A novel application of a Fourier integral representation of bound states in quantum mechanics

A method is developed to find the bound state eigenvalues and eigenfunctions of one-dimensional and rotational symmetric three-dimensional quantum mechanical problems. The method is based on the fact that eigenfunctions of bound states are square integrable. We use this property and a judicious ansatz inspired on its asymptotic behavior to obtain a differential equation that can be solved straightforwardly using a Fourier transform. The main advantage of the method is that it avoids the traditional and tedious convergence analysis using a series representation. The method leads to an integral representation of the wave function and provides insight into concepts such as energy quantization, spectrum degeneracy, and bound states. (C) 2011 American Association of Physics Teachers. [DOI: 10.1119/1.3531975]