10.4: Simple Example
- Page ID
- 14812
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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 best way to clarify this abstract discussion is to consider the quantum mechanics of the Harmonic oscillator of mass \(m\) and frequency \(\omega\),
\[\hat{H}=-\frac{\hbar^2}{2 m} \frac{d^2}{d x^2}+\frac{1}{2} m \omega^2 x^2\]
If we assume that the wave function at time \(t=0\) is a linear superposition of the first two eigenfunctions,
\[\begin{aligned}
\psi(x, t=0) & =\sqrt{\frac{1}{2}} \phi_0(x)-\sqrt{\frac{1}{2}} \phi_1(x) \\[8pt]
\phi_0(x) & =\left(\frac{m \omega}{\pi \hbar}\right)^{1 / 4} \exp \left(-\frac{m \omega}{2 \hbar} x^2\right) \\[8pt]
\phi_1(x) & =\left(\frac{m \omega}{\pi \hbar}\right)^{1 / 4} \exp \left(-\frac{m \omega}{2 \hbar} x^2\right) \sqrt{\frac{m \omega}{\hbar}} x
\end{aligned}\]
(The functions \(\phi_0\) and \(\phi_1\) are the normalised first and second states of the harmonic oscillator, with energies \(E_0=\frac{1}{2} \hbar \omega\) and \(E_1=\frac{3}{2} \hbar \omega\).) Thus we now kow the wave function for all time:
\[\psi(x, t)=\sqrt{\frac{1}{2}} \phi_0(x) e^{-\frac{1}{2} i \omega t}-\sqrt{\frac{1}{2}} \phi_1(x) e^{-\frac{3}{2} i \omega t} .\]
In figure \(\PageIndex{1}\) we plot this quantity for a few times.
The best way to visualize what is happening is to look at the probability density,
\[\begin{aligned}
\mathcal{P}(x, t) & =\psi(x, t)^* \psi(x, t) \\
& =\frac{1}{2}\left(\phi_0(x)^2+\phi_1(x)^2-2 \phi_0(x) \phi_1(x) \cos \omega t\right)
\end{aligned}\]
This clearly oscillates with frequency \(\omega\).
Question: Show that \(\int_{-\infty}^{\infty} \mathcal{P}(x, t) d x=1\).
Another way to look at that is to calculate the expectation value of \(\hat{x}\) :
\[\begin{aligned}
\langle\hat{x}\rangle & =\int_{-\infty}^{\infty} \mathcal{P}(x, t) d x \\[8pt]
& =\frac{1}{2} \underbrace{\int_{-\infty}^{\infty} \phi_0(x)^2 x d x}_{=0}+\frac{1}{2} \underbrace{\int_{-\infty}^{\infty} \phi_1(x)^2 x d x}_{=0}-\cos \omega t \int_{-\infty}^{\infty} \phi_0(x) \phi_1(x) x d x \\[8pt]
& =-\cos \omega t \sqrt{\frac{\hbar}{m \omega}} \frac{1}{\sqrt{\pi}} \int_{-\infty}^{\infty} y^2 e^{-y^2} \\[8pt]
& =-\frac{1}{2} \sqrt{\frac{\hbar}{m \omega}} \cos \omega t .
\end{aligned} \]
This once again exhibits oscillatory behaviour!