Quantum superpositions of mixed states.

[oddthingy]

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?

 

[vanesch]

There is not really something such as a "superposition of mixed states". Superposition acts in hilbert space, not in the space of density operators.

 

[Entropia]

I would first like to clarify that a quantum superposition of mixed states is a concept that arises in the field of quantum mechanics, which deals with the behavior of particles at the microscopic level. It is based on the principle of superposition, which states that a particle can exist in multiple states simultaneously until it is observed or measured. This concept is crucial for understanding the behavior of particles in quantum systems.

In the given scenario, we are dealing with a system of non-interacting particles in a potential trap, where each particle is in a different energy eigenstate. This system is considered a completely mixed system because the particles are indistinguishable and in different energy levels. Now, if we introduce a second system with particles in different energy eigenstates, we have a similar but distinct system.

A quantum superposition of mixed states is a system that combines both of these mixed systems in a single state. In other words, it is a state where the particles are in a superposition of different energy eigenstates from both systems. This can be represented mathematically using the density operator, which is a mathematical tool used to describe the state of a quantum system.

The density operator for this system would be a linear combination of the density operators for the two mixed systems. It would be a weighted sum, where the weights represent the probabilities of the particles being in a particular energy eigenstate. This density operator would fully describe the state of the quantum superposition of mixed states and can be used to calculate any observable quantities.

In conclusion, a quantum superposition of mixed states is a complex concept in quantum mechanics, where particles can exist in multiple states simultaneously. It can be represented using the density operator, which is a mathematical tool used to describe the state of a quantum system. This concept has significant implications in understanding the behavior of particles in quantum systems and has led to groundbreaking discoveries in the field of physics.