Experimental evidence from light
Interference and diffraction display wave behaviour, while light-induced electron emission reveals quantized energy exchange. These findings must be explained together. Very weak light can be detected in individual events while repeated events build a distribution shaped by the phase relationships of the experimental arrangement.
Diffraction of matter
De Broglie related a particle’s wavelength to its momentum. Electron diffraction subsequently supported this matter-wave description. Larger momentum corresponds to shorter wavelength, helping explain why everyday macroscopic objects rarely display readily observable diffraction even though quantum mechanics also applies to their constituents.
References: [1]
Probability amplitudes
A matter-wave function is not an ordinary liquid-like spreading of material. Its squared magnitude yields position probabilities in a simple particle description. Amplitudes combine before probabilities are calculated, allowing interference that would be missed by merely adding classical probabilities assigned independently to each path.
References: [2]
Specifying the measurement
An experiment must specify sources, interactions and available path information. Different outcomes are not adequately explained by saying that a person looked at the screen. Quantum states determine predictions, while uncertainty relations constrain the spreads of certain pairs of observables even with ideal measuring instruments.