Assumptions: * 1- The total energy E of a particle is

: :: This is the classical mechanics expression for a particle with mass m where the total energy E is the sum of the kinetic energy, , and the potential energy V. pis the momentum of the particle or mass times velocity. The potential energy is assumed to vary with position, and possibly time as well. Note that the energy E and momentum pappear in the following two relations: * 2- Einstein's light quanta hypothesis of 1905: : :: ::: where the frequency f of the quanta of radiation (photons) are related by Planck's constant h. * 3- The de Broglie hypothesis of 1924: : :: ::: where is the wavelength of the wave. This hypothesis also requires: * 4- The association of a wave (with wavefunction ψ) with any particle. Combining the above assumptions yields Schrödinger's equation: Expressing frequency f in terms of angular frequency and wavelength in terms of wavenumber , with we get: : :: and : :: where we have expressed p and k as vectors. Schrödinger's great insight, late in 1925, was to express the phase of a plane wave as a complex phase factor: : :: and to realize that since : :: then : :: and similarly since: : :: then : :: and hence: : :: so that, again for a plane wave, he obtained: : :: And by inserting these expressions for the energy and momentum into the classical mechanics formula we started with we get Schrödinger's famed equation for a single particle in the 3-dimensional case in the presence of a potential V: : :: Using this equation, Schrödinger computed the spectral lines for hydrogen by treating a hydrogen atom's single negatively charged electron as a wave, , moving in a potential well, V, created by the positively charged proton. This computation tallied with experiment for the Lyman, Balmer, Paschen and Brackett series, the Bohr model and also the results of Werner Heisenberg's matrix mechanics - but without having to introduce Heisenberg's concept of non-commuting observables. Schrödinger published his wave equation and the spectral analysis of hydrogen in a paper in 1926.[2] Assumptions: * 1- The total energy E of a particle is

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