QUANTUM STATES OF THE ‘ENIGMATIC’ 1-D HYDROGEN ATOM: DO THEY EXIST?

The quantum states of hydrogen atom in one dimension can be obtained by a careful application of the well-known Frobenius method. The exercise is highly educative and brings to focus the subtle aspects of quantum mechanics. The allowed states turn out to be only of odd parity and non-degenerate, having energy given by E_n=-ⅇ^2/(2n^2 a_0 ) , n= 1, 2, 3,..., and a_0 being the first Bohr radius, in exact correspondence with energy levels of 3-D H-atom. In view of odd parity of all states the spectrum of 1-D H-atom is expected to be dominated by weak electric quadrupole transitions.