Time-dependent quantum-mechanical calculation of ground and excited states of anharmonic and double-well oscillators

Ground- and excited-state energies as well as probability densities of quantum-mechanical anharmonic and double-well oscillators are obtained by numerically solving the time-dependent Schr\"odinger equation (TDSE). The methodology employed is exact in principle and is based on solving the TDSE in imaginary time, coupled with the minimization of energy expectation value. The present method works very well for deep wells and for double-well oscillators with pseudodegenerate energy levels where time-independent methods experience difficulties. The dependence of energy on anharmonicity parameter and the splitting of pseudodegeneracy by an asymmetry parameter as well as probability-density plots are discussed. Several potentials, especially those with pseudodegenerate energy levels, are reported here.