Exploring Quantum Physics - Week 3 Extra Credit Question 7

Question:

... snip ...
What is the fundamental period of the wavefunctions?

Solution:

Notice that we have $\phi(x, t) = \sum_{n=1}^{\infty} a_n \psi_n(x) \exp \left(-iE_n t/\hbar \right)$, it is just a sum of complex exponentials, so let's take a look at their periods.

The periods are just $\frac{2\pi}{E_n/\hbar} = \frac{2\pi\hbar}{E_n}$ and the energies are $\frac{\pi^2\hbar^2n^2}{2mL^2}$, so can compute the periods as $\frac{2\pi\hbar}{\frac{\pi^2\hbar^2n^2}{2mL^2}} = \frac{4mL^2}{\pi\hbar n^2} = \frac{8mL^2}{h n^2}$

Now we see the largest period happens when $n = 1$ and it is an integral multiple of all other periods, so $\frac{8mL^2}{h}$ is the answer.