NOVEL SOLUTION OF THE ONE-DIMENSIONAL QUANTUM-MECHANICAL HARMONIC-OSCILLATOR

The classical solution is used as a variable for the one-dimensional quantum-mechanical oscillator, and this leads to a separable differential equation solvable in closed form which yields a set of new quantum mechanical solutions, not energy eigenfunctions, that have the property of being localized. These new solutions are shown to be eigenfunctions of a time-independent position operator. The operator algebra of that operator and an analogous momentum operator is given, as well as the relationship of the eigenfunctions to the energy eigenfunction set. The new eigenfunctions, which are not square integrable but possess a delta-function normalization, themselves represent wavepackets that at times corresponding to when the classical position is at an extremum collapse to Dirac delta-functions.