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 $\omega$,

$$\hat{H}=-\frac{\hbar^2}{2 m} \frac{d^2}{d x^2}+\frac{1}{2} m \omega^2 x^2$$

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 $\int_{-\infty}^{\infty} \mathcal{P}(x, t) d x=1$.

Figure $\PageIndex{1}$: 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.

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!