Schrödinger’s equation is essentially an expression that the total energy of a particle is the sum of its kinetic energy and the potential energy associated with its environment - that is a very classical notion - but in a very different mathematical formalism.
One of the first examples usually presented in a quantum theory class is a particle trapped in a closed space.
Well that solution makes no sense classically. That is, it says the particle couldn’t just be placed with zero speed at the center of the space and left alone. That is, it could not simultaneously have zero kinetic and zero potential energy. That, of course is completely consistent with Heisenberg’s uncertainty principle. If it had zero kinetic energy, we would know its momentum to be exactly zero while also knowing it is contained in that space. So ∆p∆x=0 in violation of Heisenberg’s uncertainty principle.
Is there any sort of experimental verification of this? Not directly as stated. But consider this: The three dimensional version of the same problem predicts the form of the density of states function for the free electrons in a metal (within some assumptions). That density of states function allows one to predict the temperature dependence of the electronic specific heat of metals - and that is experimentally verifiable. Sometimes, experimental verification of a theory depends on examining the implications of that theory.
Another version of the one-dimensional Schrödinger problem, but with a potential barrier rather than a potential well, predicts quantum tunneling.
But the most dramatic success of this “improbable” theory was in solving for the possible energy levels and wave functions for the electron in an isolated hydrogen atom. It’s not trivial mathematically - and requires putting Schrödinger’s equation into spherical polar coordinates and solving. But it is a closed form solution that predicts the energy levels possible for an electron in the vicinity of a proton (i.e., the hydrogen atom) - and those energy levels predict the ionizaton energy of hydrogen as well as the wavelengths of the emission spectra of excited hydrogen gas … both of which had already been measured. And that information tells us that the stars are formed from hydrogen (and ultimately that those stars in distant galaxies are moving away from us). And the mathematical solution just for hydrogen, gives us the rules for the various other quantum numbers which lead to the organizing principles for more complicated atoms.
It has been nearly a century since all of that was developed. It is hard to overemphasize the significance of those developments. When I was fresh out of graduate school, my first academic appointment was at a college whose faculty included an emeritus professor (Vladimir Rojansky) who was the himself the first grad student of Nobel Laureate John Van Vleck, as young men at the time, they were friends with Paul Dirac. Rojansky talked about the excitement of the discussions (sometimes around a campfire while backpacking!), the speculations, and the pouring over the latest publications of Heisenberg and Schrödinger and the others as all of that was unfolding.
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