## Lecture ( minutes)

Discuss the two “extremes” of wave functions, namely the exponential function (extending to $\pm \infty$ in the spatial dimension) $A{e^{ikx}}$ and the delta function $A\delta \left( x \right)$. The exponential function represents a state where the momentum (wavelength) is known with certainty (students need to be reminded that it is that the infinite extent that gives this certainty) and the position is completely uncertain because the probablity density is the same everywhere. The delta function clearly is a state of definite position. To see why the momentum is completely uncertain, use a series of exponential functions (extending to $\pm \infty$) of different wavelengths and the same amplitude, and show, graphically, how adding more of them together produces a wave function that is increasingly localized.

We don't dwell on the fine points of the difficulties with normalizing these wave functions - it's the general idea that is important. In the same spirit, use sine or cosine waves for illustration and ease of plotting.

Now it's time to think about other wave functions that can be made from adding sinusoidal waves with different amplitudes - and discuss how the momentum/position trade-off might be impacted. The classic example is building a “wavepacket” with a Gaussian envelope. This example ma already have been done in the context of the non-dispersive wave wave equation, but even if so, it bears repeating.

Building a wave packet from free-particle wave functions means that the discrete → continuous limit must be taken. With the students' knowledge of projections, it is not too difficult to lead them from: $$\left| \phi \right\rangle = \sum \left \langle x| \psi \right\rangle \left| \varphi _k \right\rangle$$

and:

$$\phi \left( x \right) = \sum {c_k}{\varphi _k}\left( x \right)$$

to:

$$\phi (x) = \int\limits_{ - \infty }^\infty {dk\,A(k){e^{ikx}}}$$

with

$A(k) = \left \langle {\varphi _k}| \varphi \right\rangle _{t = 0}$

It takes some time to set up and solve this integral, but that's the crux of the lecture.

Take advantage of the students' knowledge of time development to get to

$$\varphi (x,t) = \int\limits_{ - \infty }^\infty {dk\,A(k){e^{ikx}}{e^{ - i{E_k}t/\hbar }}}$$

(though this step can be left for later)

and the same integral as above can be evaluated to see the time dependence of the dispersive packet. This is best done in Mathematica.

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