Understanding the Two Types of Density Operators in Quantum Mechanics

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Discussion Overview

The discussion centers on the two types of density operators in quantum mechanics, specifically the expressions \(\rho=\sum_{i}\delta(r-r_{i})\) and \(\rho=\sum_{i}|\psi_{i}>\rho_{ii}<\psi_{i}|\). Participants are exploring the meaning, normalization, and implications of these operators.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant expresses confusion regarding the two density operators and seeks clarification.
  • Another participant notes that the first density operator is not normalized if the sum contains more than one term.
  • A subsequent post questions whether the two density operators are equivalent and asks for further explanation regarding their connection.
  • Another participant expresses uncertainty about the meaning of the first expression, suggesting it does not make sense as an operator and provides an alternative formulation for a continuous spectrum.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the meaning or normalization of the first density operator, and multiple competing views remain regarding its interpretation and connection to the second operator.

Contextual Notes

There are unresolved questions about the definitions and implications of the density operators, particularly concerning normalization and the context in which they are applied.

Liao Chen
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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.
 
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The first one is not normalized if your sum has more than one term.
 
arkajad said:
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?
 
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

[tex](\rho\psi)(x')=\int \delta(x-x_i)\psi(x)dx=\sum_i\psi(x_i)[/tex]

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

[tex](\rho\psi)(x')=\int \rho(x',x)\psi(x)dx[/tex]

So, where did you get it from?
 

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