# Superposition and density matrix. What are these states?

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?

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