Closed quantum system -------> a state is a ray in H.
Open quantum system---------> a state is described by density operator. Not a ray.
Sorry, I couldn't resist: A semicolon, rather than a comma, should precede conjunctive adverbs when they link two complete sentences. And, if you use however at the beginning of a sentence and don't insert a comma after it, however means “in whatever manner,” “to whatever extent,” or “no matter how.”
Therefore: "The density matrix is the most general method of dealing with pure and mixed states; however, one cannot show superposition between states merely by summing density matrices."
I don't understand, by the way, what you mean by: "one cannot show superposition between states merely by summing density matrices." Density matrices cannot be summed, the trace would no longer be one.
But I do agree with the sentiment expressed in your link, although I don't think we should be grammar nazis. People should spend at least a little time in formulating the question, and indeed use a spell and grammar checker. Or maybe ask a colleague or fellow student if they understand the question. In this case I did not understand it either.
Let me nevertheless attempt an answer. I think it is correct to state that you can always use the density matrix. For a closed system (no interaction with the rest of the universe) a pure state will remain pure, so the density matrix is not mandatory and may be overkill. If you couple your system to another, or to an environment, and reduce the density matrix, or use a statistical ensemble, you get mixed states. No "ray" corresponds to mixed states. Use of the density operator and the Liouville equation is your only option (as far as I know).
Esakki,
You should be more generous with defining your problem, not let the people guess what is the problem.
I just guess that you ask whether a closed system may be in a mixture of states.
YES, it may.
Let's recall what is the definition of a closed system: it's a system that does not interact with any other system, by classical fields.
Now imagine the following scenario: an alpha particle is emitted by an unstable nucleus, 108Te. In consequence of the emission, the daughter nucleus 48Cd undergoes recoil. The alpha may fly in different directions, and the Cd nucleus recoils in the corresponding opposite direction. Thus, the total system remains in a superposition ∑n |alpha>n |Cd>n, where n is the index that keeps track of the direction of flight of alpha.
When the alpha is already far from the Cd nucleus, both the alpha and the Cd are isolated systems. But each one, taken alone, is in a mixture of states.
Thank you for all your answers. I did PhD in pure maths. I would like to work on Quantum gravity. Can I apply for another PhD for physics?
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