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10.4 Simple example

The best way to clarify this abstract discussion is to consider the quantum mechanics of the Harmonic oscillator of mass m and frequency $ \omega$,

$\displaystyle \hat{H}$ = - $\displaystyle {\frac{\hbar^{2}}{2m}}$$\displaystyle {\frac{d^{2}}{dx^{2}}}$ + $ {\frac{1}{2}}$m$\displaystyle \omega^{2}_{}$x2. (10.8)

If we assume that the wave function at time t = 0 is a linear superposition of the first two eigenfunctions,
$\displaystyle \psi$(x, t = 0) = $\displaystyle \sqrt{{\textstyle{\frac{1}{2}}}}$$\displaystyle \phi_{0}^{}$(x) - $\displaystyle \sqrt{{\textstyle{\frac{1}{2}}}}$$\displaystyle \phi_{1}^{}$(x),  
$\displaystyle \phi_{0}^{}$(x) = $\displaystyle \left(\vphantom{\frac{m\omega}{\pi \hbar}}\right.$$\displaystyle {\frac{m\omega}{\pi \hbar}}$ $\displaystyle \left.\vphantom{\frac{m\omega}{\pi \hbar}}\right)^{1/4}_{}$exp$\displaystyle \left(\vphantom{ - \frac{m \omega}{2 \hbar} x^2 }\right.$ - $\displaystyle {\frac{m \omega}{2 \hbar}}$x2$\displaystyle \left.\vphantom{ - \frac{m \omega}{2 \hbar} x^2 }\right)$,  
$\displaystyle \phi_{1}^{}$(x) = $\displaystyle \left(\vphantom{\frac{m\omega}{\pi \hbar}}\right.$$\displaystyle {\frac{m\omega}{\pi \hbar}}$ $\displaystyle \left.\vphantom{\frac{m\omega}{\pi \hbar}}\right)^{1/4}_{}$exp$\displaystyle \left(\vphantom{ - \frac{m \omega}{2 \hbar} x^2 }\right.$ - $\displaystyle {\frac{m \omega}{2 \hbar}}$x2$\displaystyle \left.\vphantom{ - \frac{m \omega}{2 \hbar} x^2 }\right)$$\displaystyle \sqrt{\frac{m\omega}{\hbar}}$x. (10.9)

(The functions $ \phi_{0}^{}$ and $ \phi_{1}^{}$ are the normalised first and second states of the harmonic oscillator, with energies E0 = $ {\frac{1}{2}}$$ \hbar$$ \omega$ and E1 = $ {\frac{3}{2}}$$ \hbar$$ \omega$.) Thus we now kow the wave function for all time:
 
$\displaystyle \psi$(x, t) = $\displaystyle \sqrt{{\textstyle{\frac{1}{2}}}}$$\displaystyle \phi_{0}^{}$(x)e- $\scriptstyle {\frac{1}{2}}$i$\scriptstyle \omega$t - $\displaystyle \sqrt{{\textstyle{\frac{1}{2}}}}$$\displaystyle \phi_{1}^{}$(x)e- $\scriptstyle {\frac{3}{2}}$i$\scriptstyle \omega$t. (10.10)

In figure 10.1 we plot this quantity for a few times.

  
Figure 10.1: The wave function (10.10) for a few values of the time t. The solid line is the real part, and the dashed line the imaginary part.
\includegraphics[width=6.0cm]{Figures/timedep1.ps}

The best way to visualize what is happening is to look at the probability density,

$\displaystyle \mathcal {P}$(x, t) = $\displaystyle \psi$(x, t)*$\displaystyle \psi$(x, t)  
  = $ {\frac{1}{2}}$$\displaystyle \left(\vphantom{ \phi_{0}(x)^{2}+\phi_{1}(x)^{2}-2\phi_{0}(x)\phi_{1}(x)
\cos\omega t }\right.$$\displaystyle \phi_{0}^{}$(x)2 + $\displaystyle \phi_{1}^{}$(x)2 - 2$\displaystyle \phi_{0}^{}$(x)$\displaystyle \phi_{1}^{}$(x)cos$\displaystyle \omega$t$\displaystyle \left.\vphantom{ \phi_{0}(x)^{2}+\phi_{1}(x)^{2}-2\phi_{0}(x)\phi_{1}(x)
\cos\omega t }\right)$. (10.11)

This clearly oscillates with frequency $ \omega$.

Question: Show that $ \int_{-\infty}^{\infty}$$ \mathcal {P}$(x, t)dx = 1.

Another way to look at that is to calculate the expectation value of $ \hat{x}$:

$\displaystyle \langle$$\displaystyle \hat{x}$$\displaystyle \rangle$ = $\displaystyle \int_{-\infty}^{\infty}$$\displaystyle \mathcal {P}$(x, t)dx  
  = $ {\frac{1}{2}}$$\displaystyle \underbrace{\int_{-\infty}^\infty \phi_{0}(x)^{2} xdx}_{=0}^{}\,$ + $ {\frac{1}{2}}$$\displaystyle \underbrace{\int_{-\infty}^\infty \phi_{1}(x)^{2}x dx}_{=0}^{}\,$ - cos$\displaystyle \omega$t$\displaystyle \int_{-\infty}^{\infty}$$\displaystyle \phi_{0}^{}$(x)$\displaystyle \phi_{1}^{}$(x)xdx  
  = -cos$\displaystyle \omega$t$\displaystyle \sqrt{\frac{\hbar}{m\omega}}$$\displaystyle {\frac{1}{\sqrt \pi}}$$\displaystyle \int_{-\infty}^{\infty}$y2e-y2  
  = - $ {\frac{1}{2}}$$\displaystyle \sqrt{\frac{\hbar}{m\omega}}$cos$\displaystyle \omega$t. (10.12)

This once again exhibits oscillatory behaviour!


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Next: 10.5 Wave packets (states Up: 10. Time dependent wave Previous: 10.3 Completeness and time-dependence

© 1998 Niels Walet, UMIST
Email Niels.Walet@umist.ac.uk