The part that has to do with quantum mechanics is the way the laser light is created in the first place. Many good explanations of this can be found by doing a web search for "how lasers work". The speckles you are seeing are an interference pattern caused by the reflection of the highly coherent, perfectly in phase, laser light as it bounces off an uneven surface, such as a wall. Here is an explanation from Paul Doherty's excellent Scientific Explorations website: "The waves of light from the laser are coherent before they scatter from the rough screen, that is, they are all in phase, the crests of one light wave line up with the crests of all the others. After scattering from the screen the crests no longer line up, the scattered waves interfere constructively to make bright points and destructively to make dark points. We say that the speckles are caused by interference of the laser light scattered from the rough surface of the screen."
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