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We know that Young's double slit experiment shows that light is a wave. On the other hand photoelectric effect shows that light is made up of photons. How can light be both at the same time?

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Hi mhmd. Welcome to Physics.SE. You may enjoy reading this intro to QM especially the Copenhagen interpretation ;-) –  Waffle's Crazy Peanut Jun 15 '13 at 10:39
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The light we see is composed of photons, elementary particles of light, zillions of them. In a similar way this keyboard I am typing on is composed of atoms, which are composed of nucleons and electrons which nucleons are composed of quarks. A solid we can weight and feel is much more nested than the light we see . –  anna v Jun 15 '13 at 10:40
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continued: How can the keyboard be a keyboard and also be composed of atoms is the corresponding question. Now if you ask why photons and electrons are also waves and particles that is another question. you can read an answer to this last in physics.stackexchange.com/questions/46237/… –  anna v Jun 15 '13 at 10:40

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It act as a wave, but this wave is a probability density function, which describes the chances to hit a particular location of the detector.

You cannot predict where it strikes, but if you fire enough photons or electrons, you will see an interference pattern. Described by this function.

If you fire one particle at a time, then you will see individual particles hit the detector. They hit with higher chances where the interference pattern is strong, but you won't see a hit where the interfering waves completely cancel out each other.

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It act as a wave, but this wave is a probability density function It would be more accurate to say that the square of the wave is an energy density, which can be converted into a probability density function by dividing by $h\nu$. Even that statement is somewhat of an oversimplification: physics.stackexchange.com/q/66977/4552 –  Ben Crowell Jun 15 '13 at 21:27

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