I have a rather general question regarding the theory of Quantum Mechanics. To preface this question, consider a violin string. When a violinist exposes the string to a bow, this is exposing the string to a wide range of frequencies. In response to this excitation, the string resonates at rather distinct frequencies. In other words, the energy spectrum of a violin string can be understood as a linear combination of distinct energies(frequencies). Why would it not be possible to develop an accurate atomic physics model based on resonant orbital electronic frequencies including possibly spin-orbit resonances as well?
Classical wave equations cannot describe an electron around a hydrogen atom etc because classically the electron in a circular orbit would loose energy by radiation and spiral down into the nucleus. ( see paragraph 20-4 in "classical electricity and magnetism" by Panofski and Philips, radiation from circular orbits)
Any classical solution would have the same fate, whether superpositions or what not. This is the basic reason why quantum mechanical intuition was developed.
After the Bohr model which constrained axiomatically the electrons into orbits the Schrodinger equation merged the concepts of resonance, so evident in the atomic spectra measurements, with the concept of probability distributions: The electrons are in orbitals, not orbits, and the wavefunction of the solution of the quantum mechanical equation gives the probability of finding a particle with (x,y,z,t) or (p_x,p_y,p_z,E). The solution does not describe the coordinate position as a classical wave equation, there is no correspondence.
Nevertheless, the classical string has a comeback into the theory of elementary particles as string theory, with ten or eleven dimensions:
The standard model of elementary particles is embedded into the string group symmetries structure, where the particles themselves are excitation modes of a string.