The Schrödinger Equation
In 1926, the Austrian physicist Erwin Schrodinger derived an equation of waves from the Hamilton variational principle inspired by the analogy between Mechanics and Optics. This equation explained much of quantum phenomenology was known at that time. Physical laws must be invariant under certain symmetries, represented by the transformations of the mathematical object that define these laws. In particular in quantum mechanics one can ask for the transformations of states and observables. These transformations must fulfill what constitutes the essence of the so-called Wigner's theorem, that is, that the transformed observables must possess the same possible sets of eigenvalues as the old ones and that the transformations of the states must give the same probabilities. The temporal evolution of the states must always be given by unit operators that conserve scalar products.