Understanding the Two Types of Density Operators in Quantum Mechanics

[Liao Chen]

I'm confused about the two density operators:

\rho=\sum_{i}\delta(r-r_{i}) and \rho=\sum_{i}|\psi_{i}>\rho_{ii}<\psi_{i}|

Is there anyone explaining this question to me? Thanks very much.

 

[arkajad]

The first one is not normalized if your sum has more than one term.

 

[Liao Chen]

The first one is not normalized if your sum has more than one term.

Thanks a lot. Do you mean the two density operators are the same and connected through some transformations? Could you explain with a little more details?

 

[arkajad]

After some thinking I really do not know what the first expression could mean. It does not make any sense to me. If I consider it as an operator, it would act as

$$(\rho\psi)(x')=\int \delta(x-x_i)\psi(x)dx=\sum_i\psi(x_i)$$

which is a number and not a function. For a continuous spectrum the formula should look like

$$(\rho\psi)(x')=\int \rho(x',x)\psi(x)dx$$

So, where did you get it from?