Is the Product of Two Normalized Wavefunctions Also Normalized?

[Niles]

Homework Statement

Hi all.

If I have two normalized wavefunctions f and g, will their product fg also be a normalized wavefunction?

Thanks in advance.


Niles.

 

[CompuChip]

Nope.
Consider this wavefunction from Wikipedia :)

$$\psi(x, t) = \begin{cases} \frac{1}{\sqrt{L}} \exp\left[ i(kx - \omega t) \right] & \text{ if } 0 \le x \le L \\ 0 & \text{otherwise} \end{cases}$$
(this is actually the wavefunction of a particle in a 1-D box of size L).

Now you can check that $\psi$ is normalized (by the 1/L^(1/2) in front) and take $f = g = \psi$