The development of quantum theory was indeed motivated by the need to explain experimental results and observations that could not be accounted for by classical physics. Physicists at the turn of the 20th century encountered phenomena such as the blackbody radiation spectrum, the photoelectric effect, and the behavior of atomic spectra, which could not be explained by classical physics.
In their attempts to understand these phenomena, physicists like Max Planck, Albert Einstein, Niels Bohr, Louis de Broglie, Erwin Schrödinger, and Werner Heisenberg, among others, proposed and developed the mathematical framework of quantum theory. They aimed to find equations and models that would provide a better description of the observed phenomena and make accurate predictions.
The development of quantum theory involved a combination of experimental observations, theoretical analysis, and mathematical formulations. Scientists studied the behavior of particles and electromagnetic radiation, conducted experiments, and analyzed the experimental data. In doing so, they recognized that the classical concepts and equations of physics were inadequate to explain certain observations.
To bridge the gap between theory and experiment, physicists introduced new mathematical formalisms and concepts. For example, Max Planck introduced the concept of quantized energy levels to explain the blackbody radiation spectrum. Albert Einstein proposed that light exists in discrete packets of energy, now known as photons, to explain the photoelectric effect. Louis de Broglie suggested that particles such as electrons can exhibit wave-like properties.
These initial ideas and proposals led to the development of more comprehensive mathematical formalisms, such as Schrödinger's wave equation and Heisenberg's matrix mechanics, which provided a framework for describing the behavior of quantum systems. The mathematical equations and models of quantum theory were refined and further developed over time through both theoretical investigations and experimental tests.
So, while the development of quantum theory was driven by the need to explain experimental results, it was not a simple process of matching equations to observations. It involved a deep exploration of the fundamental nature of matter and energy, and the formulation of new mathematical frameworks that could consistently describe the behavior of quantum systems.