I Can two particles be in the same place at the same time?

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Can two particles ever be said to occupy exactly the same space at the same time while remaining distinct objects?
 
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It depends. There are two kinds of particles in QM classified as fermions or bosons. Fermions cannot occupy the same energy state in a system whereas bosons can.

The famous Pauli Exclusion principle describes it well:


Electrons are examples of fermions and a photon of light is an example of a boson.



and a nice video on the topic:

 
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occupy exactly the same space at the same time
As you state it, this is really too vague to answer. A better way to ask would be: can two particles ever be in exactly the same quantum state at the same time. The answer to that question is what @jedishrfu is giving in post #2.

An example of why "the same quantum state" is not the same as "the same space" is different energy levels in an atom. In an atom with multiple electrons, the wave functions of those electrons overlap in space, so they can all be viewed as occupying "the same space" at the same time. But electrons are fermions, so no two of them can be in the same quantum state at the same time. That is the underlying reason for the different chemical properties of different elements that are described in the periodic table.
 

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