Proving the Schrodinger Equation

[thegroundhog]

TL;DR

How were scientists able to do experiments to prove that the Schrodinger equation was correct?

How did scientists prove the accuracy of Schrödinger's equation to describe the behaviour of subatomic particles, especially in the 1920s? How do you monitor an electron's momentum and position when they are so small? Also, if the Schrödinger equation just describes probabilities, is the accuracy of the experimenting based on doing the same measurements over and over to make sure the probabilities are correct (in the same way to check the heads/tails probability of a coin, the more measurements the better, as it will tend towards 50%)?

 

[PeroK]

Summary:: How were scientists able to do experiments to prove that the Schrödinger equation was correct?

How did scientists prove the accuracy of Schrödinger's equation to describe the behaviour of subatomic particles, especially in the 1920s? How do you monitor an electron's momentum and position when they are so small? Also, if the Schrödinger equation just describes probabilities, is the accuracy of the experimenting based on doing the same measurements over and over to make sure the probabilities are correct (in the same way to check the heads/tails probability of a coin, the more measurements the better, as it will tend towards 50%)?

You probably need to read the history of QM to clarify the exact sequence of events. In modern QM, the Schrödinger equation can be derived from some expected properties of the time evolution operator: see, for example, Modern Quantum Mechanics by J..J. Sakurai.

The acid test is not direct measurements of an electron, but the prediction of the atomic spectra for hydrogen etc. In this case, even the different energy levels themselves are not measured directly but inferred from the atom's spectrum (i.e. the measurement is of the wavelength (or frequency) of the emission and absorption spectrum).