Normalizing Wave Functions

“Normalize one of the two following wave functions:”

$$\psi(x) = \begin{cases} 0, & \text{if $x$ < 0} \\ sin(\frac{\pi{x}}{L}), & \text{if 0 < $x$ < $L$}\\ 0, & \text{if $x$ > $L$} \end{cases} $$

$$\psi(x) = \begin{cases} 0, & \text{if} \; x < -\frac{L}{2} \\ cos(\frac{\pi{x}}{L}), & \text{if} \; -\frac{L}{2} < x < \frac{L}{2}\\ 0, & \text{if} \; x > \frac{L}{2} \end{cases} $$

This SWBQ can be used to open up discussions about the normalizing factor and why it is there.

• Discussion of the normalizing factor $\sqrt {\frac{2}{L}} $ (which has been encountered before in a Modern Physics course) and a realization of why it is there. Also discuss its dimensions.
• Discussion of the need for specifying the wave function outside the box.
• Discussion after the report back of the fact that the two forms represent the same wave function, but with a different origin of the well. Parity can be discussed here.

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