Numerical solution of the two-dimensional time independent Schrödinger equation by third order symplectic schemes

The solution of the two-dimensional time-independent Schrödinger equation is considered by partial discretisation. The discretized problem is treated as an ordinary differential equation problem and solved numerically by symplectic methods. The problem is then transformed into an algebraic eigenvalue problem involving real, symmetric, large sparse matrices. The eigenvalues of the two-dimensional harmonic oscillator the two-dimensional Henon-Heils potential and the helium atom are computed by the application of the methods developed.