Origin of the schroedinger's wave equation

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Schrödinger arrived at his wave equation through heuristic reasoning rather than formal derivation, influenced by the concept of "matter waves" proposed by de Broglie. He sought an equation of motion for these waves to describe particles, initially exploring relativistic cases before focusing on non-relativistic scenarios. His work led to the formulation of the Schrödinger equation, which successfully described the energy levels of electrons in atoms, although it lacked a formal proof and is treated as a postulate in quantum mechanics. The equation's interpretation evolved, with Born introducing the probability interpretation to reconcile wave-like and particle-like behaviors. Despite ongoing debates about its foundations, the Schrödinger equation remains a cornerstone of non-relativistic quantum mechanics.
  • #31
Schrodinger's equation can be arrived at using symmetry arguments. Specifically, you write out the commutation relations that exist between the generators of the Galilei group and then choose the position representation. What you find is that the generator of time translations has the same form that is usually ascribed to the Hamiltonian. Details can be found, for example, in the third chapter of Ballentine's "Quantum Mechanics: A Modern Development".
 

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