From c4ec16c8b34f84f0e95d2988df083c6d31ba21ef Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Wed, 10 Mar 2021 15:26:15 +0100
Subject: Expand knowledge base
---
content/know/concept/density-operator/index.pdc | 26 +++++++++++++------------
1 file changed, 14 insertions(+), 12 deletions(-)
(limited to 'content/know/concept/density-operator')
diff --git a/content/know/concept/density-operator/index.pdc b/content/know/concept/density-operator/index.pdc
index 39c2e85..5126f31 100644
--- a/content/know/concept/density-operator/index.pdc
+++ b/content/know/concept/density-operator/index.pdc
@@ -81,15 +81,17 @@ $$\begin{aligned}
This can be used to find out whether a given $\hat{\rho}$
represents a pure or mixed ensemble.
-Next, we define the ensemble average $\expval*{\expval*{\hat{L}}}$
-as the mean of the expectation values for states in the ensemble,
-which can be calculated like so:
+Next, we define the ensemble average $\expval*{\hat{O}}$
+as the mean of the expectation values of $\hat{O}$ for states in the ensemble.
+We use the same notation as for the pure expectation value,
+since this is only a small extension of the concept to mixed ensembles.
+It is calculated like so:
$$\begin{aligned}
\boxed{
- \expval*{\expval*{\hat{L}}}
- = \sum_{n} p_n \matrixel{\Psi_n}{\hat{L}}{\Psi_n}
- = \mathrm{Tr}(\hat{L} \hat{\rho})
+ \expval*{\hat{O}}
+ = \sum_{n} p_n \matrixel{\Psi_n}{\hat{O}}{\Psi_n}
+ = \mathrm{Tr}(\hat{\rho} \hat{O})
}
\end{aligned}$$
@@ -97,13 +99,13 @@ To prove the latter,
we write out the trace $\mathrm{Tr}$ as the sum of the diagonal elements, so:
$$\begin{aligned}
- \mathrm{Tr}(\hat{L} \hat{\rho})
- &= \sum_{j} \matrixel{j}{\hat{L} \hat{\rho}}{j}
- = \sum_{j} \sum_{n} p_n \matrixel{j}{\hat{L}}{\Psi_n} \braket{\Psi_n}{j}
+ \mathrm{Tr}(\hat{\rho} \hat{O})
+ &= \sum_{j} \matrixel{j}{\hat{\rho} \hat{O}}{j}
+ = \sum_{j} \sum_{n} p_n \braket{j}{\Psi_n} \matrixel{\Psi_n}{\hat{O}}{j}
\\
- &= \sum_{n} \sum_{j} p_n \braket{\Psi_n}{j} \matrixel{j}{\hat{L}}{\Psi_n}
- = \sum_{n} p_n \matrixel{\Psi_n}{\hat{I} \hat{L}}{\Psi_n}
- = \expval*{\expval*{\hat{L}}}
+ &= \sum_{n} \sum_{j} p_n\matrixel{\Psi_n}{\hat{O}}{j} \braket{j}{\Psi_n}
+ = \sum_{n} p_n \matrixel{\Psi_n}{\hat{O} \hat{I}}{\Psi_n}
+ = \expval*{\hat{O}}
\end{aligned}$$
In both the pure and mixed cases,
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