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The wave properties of electrons were discovered through the results of the photoelectric effect and diffraction experiments. Not only did they discover the wave-like properties of electrons, but they also discovered that electromagnetic radiation is particlelike and wavelike. This led towards the acceptance of wave-particle duality as the properties of waves and particles blend together. This led towards Broglie's proposal that all particles have wavelike properties.
If an electron didn't have wavelike properties, in the photoelectric experiment, the electron would go through the small hole and travel straight. However, because the electron has wavelike properties, it hit several different locations. An example of wavelike properties would be fields oscillating in direction and strength with a frequency.
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