A Survey of Computational Physics

Numerical approach to the Schrödinger equation in momentum space

“…We have presented an approach to the real-space Schrödinger equation via the Hamiltonian matrix in momentum-space that is obtained after a suitable discretization. 6 This method does not suffer from the instability associated with discretization of the real-space localized (an infinite quantum well or a simple harmonic oscillator) cannot be studied using our approach because the Fourier transform is ill-defined. However, as we have discussed in Sec.…”

“…III we discuss the generalization of our approach to the Schrödinger equation in higher dimensions. 6 We conclude in Sec. IV with a discussion and suggested problems.…”

“…Instead, the Numerov method has to be used to numerically obtain the eigenvalues and eigenfunctions. 5,6 Another approach is to use the eigenfunction expansion method, 7 which results in a matrix equation for the expansion coefficients. 8…”

The hardwall method of solving the radial Schrödinger equation and unmasking hidden symmetries

“…where u(r) = rR nl (r), has been considered an important teaching tool in this Journal 1 and a standard subject in Koonin's early computational physics text. 2 However, in recent years, this topic is not discussed in most computational physics texts, [3][4][5][6][7] or only alluded to briefly. 5 Perhaps, this is related to the difficulty of implementing the conventional shooting or matching method 1,3-7 of solving the eigenvalue problem.…”

“…Our simplification follows from two key ideas: 1) The eigenvalue E is usually determined by requiring the eigenfunction to satisfy certain boundary conditions. The adjustment of the eigenvalue by shooting or matching the eigenfunction [3][4][5][6][7] to satisfy the boundary condition is a tedious iterative process. In this work, we completely by-pass this adjustment process by sweeping through all values of E to a desire accuracy.…”