The Schrödinger equation, which is a central equation in quantum mechanics, was not empirically derived but was rather formulated as a fundamental, axiomatic mathematical construction. Erwin Schrödinger developed the equation in 1925 as part of his efforts to describe the behavior of quantum systems.
Schrödinger's approach was influenced by Louis de Broglie's proposal that particles, such as electrons, exhibit wave-like properties. Schrödinger sought an equation that would describe the wave nature of particles and provide a framework for calculating their behavior. He arrived at the Schrödinger equation through a series of mathematical considerations and analogies to classical wave equations.
The equation itself is a partial differential equation that relates the time evolution of a quantum system's wave function to its energy and other relevant physical quantities. The wave function represents the state of the system and contains information about its probabilities and properties.
Schrödinger's formulation of the equation was a significant step forward in quantum mechanics, as it provided a mathematical framework for describing the behavior of particles at the microscopic level. The Schrödinger equation has since played a crucial role in numerous applications and calculations in quantum mechanics.
It is worth noting that the Schrödinger equation has been extensively tested and confirmed through its ability to accurately predict experimental results in various quantum systems. Its success in explaining and predicting a wide range of phenomena provides strong support for its validity as a fundamental equation in quantum mechanics.