The expression for completeness relation in quantum mechanics is -
Σ |ψ_n><ψ_n| = 1
where the expression for density matrix in statistical mechanics is -
ρ = Σ p_n |ψ_n><ψ_n|
Both of the equation looks the same. So what are the differences between the density matrix and completeness relation?
What is the basic difference between them?
Formally the difference is that for the density matrix there are pre-factors p_n which sum up to 1 rather than all being 1 as in the completeness relation.
The meaning is also quite different.
Here is a rough illustration what they mean:
This object is a projection operator:
|ψ_n><ψ_n|
It projects on the n-th basis vector.
For simplicity lets take a simple example. Say our Hilbert space is 3 dimensional. Then the sum runs from 1 to 3. Each so-called pure state can be represented by a vector of length 1 in a 3 dimensional space like these examples:
|ψ_1> = (1, 0, 0)T
|ψ_2> = (0, 1, 0)T
|ψ_3> = (0, 0, 1)T
|φ> := (0, 1/2^0.5, 1/2^0.5)T
(The "T" stands for transposed) These projection operators can be written as a matrix like for example:
/ 0 0 0 \
|ψ_2><ψ_2| = | 0 1 0 |
\ 0 0 0 /
Now what these projection operators do is projecting a vector on one of the coordinate axes. E.g. for n=2 we project to the y-axis.
|ψ_2><ψ_2|φ> = (0, 1/2^0.5, 0)
Now what the completeness relation says is that the sum of those 3 vectors you get when projection on each coordinate axis is once again the original vector (see Basis Decomposition). As this is true for any vector, this means the operation is the identity matrix:
#/media/File:3d_two_bases_same_vector.svg){kind=link}
/ 1 0 0 \ + / 0 0 0 \ + / 0 0 0 \ / 1 0 0 \
|ψ_1><ψ_1| + |ψ_2><ψ_2| + |ψ_3><ψ_3| = | 0 0 0 | + | 0 1 0 | + | 0 0 0 | = | 0 1 0 | = 1
\ 0 0 0 / + \ 0 0 0 / + \ 0 0 1 / \ 0 0 1 /
Now the density matrix is a completely different matter. The weights p_n describe how one state is a mixture of several "pure" states. See e.g. https://en.wikipedia.org/wiki/Density_matrix