In the 20th century quantum physics was discovered/invented.
It started with Max Planck who introduced quantised energy levels because it was the only logical way to be able to give a sensible prediction for the observed radiation of light from hot black bodies.
Then Louis deBroglie proposed the strange though logical idea of associating momentum intimately with waves.
Then Erwin Schrödinger used these principles, the conservation of energy & some others to derive an equation that described the evolution of the waves associated with quantum particles.
Here's one way of getting Schrödinger's equation.
The applet solves Schrödinger's equation numerically using the simplest time stepping method.
Clicking in the wave function window places a quantum particle (e.g. an electron) at that point.
The initial frequency (& hence momentum), direction & width (uncertainty in location) are adjusted using the sliders.
Conditions affecting the particle (the electric field) include a well like that from a hydrogen atom's nucleus & a boundary similar to a Young's slit experiment.
Using the V slider adjusts the the height/depth of the potential
The real part of the quantum waves or the modulus of the wave function (which gives the probability of finding the particle) can be displayed.