. In addition, we propose a modified version of the classical fourth order Runge--Kutta method. Numerical experiments explore parameter convergence and geometric properties. Moreover, we benchmark against the analytical solution of the Penning trap.

Boris V. Alekseev, A. I. Аbakumov