Question

How do we know that light can be both a particle and a wave?

Answer 1

It isn't. You've been misinformed. What you can say is that light is a phenomenon that has some of the characteristics of a particle, and of a wave. It is neither -- but rather some odd kind of hybrid of both, somewhat in the way someone of mixed race is neither one nor the other, but a hybrid of both. Depending on the experiment you do, you may end up seeing the wavelike properties of light, or the particlelike, or something not really like either.

The experiments you can do that turn up very particle-like properties are when you measure the amount of light. What you find is that the amount of light is quantized, comes in small discrete chunks, like a particle. You can do experiments as simple as counting the flashes of light on a scintillation screen in a darkened room. The wave-like properties occur when you allow light to interfere with itself, which also allows it to refract (be focussed).

If you can stand the quantum mysticism, here's an amusing illustration of the duality for electrons, which is identical for photons (particles of light):

https://www.youtube.com/watch?v=DfPeprQ7oGc

You may wonder why we can't do an experiment that would "catch" light (or electrons) being both particle- and wave-like at once, so we could settle, once and for all, just what kind of hybrid they are. The reason is essentially what's called the uncertainty principle, which puts strict limits on what kinds of simultaneous measurements we can do. It turns out that measurements of very particle-like behaviour (localization in space) are completely incompatable with measurements of very wave-like behaviour (localization in momentum, or energy). You can only make one or the other, never both. So on general theoretical grounds, there will never be an experiment that could pin down light or electrons exactly and reveal just what kind of hybrid they are.