On the Foundations of Quantum Mechanics

Matrix mechanics describes the spectral emissions of hydrogen gas, but quantum mechanics concerns the emission and absorption of radiation by a single atom. To obtain the spectral emissions of a single hydrogen atom we introduce thermal energy in infinitesimal increments registering the spectral lines one by one until a complete spectrum is obtained. We describe the gradual increase of the internal quantum mechanical energy in a non-conservative system by integrating between two specific endpoints, the atomic orbitals, and two specific times, the period τ, which are the conditions necessary for Hamilton’s principle. Wave mechanics is shown to be limited to conservative systems. A complete quantum mechanics is formulated using energy-time conjugate variables as the time integral of a Lagrangian independently of Hamiltonian formulations that use position-momentum conjugate variables.