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I just wanted to understand the following. Let's stay with the harmonic oscillator in QM, just to have an example at hand. First, there are all the different states for $n=1,2,...$. (Let's call them $\psi_n$).

Then, the superposition of a state, for example $$\frac{1}{\sqrt{2}}(\psi_1+ \psi_2)$$ is also a solution to the Schrödinqer equation.

But then, there is also the concept of a density matrix, for example

$$\rho= \frac{1}{2} |\psi_1 \rangle \langle \psi_1| + \frac{1}{2} |\psi_2 \rangle \langle \psi_2|.$$

My question is: What is the meaning of the concept of superposition and this density matrix state?

What I know so far is somewhat vague:I would say I can only measure whether a particle is in one of the $\psi_i$. The density matrix tells me that the particle is equally likely in one of the two states written down there, but if I would measure, then I would get one of them. But I don't really know what the superposition tells me? Which experimental setting corresponds to a superposition and what is the meaning of it?

If anything is unclear about my question, please let me know.

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The superposition $1/\sqrt{2} (\psi_1 + \psi_2)$ is not equivalent to $\rho= \frac{1}{2} |\psi_1 \rangle \langle \psi_1| + \frac{1}{2} |\psi_2 \rangle \langle \psi_2|$. Density operator corresponding to $\psi = 1/\sqrt{2} (\psi_1 + \psi_2)$ is $\rho_\psi = |\psi \rangle \langle \psi |$ which can be expanded into $\frac{1}{2}(|\psi_1\rangle \langle\psi_1| + |\psi_1\rangle \langle\psi_2| + |\psi_2\rangle \langle\psi_1| + |\psi_2\rangle \langle\psi_2|)$ – Ján Lalinský Feb 3 '14 at 13:41
try to understand the example here… which clarifies the distinction between pure states with a superposition and mixed states. Does that answer your question? – Martin Feb 3 '14 at 13:49
i read it, but the problem is that I can never see a harmonic oscillator in this superposition state, right? but why is this so and how is such a superposition state generated? – Xin Wang Feb 3 '14 at 14:20
i understood that mixed states come from having a percentage of state one and of state 2, but a superposition? how do you create a superposition in a harmonic oscillator? – Xin Wang Feb 3 '14 at 14:21
that's exactly the idea: you CAN see a Quantum harmonic oscillator in such a superposition state - for example, a coherent state (eigenstate of the annihilation operator) is a superposition of energy eigenstates. Preparation: e.g. photonic states. However, you COULDN'T see a classical harmonic oscillator in a superposition state - here, you can just have a mixed state (owing to the fact that you don't have enough knowledge about which state the system is in). – Martin Feb 3 '14 at 15:27

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