I'd like to get an answer to this question from someone who knows his density matrix theory. I want to compare two different systems and ask how their density matrix representation looks.
First look at an ensemble of 100 hydrogen atoms, and let us suppose that one of those atoms is excited atoms to 2p state. Let's call this Case 1.
Now for Case 2, let us consider an alternate system where all 100 atoms are in the mixed state, each of them excited to 1% of the same 2p state as Case 1.
Two questions, really: first what is the density matrix representation of these two states? and second, how would we distinguish these two states experimentally?
EDIT: There is a thoughtful and well-reasoned answer by Mikael Kuisma which I believe is incorrect. However, it made me think about the following alternative question which I might have asked, that appears to have the same form as the question I actually did ask: yet for this modified question, Mikael's answer appeaars to be correct. Here is the alternative question:
"A beam of 100 silver atoms is shot throught a Stern-Gerlach magnet. In Case 1, there are 99 atoms spin-up and one atom spin-down. In Case 2, all the atoms are in a superposition 0.995|up> + 0.1|down>. Are these two beams the same or different; and if so, how could you distinguish them?
So the new question is: are these cases perfectly analogous...the bottle of hydrogen atoms versus the beam of silver atoms...and does the same reasoning apply to both?