7.10: Using the Trace Function to Calculate Expectation Values
- Page ID
- 138798
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Starting with the traditional expression for the calculation of the expectation value, the identity operator is inserted between the measurement operator and the ket containing the wave function. Rearranging terms gives the trace function operating on the product of the state's density operator and the measurement operator.
\[ \begin{align*} \langle \psi | \hat{O} | \psi \rangle &= \sum_i \langle \psi | \hat{O} | i \rangle \langle i | \psi \rangle \\[4pt] &= \sum_i \langle i | \psi \rangle \langle \psi | \hat{O} | i \rangle \\[4pt] &= Trace \left( | \psi \rangle \langle \psi | \hat{O} \right) \end{align*} \]
where
\[ \sum_i | i \rangle \langle i | = identity \nonumber \]
Next this transition is carried out in detail using matrix algebra.
\[ \begin{pmatrix}
a & b
\end{pmatrix}\begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix}\begin{pmatrix}
a\\
b
\end{pmatrix} = \begin{pmatrix}
a & b
\end{pmatrix}\begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix}\begin{pmatrix}
1 & 0\\
0 & 1
\end{pmatrix}\begin{pmatrix}
a\\
b
\end{pmatrix} = \begin{pmatrix}
a & b
\end{pmatrix}\begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix} \left[ \begin{pmatrix}
1\\
0
\end{pmatrix}\begin{pmatrix}
1 & 0
\end{pmatrix} + \begin{pmatrix}
0\\
1
\end{pmatrix}\begin{pmatrix}
0 & 1
\end{pmatrix} \right] \begin{pmatrix}
a\\
b
\end{pmatrix} = 2ab \nonumber \]
\[ \begin{pmatrix}
a & b
\end{pmatrix} \begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix} \begin{pmatrix}
1\\
0
\end{pmatrix}\begin{pmatrix}
1 & 0
\end{pmatrix} \begin{pmatrix}
a\\
b
\end{pmatrix} = \begin{pmatrix}
a & b
\end{pmatrix} \begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix} \begin{pmatrix}
0\\
1
\end{pmatrix}\begin{pmatrix}
0 & 1
\end{pmatrix}\begin{pmatrix}
a\\
b
\end{pmatrix} = 2ab \nonumber \]
\[ \begin{pmatrix}
1 & 0
\end{pmatrix} \begin{pmatrix}
a \\
b
\end{pmatrix} \begin{pmatrix}
a & b
\end{pmatrix}\begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix} \begin{pmatrix}
1\\
0
\end{pmatrix} + \begin{pmatrix}
0 & 1
\end{pmatrix} \begin{pmatrix}
a\\
b
\end{pmatrix} \begin{pmatrix}
a & b
\end{pmatrix} \begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix} \begin{pmatrix}
0\\
1
\end{pmatrix} = 2ab \nonumber \]
\[ \begin{pmatrix}
1 & 0
\end{pmatrix} \begin{pmatrix}
a^2 & ab \\
ab & b^2
\end{pmatrix} \begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix} \begin{pmatrix}
1\\
0
\end{pmatrix} + \begin{pmatrix}
0 & 1
\end{pmatrix} \begin{pmatrix}
a^2 & ab\\
ab & b^2
\end{pmatrix} \begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix} \begin{pmatrix}
0\\
1
\end{pmatrix} = 2ab \nonumber \]
\[ \begin{pmatrix}
1 & 0
\end{pmatrix} \begin{pmatrix}
ab & a^2 \\
b^2 & ab
\end{pmatrix} \begin{pmatrix}
1\\
0
\end{pmatrix} + \begin{pmatrix}
0 & 1
\end{pmatrix} \begin{pmatrix}
ab & a^2\\
b^2 & ab
\end{pmatrix} \begin{pmatrix}
0\\
1
\end{pmatrix} = ab + ab = Trace \begin{pmatrix}
ab & a^2 \\
b^2 & ab\\
\end{pmatrix} \nonumber \]
\[ \begin{pmatrix}
a & b
\end{pmatrix} \begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix} \begin{pmatrix}
a\\
b
\end{pmatrix} \rightarrow 2ab ~~~ \begin{pmatrix}
a & b
\end{pmatrix} \begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix} \begin{pmatrix}
1 & 0\\
0 & 1
\end{pmatrix} \begin{pmatrix}
a\\
b
\end{pmatrix} \rightarrow 2ab \nonumber \]
\[ \begin{pmatrix}
1\\
0
\end{pmatrix} \begin{pmatrix}
1 & 0
\end{pmatrix} + \begin{pmatrix}
0\\
1
\end{pmatrix} \begin{pmatrix}
0 & 1
\end{pmatrix} \rightarrow \begin{pmatrix}
1 & 0\\
0 & 1
\end{pmatrix} \nonumber \]
\[ \begin{pmatrix}
a & b
\end{pmatrix} \begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix} \begin{pmatrix}
1\\
0
\end{pmatrix} \begin{pmatrix}
1 & 0
\end{pmatrix} \begin{pmatrix}
a\\
b
\end{pmatrix} + \begin{pmatrix}
a & b
\end{pmatrix} \begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix} \begin{pmatrix}
0 \\
1
\end{pmatrix} \begin{pmatrix}
0 & 1 \end{pmatrix} \begin{pmatrix}
a \\
b\end{pmatrix} \rightarrow 2ab \nonumber \]
\[ \begin{pmatrix}
1 & 0
\end{pmatrix} \begin{pmatrix}
a \\
b
\end{pmatrix} \begin{pmatrix}
a & b
\end{pmatrix}
\begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix} \begin{pmatrix}
1\\
0
\end{pmatrix} + \begin{pmatrix}
0 & 1
\end{pmatrix} \begin{pmatrix}
a\\
b
\end{pmatrix} \begin{pmatrix}
a & b
\end{pmatrix} \begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix} \begin{pmatrix}
0 \\
1
\end{pmatrix} \rightarrow 2ab \nonumber \]
\[ tr \left[ \begin{pmatrix}
a \\
b
\end {pmatrix} \begin{pmatrix} a & b \end{pmatrix} \begin{pmatrix}
0 & 1 \\
1 & 0 \\
\end{pmatrix} \right] \rightarrow 2ab ~~~ tr \left[ \begin{pmatrix}
0 & 1 \\
1 & 0 \\
\end{pmatrix} \begin{pmatrix}
a \\
b
\end {pmatrix} \begin{pmatrix} a & b \end{pmatrix} \right] \rightarrow 2ab \nonumber \]
The last calculation on the right is justified by the following:
\[ \langle \psi | \hat{O} | \psi \rangle = \sum_i \langle \psi | i \rangle \langle i | \hat{O} | \psi \rangle = \sum_i \langle i | \hat{O} | \psi \rangle \langle \psi | i \rangle = Trace \left( \hat{O} | \psi \rangle \langle \psi | \right) \nonumber \]