#density operator

Ensembles for spin one half

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Mixed ensemble averages

In [1], Sakurai leaves it to the reader to verify that knowledge of the three ensemble averages [S_x], [S_y],[S_z] is sufficient to reconstruct the density operator for a spin one half system.

I’ll do this in two parts, the first using a spin-up/down ensemble to see what form this has, then the general case. The…

PHY1520H Graduate Quantum Mechanics. Lecture 3: Density matrix (cont.). Taught by Prof. Arun Paramekanti

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Disclaimer

Peeter’s lecture notes from class. These may be incoherent and rough.

These are notes for the UofT course PHY1520, Graduate Quantum Mechanics, taught by Prof. Paramekanti, covering [1] chap. 3 content.

Density matrix (cont.)

An example of a partitioned system with four total states (two spin 1/2 particles) is sketched in fig. 1.

PHY1520H Graduate Quantum Mechanics. Lecture 2: Basic concepts, time evolution, and density operators. Taught by Prof. Arun Paramekanti

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Disclaimer

Peeter’s lecture notes from class. These may be incoherent and rough.

These are notes for the UofT course PHY1520, Graduate Quantum Mechanics, taught by Prof. Paramekanti, covering chap 1 (basic concepts),3 (density operator). content from [1].

Basic concepts

We’ve reviewed the basic concepts that we will encounter in Quantum…

Entropy when density operator has zero eigenvalues

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In the class notes and the text [1] the Von Neumann entropy is defined as

\begin{equation}\label{eqn:densityMatrixEntropy:20} S = -\textrm{Tr} \rho \ln \rho. \end{equation}

In one of our problems I had trouble evaluating this, having calculated a density operator matrix representation

\begin{equation}\label{eqn:densityMatrixEntropy:40} \rh…