Quantum oscillator as a minimization problem

The ground state of the linear harmonic oscillator is the solution to a minimization problem, and this opens up two new ways to completely solve Schrödinger’s time-independent equation. The first exploits systematic patterns that emerge from a formal relationship between two different integral representations of the energy. A remarkable invariant form of the ground-state probability-density equation under specific conditions is the leading principle of the second procedure, where a uniform notation for all mathematical object involved facilitates calculations and generates useful schemes to quickly deduce the main functional relationships of the problem. We believe these procedures may represent useful educational alternatives to standard treatments that address this fundamental problem.