Interdimensional degeneracies, near degeneracies, and their applications Available to Purchase

Recently developed approximation methods for quantum mechanical problems which treat the spatial dimension D as an expansion parameter offer approximations to energy levels at arbitrary D. Rather than simply being a detour to the D=3 case, there is physical interest in nonphysical values of D due to degeneracies between states in different dimensions. For example, such degeneracies make it possible to calculate some excited states of two‐electron atoms in three dimensions from the ground state energy at nonphysical values of D. Such relationships can be exploited in a simple derivation of the hydrogen atom spectrum in arbitrary D, using only the solution at D=1 and a combination of inter‐ and intradimensional symmetry arguments. Applications to the Yukawa potential and an anharmonic oscillator are also presented. A large class of interdimensional degeneracies is found for two‐electron atoms. Approximate degeneracies are also identified for these atoms which allow highly excited D=3 states to be treated as perturbed low‐lying states in another dimension. The approximate degeneracies also serve to generalize the treatment of the hydrogen atom spectrum in a way appropriate to two‐electron atoms.