Quantum superpositions of mixed states.

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SUMMARY

The discussion focuses on the concept of quantum superpositions of mixed states, particularly in systems of non-interacting particles within potential traps. It establishes that a completely mixed system, or mixed ensemble, consists of identical particles in different energy eigenstates. The conversation emphasizes that while one can envision a superposition of systems with different energy eigenstates, defining a superposition of mixed states is problematic within quantum mechanics. Specifically, it concludes that superposition operates in Hilbert space rather than in the space of density operators.

PREREQUISITES
  • Understanding of quantum mechanics principles, particularly superposition and mixed states.
  • Familiarity with Hilbert space concepts and their application in quantum systems.
  • Knowledge of density operators and their role in describing quantum states.
  • Basic grasp of energy eigenstates and their significance in quantum systems.
NEXT STEPS
  • Research the mathematical formulation of density operators in quantum mechanics.
  • Explore the implications of superposition in Hilbert space for quantum systems.
  • Study the differences between pure states and mixed states in quantum mechanics.
  • Investigate the role of non-interacting particles in potential traps and their energy eigenstates.
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Quantum physicists, students of quantum mechanics, and researchers exploring the foundations of quantum theory will benefit from this discussion.

oddthingy
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Suppose you have a system which consists of a number of non-interacting particles in some potential trap. These particles are essentially identical; but each is in a different energy level (in a different energy eigenstate). Such a system is a completely mixed system (also called a 'mixed ensemble'). Then suppose you can envisage a second system which is similar to the first, except that the energy eigenstates which each of the particles are in, are different.
Now finally imagine a system which is a quantum superposition of the following two systems; that is, it is a superposition of mixed states.
How would you define such a system in the language of quantum mechanics? For example, what would the density operator for this system be?
 
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There is not really something such as a "superposition of mixed states". Superposition acts in hilbert space, not in the space of density operators.
 

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