1.15: Expectation values of k and \(\omega\)
- Page ID
- 50116
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The expectation value of k is obtained by integrating the wavefunction over all k. This must be performed in k-space
\[ \langle k\rangle = \frac{\frac{1}{2\pi}\int^{+\infty}_{-\infty}A(k)^{*}kA(k)dk}{\frac{1}{2\pi}\int^{+\infty}_{-\infty}A(k)^{*}A(k)dk} = \frac{\langle A|k|A\rangle}{\langle A|A\rangle} \nonumber \]
From the Inverse Fourier transform in k-space
\[ \psi(x)=\frac{1}{2\pi}\int^{+\infty}_{-\infty}A(k)e^{ikx}dx \nonumber \]
note that
\[ -i\frac{d}{dx}\psi(x)=\frac{1}{2\pi}\int^{+\infty}_{-\infty}kA(k)e^{ikx}dx \nonumber \]
Thus, we have the following Fourier transform pair:
\[ -i \frac{d}{dx}\psi(x) \Leftrightarrow kA(k) \nonumber \]
It follows that\(^{†}\)
\[ \langle k\rangle = \frac{\langle A\rangle|k|\langle A\rangle}{\langle A\rangle|\langle A\rangle} = \frac{\langle \psi | -i\frac{d}{dx}|\psi\rangle}{\langle \psi|\psi\rangle} \nonumber \]
Similarly, from the Inverse Fourier transform in the frequency domain
\[ \psi(t)=\frac{1}{2\pi}\int^{+\infty}_{-\infty} A(\omega)e^{-i\omega t} d\omega \nonumber \]
We can derive the Fourier transform pair:
\[ i \frac{d}{dt} \psi(t) \Leftrightarrow \omega A(\omega) \nonumber \]
It follows that
\[ \langle \omega \rangle = \frac{\langle A\rangle|\omega |\langle A\rangle}{\langle A\rangle|\langle A\rangle} = \frac{\langle \psi | i\frac{d}{dt}|\psi\rangle}{\langle \psi|\psi\rangle} \nonumber \]
We define two operators
\[ \hat{k} = -i \frac{d}{dx} \nonumber \]
and
\[ \hat{\omega} = i \frac{d}{dt} \nonumber \]
Operators only act on functions to the right. To signify this difference we mark them with a caret.
We could also define the (somewhat trivial) position operator
\[ \hat{x} = x \nonumber \]